Defs

📁 Source: Mathlib/Algebra/Group/Hom/Defs.lean

Statistics

Metric	Count
Definitions `AddHom`, `comp`, `copy`, `funLike`, `id`, `inverse`, `toFun`, `AddHomClass`, `toAddHom`, `instFunLike`, `instInhabited`, `instMonoid`, `instMul`, `instOne`, `coeToAddHom`, `coeToZeroHom`, `comp`, `copy`, `id`, `instFunLike`, `inverse`, `mk'`, `toAddHom`, `toZeroHom`, `AddMonoidHomClass`, `toAddMonoidHom`, `hom_simp_lemma_priority`, `instFunLike`, `instInhabited`, `instMonoid`, `instMul`, `instOne`, `coeToMulHom`, `coeToOneHom`, `comp`, `copy`, `id`, `instFunLike`, `inverse`, `mk'`, `toMulHom`, `toOneHom`, `MonoidHomClass`, `toMonoidHom`, `MulHom`, `comp`, `copy`, `funLike`, `id`, `inverse`, `toFun`, `MulHomClass`, `toMulHom`, `OneHom`, `comp`, `copy`, `funLike`, `id`, `inverse`, `toFun`, `OneHomClass`, `toOneHom`, `ZeroHom`, `comp`, `copy`, `funLike`, `id`, `inverse`, `toFun`, `ZeroHomClass`, `toZeroHom`, `instCoeTCAddHomOfAddHomClass`, `instCoeTCAddMonoidHomOfAddMonoidHomClass`, `instCoeTCMonoidHomOfMonoidHomClass`, `instCoeTCMulHomOfMulHomClass`, `instCoeTCOneHomOfOneHomClass`, `instCoeTCZeroHomOfZeroHomClass`, `instInhabitedAddHom`, `instInhabitedAddMonoidHom`, `instInhabitedMonoidHom`, `instInhabitedMulHom`, `instInhabitedOneHom`, `instInhabitedZeroHom`, `instOneMonoidHom`, `instOneMulHom`, `instOneOneHom`, `instZeroAddHom`, `instZeroAddMonoidHom`, `instZeroZeroHom`, `«term_→_»`, `«term_→+_»`, `«term_→ₙ_»`, `«term_→ₙ+_»`	93
Theorems `addHomClass`, `cancel_left`, `cancel_right`, `coe_coe`, `coe_comp`, `coe_copy`, `coe_copy_eq`, `coe_id`, `coe_mk`, `comp_apply`, `comp_assoc`, `comp_id`, `ext`, `ext_iff`, `id_apply`, `id_comp`, `inverse_apply`, `map_add`, `map_add'`, `mk_coe`, `toFun_eq_coe`, `map_add`, `coe_mul`, `coe_one`, `coe_pow`, `instAddMonoidHomClass`, `cancel_left`, `cancel_right`, `coe_coe`, `coe_comp`, `coe_copy`, `coe_id`, `coe_mk`, `comp_apply`, `comp_assoc`, `comp_id`, `comp_zero`, `copy_eq`, `ext`, `ext_iff`, `id_apply`, `id_comp`, `instAddMonoidHomClass`, `inverse_apply`, `map_add`, `map_add'`, `map_add_neg`, `map_exists_left_neg`, `map_exists_right_neg`, `map_neg`, `map_nsmul`, `map_sub`, `map_zero`, `map_zsmul`, `map_zsmul'`, `mk'_apply`, `mk_coe`, `toAddHom_coe`, `toAddHom_injective`, `toFun_eq_coe`, `toZeroHom_coe`, `toZeroHom_injective`, `zero_apply`, `zero_comp`, `toAddHomClass`, `toZeroHomClass`, `isAddTorsionFree`, `isMulTorsionFree`, `add_comm`, `mul_comm`, `of_injective`, `of_injective`, `coe_mul`, `coe_one`, `coe_pow`, `instMonoidHomClass`, `cancel_left`, `cancel_right`, `coe_coe`, `coe_comp`, `coe_copy`, `coe_id`, `coe_mk`, `comp_apply`, `comp_assoc`, `comp_id`, `comp_one`, `copy_eq`, `ext`, `ext_iff`, `id_apply`, `id_comp`, `instMonoidHomClass`, `inverse_apply`, `isDedekindFiniteMonoid_of_injective`, `map_div`, `map_exists_left_inv`, `map_exists_right_inv`, `map_inv`, `map_mul`, `map_mul'`, `map_mul_inv`, `map_one`, `map_pow`, `map_zpow`, `map_zpow'`, `mk'_apply`, `mk_coe`, `one_apply`, `one_comp`, `toFun_eq_coe`, `toMulHom_coe`, `toMulHom_injective`, `toOneHom_coe`, `toOneHom_injective`, `toMulHomClass`, `toOneHomClass`, `cancel_left`, `cancel_right`, `coe_coe`, `coe_comp`, `coe_copy`, `coe_copy_eq`, `coe_id`, `coe_mk`, `comp_apply`, `comp_assoc`, `comp_id`, `ext`, `ext_iff`, `id_apply`, `id_comp`, `inverse_apply`, `map_mul`, `map_mul'`, `mk_coe`, `mulHomClass`, `toFun_eq_coe`, `map_mul`, `cancel_left`, `cancel_right`, `coe_coe`, `coe_comp`, `coe_copy`, `coe_copy_eq`, `coe_id`, `coe_mk`, `comp_apply`, `comp_assoc`, `comp_id`, `comp_one`, `ext`, `ext_iff`, `id_apply`, `id_comp`, `inverse_apply`, `map_one`, `map_one'`, `mk_coe`, `oneHomClass`, `one_apply`, `one_comp`, `toFun_eq_coe`, `map_one`, `of_oneHomClass`, `of_zeroHomClass`, `cancel_left`, `cancel_right`, `coe_coe`, `coe_comp`, `coe_copy`, `coe_copy_eq`, `coe_id`, `coe_mk`, `comp_apply`, `comp_assoc`, `comp_id`, `comp_zero`, `ext`, `ext_iff`, `id_apply`, `id_comp`, `inverse_apply`, `map_zero`, `map_zero'`, `mk_coe`, `toFun_eq_coe`, `zeroHomClass`, `zero_apply`, `zero_comp`, `map_zero`, `instSubsingletonAddMonoidHom`, `instSubsingletonMonoidHom`, `instSubsingletonOneHom`, `instSubsingletonZeroHom`, `iterate_map_add`, `iterate_map_div`, `iterate_map_inv`, `iterate_map_mul`, `iterate_map_neg`, `iterate_map_nsmul`, `iterate_map_one`, `iterate_map_pow`, `iterate_map_sub`, `iterate_map_zero`, `iterate_map_zpow`, `iterate_map_zsmul`, `map_add`, `map_add_eq_zero`, `map_add_neg`, `map_comp_add`, `map_comp_add_neg`, `map_comp_div`, `map_comp_div'`, `map_comp_inv`, `map_comp_mul`, `map_comp_mul_inv`, `map_comp_neg`, `map_comp_nsmul`, `map_comp_one`, `map_comp_pow`, `map_comp_sub`, `map_comp_sub'`, `map_comp_zero`, `map_comp_zpow`, `map_comp_zpow'`, `map_comp_zsmul`, `map_comp_zsmul'`, `map_div`, `map_div'`, `map_eq_one_iff`, `map_eq_zero_iff`, `map_inv`, `map_mul`, `map_mul_eq_one`, `map_mul_inv`, `map_ne_one_iff`, `map_ne_zero_iff`, `map_neg`, `map_nsmul`, `map_one`, `map_pow`, `map_sub`, `map_sub'`, `map_zero`, `map_zpow`, `map_zpow'`, `map_zsmul`, `map_zsmul'`, `ne_one_of_map`, `ne_zero_of_map`	251
Total	344

AddHom

Definitions

Name	Category	Theorems
`comp` 📖	CompOp	36 math math: `AddMagmaCat.ofHom_comp`, `AddMagma.FreeAddSemigroup.lift_symm_apply`, `WithZero.mapAddHom_comp`, `Matrix.entryAddHom_eq_comp`, `comp_noncommCoprod`, `AddSubsemigroup.comap_comap`, `AddMagma.FreeAddSemigroup.lift_comp_of`, `cancel_left`, `comp_add`, `prodMap_def`, `comp_assoc`, `FreeAddMagma.toFreeAddSemigroup_comp_map`, `AddMagma.FreeAddSemigroup.lift_comp_of'`, `AddEquiv.entryAddHom_comp_mapMatrix`, `AddSemigrp.hom_comp`, `prod_unique`, `coe_comp`, `comp_apply`, `comp_coprod`, `AddMonoidAlgebra.mapDomainNonUnitalRingHom_comp`, `fst_comp_prod`, `AddSubsemigroup.map_map`, `map_srange`, `WithZero.lift_unique`, `AddSemigrp.ofHom_comp`, `AddMagma.FreeAddSemigroup.hom_ext_iff`, `cancel_right`, `WithZero.map_map`, `WithZero.map_comp`, `comp_id`, `id_comp`, `add_comp`, `snd_comp_prod`, `prod_comp_prodMap`, `WithZero.mapAddHom_mapAddHom`, `AddMagmaCat.hom_comp`
`copy` 📖	CompOp	2 math math: `coe_copy_eq`, `coe_copy`
`funLike` 📖	CompOp	123 math math: `AddMagmaCat.coe_id`, `coe_prodMap`, `Matrix.entryAddHom_apply`, `LinearMap.coe_mk`, `coeFn_apply`, `coe_srangeRestrict`, `subsemigroupMap_apply_coe`, `FreeAddSemigroup.lift_comp_of`, `PNat.coeAddHom_apply`, `withTopMap_apply`, `toAddEquiv_apply`, `AddMagmaCat.forgetReflectsIsos`, `toAddEquiv_symm_apply`, `FreeAddSemigroup.lift_symm_apply`, `coe_prod`, `srangeRestrict_surjective`, `subsemigroupMap_surjective`, `compLeft_apply`, `AddSemigrp.coe_id`, `ENatMap_apply`, `op_apply_apply`, `AddSubsemigroup.mem_map_of_mem`, `FreeAddMagma.toFreeAddSemigroup_comp_of`, `AddCon.ker_coeAddHom`, `AddMagmaCat.forget_map`, `srange_eq_top_iff_surjective`, `AddMagmaCat.ofHom_apply`, `FreeAddMagma.lift_comp_of`, `AddCon.ker_mkAddHom_eq`, `FreeAddSemigroup.lift_of`, `addHomClass`, `AddMonoidAlgebra.mapDomainNonUnitalRingHom_apply`, `toFun_eq_coe`, `TensorProduct.lift.tmul'`, `mulOp_apply_apply`, `mulLeft_apply`, `prod_apply`, `isClosed_range_coe`, `FreeAddMagma.map_of`, `AddMemClass.subtype_apply`, `AddMagmaCat.coe_comp`, `op_symm_apply_apply`, `Pi.addHom_apply`, `WithZero.lift_coe`, `AddSemigrp.comp_apply`, `id_apply`, `Set.coe_singletonAddHom`, `AddSubsemigroup.coe_map`, `coe_id`, `ext_iff`, `AddSemigrp.neg_hom_apply`, `coe_comp`, `coe_fst`, `AddSemigrp.hom_neg_apply`, `WithZero.mapAddHom_coe`, `FreeAddMagma.lift_comp_of'`, `comp_apply`, `AddSemigrp.ofHom_apply`, `AddMagmaCat.ext_iff`, `coe_add`, `FreeAddSemigroup.length_map`, `coe_coe`, `Finset.singletonAddHom_apply`, `AddMemClass.coe_subtype`, `AddSemigrp.forget₂_full`, `Finset.imageAddHom_apply`, `AddSubsemigroup.mem_comap`, `AddSubsemigroup.mem_map`, `AddCon.mkAddHom_apply`, `AddMonoidAlgebra.le_infDegree_mul`, `mulOp_symm_apply_apply`, `FreeAddSemigroup.map_of`, `FreeAddMagma.toFreeAddSemigroup_of`, `FreeAddMagma.toFreeAddSemigroup_map`, `Set.singletonAddHom_apply`, `nhdsAddHom_apply`, `addAddHom_apply`, `AddMagma.FreeAddSemigroup.lift_of`, `AddSemigrp.coe_comp`, `subsemigroupComap_apply_coe`, `LinearMap.coe_toAddHom`, `AddSubsemigroup.coe_comap`, `addHomOfMemClosureRangeCoe_apply`, `Filter.pureAddHom_apply`, `FreeAddMagma.lift_symm_apply`, `Pi.constAddHom_apply`, `FreeAddMagma.hom_ext_iff`, `mulRight_apply`, `coprod_apply`, `Finset.coe_singletonAddHom`, `AddMemClass.subtype_injective`, `mem_eqLocus`, `AddMagmaCat.comp_apply`, `AddMagmaCat.id_apply`, `FreeAddSemigroup.hom_ext_iff`, `AddSemigrp.id_apply`, `restrict_apply`, `coe_snd`, `FreeAddMagma.lift_of`, `mclosure_preimage_le`, `FreeAddMagma.length_toFreeAddSemigroup`, `add_apply`, `coe_mk`, `addHomOfTendsto_apply`, `FreeAddSemigroup.lift_of_add`, `coe_ofDense`, `mem_srange`, `AddEquiv.coe_toAddHom`, `AddMagma.FreeAddSemigroup.map_of`, `AddSubsemigroup.apply_coe_mem_map`, `AddMagmaCat.neg_hom_apply`, `Pi.evalAddHom_apply`, `WithZero.coeAddHom_apply`, `withBotMap_apply`, `FreeAddSemigroup.lift_comp_of'`, `Filter.coe_pureAddHom`, `coe_srange`, `map_multiset_ne_zero_sum`, `map_add`, `AddSemigrp.forgetReflectsIsos`, `map_mclosure`, `AddMagmaCat.hom_neg_apply`, `AddSemigrp.ext_iff`
`id` 📖	CompOp	13 math math: `AddMonoidAlgebra.mapDomainNonUnitalRingHom_id`, `WithZero.mapAddHom_id`, `AddSemigrp.hom_id`, `WithZero.map_id`, `id_apply`, `coe_id`, `AddSubsemigroup.comap_id`, `AddMagmaCat.ofHom_id`, `AddSubsemigroup.map_id`, `AddMagmaCat.hom_id`, `comp_id`, `id_comp`, `AddSemigrp.ofHom_id`
`inverse` 📖	CompOp	1 math math: `inverse_apply`
`toFun` 📖	CompOp	44 math math: `PositiveLinearMap.monotone'`, `ModuleCat.linearIndependent_shortExact`, `LinearEquiv.right_inv`, `CoalgEquiv.left_inv`, `ModuleCat.RestrictionCoextensionAdj.counit'_app`, `LieModuleEquiv.left_inv`, `Derivation.toFun_eq_coe`, `AddMonoidHom.toAddHom_coe`, `FGModuleCat.FGModuleCatEvaluation_apply`, `LinearEquiv.left_inv`, `LinearMap.toFun_eq_coe`, `toFun_eq_coe`, `LieEquiv.right_inv`, `Representation.IntertwiningMap.ext_iff`, `map_add'`, `addMonoidEndRingEquivInt_apply`, `BialgEquiv.map_mul'`, `LinearMap.toMatrixOrthonormal_apply`, `LieHom.toFun_eq_coe`, `Representation.IntertwiningMap.toFun_injective`, `Ideal.constr_basisSpanSingleton`, `Representation.IntertwiningMap.isIntertwining'`, `QuadraticMap.Isometry.map_app'`, `BialgHom.map_mul'`, `Rep.unit_iso_comm`, `LieModuleHom.map_lie'`, `RootPairing.flip_toFun_apply`, `LinearEquiv.toFun_eq_coe`, `ContinuousLinearMap.cont`, `QuadraticMap.IsometryEquiv.map_app'`, `LinearMap.map_smul'`, `AddSubmonoid.LocalizationMap.isLocalizationMap`, `LieHom.map_lie'`, `FGModuleCat.FGModuleCatEvaluation_apply'`, `addAddMonoidHom_apply`, `ContinuousLinearEquiv.continuous_toFun`, `ModuleCat.span_rightExact`, `LieEquiv.left_inv`, `Finset.imageAddMonoidHom_apply`, `LeftInvariantDerivation.toFun_eq_coe`, `LieDerivation.toFun_eq_coe`, `CoalgEquiv.right_inv`, `LieModuleEquiv.right_inv`, `BialgHom.map_one'`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`addHomClass` 📖	mathematical	—	`AddHomClass` `AddHom` `funLike`	—	`map_add'`
`cancel_left` 📖	mathematical	`DFunLike.coe` `AddHom` `funLike`	`comp`	—	`ext` `comp_apply`
`cancel_right` 📖	mathematical	`DFunLike.coe` `AddHom` `funLike`	`comp`	—	`ext` `Function.Surjective.forall` `DFunLike.ext_iff`
`coe_coe` 📖	mathematical	—	`DFunLike.coe` `AddHom` `funLike` `AddHomClass.toAddHom`	—	—
`coe_comp` 📖	mathematical	—	`DFunLike.coe` `AddHom` `funLike` `comp`	—	—
`coe_copy` 📖	mathematical	`DFunLike.coe` `AddHom` `funLike`	`copy`	—	—
`coe_copy_eq` 📖	mathematical	`DFunLike.coe` `AddHom` `funLike`	`copy`	—	`DFunLike.ext'`
`coe_id` 📖	mathematical	—	`DFunLike.coe` `AddHom` `funLike` `id`	—	—
`coe_mk` 📖	mathematical	—	`DFunLike.coe` `AddHom` `funLike`	—	—
`comp_apply` 📖	mathematical	—	`DFunLike.coe` `AddHom` `funLike` `comp`	—	—
`comp_assoc` 📖	mathematical	—	`comp`	—	—
`comp_id` 📖	mathematical	—	`comp` `id`	—	`ext`
`ext` 📖	—	`DFunLike.coe` `AddHom` `funLike`	—	—	`DFunLike.ext`
`ext_iff` 📖	mathematical	—	`DFunLike.coe` `AddHom` `funLike`	—	`ext`
`id_apply` 📖	mathematical	—	`DFunLike.coe` `AddHom` `funLike` `id`	—	—
`id_comp` 📖	mathematical	—	`comp` `id`	—	`ext`
`inverse_apply` 📖	mathematical	`DFunLike.coe` `AddHom` `funLike`	`inverse`	—	—
`map_add` 📖	mathematical	—	`DFunLike.coe` `AddHom` `funLike`	—	`map_add'`
`map_add'` 📖	mathematical	—	`toFun`	—	—
`mk_coe` 📖	—	`DFunLike.coe` `AddHom` `funLike`	—	—	`ext`
`toFun_eq_coe` 📖	mathematical	—	`toFun` `DFunLike.coe` `AddHom` `funLike`	—	—

AddHomClass

Definitions

Name	Category	Theorems
`toAddHom` 📖	CompOp	20 math math: `Matrix.entryAddHom_eq_comp`, `AddSubsemigroup.mem_map_equiv`, `AddMagmaCat.addEquiv_coe_eq`, `AddSemigrp.addEquiv_coe_eq`, `AddSubsemigroup.comap_equiv_eq_map_symm`, `AddCon.ker_coeAddHom`, `AddEquiv.subsemigroupMap_apply_coe`, `AddEquiv.entryAddHom_comp_mapMatrix`, `AddSubsemigroup.map_equiv_eq_comap_symm`, `Matrix.entryAddMonoidHom_toAddHom`, `AddSubsemigroup.map_equiv_top`, `NonUnitalAlgHom.addHomMk_coe`, `AddHom.coe_coe`, `AddEquiv.toAddHom_eq_coe`, `LinearMap.coe_addHom_mk`, `AlgHom.addHomMk_coe`, `AddEquiv.subsemigroupMap_symm_apply_coe`, `AddSubsemigroup.topEquiv_toAddHom`, `AlgHom.linearMapMk_toAddHom`, `Matrix.entryLinearMap_toAddHom`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`map_add` 📖	mathematical	—	`DFunLike.coe`	—	—

AddMonoid.End

Definitions

Name	Category	Theorems
`instFunLike` 📖	CompOp	17 math math: `natCast_apply`, `WithLp.prod_norm_eq_idemFst_of_L1`, `ofNat_apply`, `instAddMonoidHomClass`, `intCast_apply`, `smul_apply`, `WithLp.prod_norm_eq_idemFst_sup_idemSnd`, `smul_def`, `WithLp.prod_norm_eq_add_idemFst`, `coe_mul`, `coe_pow`, `coe_one`, `WithLp.idemSnd_apply`, `WithLp.idemFst_apply`, `one_apply`, `coe_smul`, `zero_apply`
`instInhabited` 📖	CompOp	—
`instMonoid` 📖	CompOp	8 math math: `CentroidHom.toEnd_pow`, `DistribMulAction.toAddMonoidEnd_apply`, `smul_def`, `coe_pow`, `natCast_def`, `applyFaithfulSMul`, `intCast_def`, `AddSubgroup.pointwise_smul_def`
`instMul` 📖	CompOp	16 math math: `commute_lmul_rmul`, `commute_lmul_sq_rmul`, `monoidEndToAdditive_apply_apply`, `commute_lmul_rmul_sq`, `MulEquiv.AddMonoid.End_apply`, `NonUnitalNonAssocCommSemiring.mem_center_iff`, `coe_mul`, `addMonoidEndRingEquivInt_apply`, `commute_lmul_lmul_sq`, `monoidEndToAdditive_symm_apply_apply`, `CentroidHom.toEnd_mul`, `MulEquiv.Monoid.End_apply`, `addMonoidEndRingEquivInt_symm_apply`, `addMonoidEndToMultiplicative_apply_apply`, `commute_rmul_rmul_sq`, `addMonoidEndToMultiplicative_symm_apply_apply`
`instOne` 📖	CompOp	6 math math: `CentroidHom.toEnd_one`, `WithLp.idemSnd_compl`, `WithLp.idemFst_compl`, `coe_one`, `WithLp.idemFst_add_idemSnd`, `one_apply`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`coe_mul` 📖	mathematical	—	`DFunLike.coe` `AddMonoid.End` `instFunLike` `instMul`	—	—
`coe_one` 📖	mathematical	—	`DFunLike.coe` `AddMonoid.End` `instFunLike` `instOne`	—	—
`coe_pow` 📖	mathematical	—	`DFunLike.coe` `AddMonoid.End` `instFunLike` `Monoid.toNatPow` `instMonoid` `Nat.iterate`	—	—
`instAddMonoidHomClass` 📖	mathematical	—	`AddMonoidHomClass` `AddMonoid.End` `instFunLike`	—	`AddMonoidHom.instAddMonoidHomClass`

AddMonoidHom

Definitions

Name	Category	Theorems
`coeToAddHom` 📖	CompOp	—
`coeToZeroHom` 📖	CompOp	—
`comp` 📖	CompOp	210 math math: `AddMonoid.Coprod.swap_comp_inl`, `AddMonoid.Coprod.lift_comp_inr`, `mapMatrix_comp`, `AddSubmonoid.LocalizationMap.ofAddEquivOfLocalizations_eq`, `IsAddIndecomposable.image_baseOf_neg_comp_eq`, `Finsupp.mapDomain.addMonoidHom_comp`, `AddEquiv.addMonoidHomCongrLeftEquiv_apply`, `comp_coprod`, `AddMonCat.uliftFunctor_map`, `AddCommGrpCat.ofHom_comp`, `AddMonoid.Coprod.snd_comp_toProd`, `range_comp`, `AddMonoid.Coprod.toProd_comp_inr`, `ker_comp_addEquiv`, `WithZero.mapAddHom_comp`, `prod_comp_prodMap`, `map_dfinsuppSumAddHom`, `AddMonoid.Coprod.map_map`, `DirectSum.toAddMonoid.unique`, `AddMonoid.Coprod.snd_comp_inr`, `fst_comp_inr`, `Set.mem_center_iff_addMonoidHom`, `AddSubgroup.map_map`, `DirectSum.fromAddMonoid_of`, `comp_noncommPiCoprod`, `CharacterModule.curry_apply_apply`, `coprod_unique`, `AddSubmonoid.subtype_comp_inclusion`, `comp_noncommCoprod`, `AddMonoid.Coprod.fst_comp_swap`, `AddGrpCat.uliftFunctor_map`, `AddMonCat.hom_comp`, `AddGrpCat.hom_comp`, `Finsupp.liftAddHom_comp_single`, `AddCommGrpCat.Colimits.colimitCocone_ι_app`, `AddMonoid.Coprod.clift_comp_mk`, `DFinsupp.mapRange.addMonoidHom_comp`, `FreeAddMonoid.map_comp`, `cancel_left`, `FreeAbelianGroup.map_comp`, `coe_dfinsuppSumAddHom`, `AddCommMonCat.coyonedaType_obj_map`, `AddEquiv.coe_addMonoidHom_comp_coe_addMonoidHom_symm`, `AddMonoid.Coprod.fst_comp_toProd`, `AddCon.hom_ext_iff`, `DirectSum.algebraMap_toAddMonoid_hom`, `AddCommMonCat.coyonedaType_map_app`, `AddMonoidAlgebra.mapDomainBialgHom_mapDomainBialgHom`, `prod_unique`, `AddMonoid.Coprod.map_comp_map`, `comp_zero`, `Matrix.entryAddMonoidHom_eq_comp`, `AddMonoid.Coprod.lift_comp_inl`, `map_mrange`, `AddMonoid.Coprod.hom_ext_iff`, `AddEquiv.map_dfinsuppSumAddHom`, `Finsupp.toFreeAbelianGroup_comp_singleAddHom`, `AddEquiv.addMonoidHomCongrLeft_apply`, `noncommCoprod_comp_inl`, `comp_assoc`, `noncommCoprod_unique`, `Finsupp.comp_liftAddHom`, `Polynomial.addHom_ext'_iff`, `map_range`, `subtype_comp_rangeRestrict`, `AddEquiv.comp_left_injective`, `FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup`, `AddSubmonoid.LocalizationMap.AwayMap.lift_comp`, `AddSubgroup.goursat_surjective`, `AddMonoid.Coprod.snd_comp_swap`, `AddSubmonoid.LocalizationMap.isAddUnit_comp`, `entryAddMonoidHom_comp_mapMatrix`, `comap_ker`, `smul_comp`, `AddSubmonoid.LocalizationMap.lift_of_comp`, `OpenAddSubgroup.comap_comap`, `AddMonoid.Coprod.range_eq`, `AddMonoid.Coprod.fst_comp_inl`, `QuotientAddGroup.ker_le_range_iff`, `LinearMap.map_dfinsuppSumAddHom`, `AddCon.lift_comp_mk'`, `QuotientAddGroup.mk'_comp_subtype`, `AddCommGrpCat.Colimits.Quot.desc_toCocone_desc`, `AddMonoid.Coprod.map_comp_inl`, `zero_comp`, `liftOfRightInverse_comp`, `AddCon.comap_eq`, `AddMonoid.Coprod.fst_comp_inr`, `AddEquiv.comp_right_injective`, `AddGroupSeminorm.comp_assoc`, `compHom_apply_apply`, `QuotientAddGroup.homQuotientZSMulOfHom_comp_of_rightInverse`, `AddMonoid.Coprod.comp_lift`, `SkewMonoidAlgebra.addHom_ext'_iff`, `Matrix.evalAddMonoidHom_comp_diagAddMonoidHom`, `AddMonoidAlgebra.mapDomainRingHom_comp`, `AddSubmonoid.LocalizationMap.map_comp`, `AddEquiv.addMonoidHomCongrRightEquiv_symm_apply`, `Finsupp.mapRange.addMonoidHom_comp`, `comp_sub`, `DFinsupp.sumAddHom_piSingle`, `QuotientAddGroup.lift_comp_mk'`, `fst_comp_prod`, `coprod_comp_inl`, `AddCommGrpCat.coyonedaType_obj_map`, `negAddMonoidHom_comp_negAddMonoidHom`, `AddEquiv.addMonoidHomCongrLeftEquiv_symm_apply`, `ZMod.lift_comp_castAddHom`, `AddCommGrpCat.Colimits.toCocone_ι_app`, `AddMonoid.Coprod.swap_comp_swap`, `LinearEquiv.map_dfinsuppSumAddHom`, `snd_comp_inl`, `AddSubmonoid.bsupr_eq_mrange_dfinsuppSumAddHom`, `AddEquiv.coe_addMonoidHom_trans`, `AddMonoid.Coprod.lift_comp_swap`, `mul_op_ext_iff`, `OrderAddMonoidHom.coe_comp_addMonoidHom`, `Finsupp.addHom_ext'_iff`, `AddSubmonoid.LocalizationMap.lift_comp`, `noncommCoprod_comp_inr`, `AddGrpCat.ofHom_comp`, `AddSubmonoid.LocalizationMap.lift_comp_lift`, `AddMonoidAlgebra.mapDomainAlgHom_comp`, `AddCommMonCat.ofHom_comp`, `AddSubmonoid.LocalizationMap.lift_comp_lift_eq`, `neg_comp`, `snd_comp_prod`, `DFinsupp.comp_liftAddHom`, `AddSubmonoid.LocalizationMap.ofAddEquivOfLocalizations_comp`, `AddMonCat.ofHom_comp`, `AddCommGrpCat.uliftFunctor_map`, `DirectSum.addHom_ext'_iff`, `AddEquiv.addMonoidHomCongrRightEquiv_apply`, `AddCommGroup.DirectLimit.lift_comp_of`, `AddCommGrpCat.hom_comp`, `AddSubmonoid.LocalizationMap.ofAddEquivOfDom_eq`, `DirectSum.ringHom_ext'_iff`, `DirectSum.map_comp`, `comp_id`, `coe_comp`, `DFinsupp.sumAddHom_comp_single`, `AddCon.lift_apply_mk'`, `QuotientAddGroup.addMonoidHom_ext_iff`, `AddMonoidAlgebra.mapDomainBialgHom_comp`, `DirectSum.liftRingHom_symm_apply_coe`, `AddSubmonoid.LocalizationMap.map_comp_map`, `CategoryTheory.Discrete.addMonoidalFunctorComp_isMonoidal`, `NonUnitalRingHom.coe_comp_addMonoidHom`, `snd_comp_inr`, `AddEquiv.coe_addMonoidHom_symm_comp_coe_addMonoidHom`, `AddSubmonoid.LocalizationMap.map_map`, `FreeAddMonoid.comp_lift`, `AddMonoid.Coprod.map_comp_inr`, `AddCommGrpCat.Colimits.Quot.desc_quotQuotUliftAddEquiv`, `range_le_ker_iff`, `AddSubgroup.subtype_comp_inclusion`, `CentroidHom.coe_comp_addMonoidHom`, `sub_comp`, `FiniteIndexNormalAddSubgroup.comap_comp`, `id_comp`, `AddCommGroup.DirectLimit.map_id`, `AddMonoid.Coprod.mrange_eq`, `comap_mker`, `AddMonoid.Coprod.toProd_comp_inl`, `functions_ext'_iff`, `Finsupp.liftAddHom_symm_apply`, `RingEquiv.coe_addMonoidHom_trans`, `AddSubmonoid.LocalizationMap.of_addEquivOfAddEquiv`, `AddCommGroup.DirectLimit.hom_ext_iff`, `AddSubmonoid.comap_comap`, `DFinsupp.liftAddHom_comp_single`, `DFinsupp.comp_sumAddHom`, `AddSubgroupClass.subtype_comp_inclusion`, `AddMonoidAlgebra.addHom_ext'_iff`, `dfinsuppSumAddHom_apply`, `cancel_right`, `AddSubgroup.comap_comap`, `coprod_comp_inr`, `RingHom.map_dfinsuppSumAddHom`, `WithZero.map_comp`, `MonoidAlgebra.addHom_ext'_iff`, `comp_apply`, `AddMonoid.Coprod.snd_comp_inl`, `comp_neg`, `AddGroupExtension.rightHom_comp_inl`, `Function.Exact.addMonoidHom_comp_eq_zero`, `Finsupp.mapDomain.addMonoidHom_comp_mapRange`, `map_mul_iff`, `DFinsupp.liftAddHom_symm_apply`, `AddEquiv.addMonoidHomCongrRight_apply`, `Algebra.GrothendieckAddGroup.lift_symm_apply`, `DFinsupp.addHom_ext'_iff`, `AddMonoid.Coprod.swap_comp_map`, `CharacterModule.dual_apply`, `AddCommMonCat.hom_comp`, `fst_comp_inl`, `comp_add`, `AddCommMonCat.uliftFunctor_map`, `AddUnits.map_comp`, `add_comp`, `QuotientAddGroup.homQuotientZSMulOfHom_comp`, `AddMonoid.Coprod.swap_comp_inr`, `Matrix.ext_addMonoidHom_iff`, `AddCommGrpCat.coyonedaType_map_app`, `AddSubmonoid.map_map`, `Finsupp.toFreeAbelianGroup_comp_toFinsupp`, `Module.Baer.extension_property_addMonoidHom`, `mrange_comp`, `AddGroupExtension.Splitting.rightHom_comp_splitting`, `prodMap_def`
`copy` 📖	CompOp	2 math math: `coe_copy`, `copy_eq`
`id` 📖	CompOp	72 math math: `mgraph_eq_mrange_prod`, `mapMatrix_id`, `AddMonCat.ofHom_id`, `MeasureTheory.VectorMeasure.mapRange_id`, `coe_id`, `AddMonoid.Coprod.snd_comp_inr`, `AddCommGroup.DirectLimit.lift_of'`, `AddGroupSeminorm.comp_id`, `AddMonoidAlgebra.mapDomainBialgHom_id`, `RingEquiv.toAddMonoidHom_refl`, `AddMonoidAlgebra.mapDomainAlgHom_id`, `AddEquiv.coe_addMonoidHom_refl`, `id_apply`, `WithZero.mapAddHom_id`, `FreeAbelianGroup.map_id`, `AddEquiv.coe_addMonoidHom_comp_coe_addMonoidHom_symm`, `noncommCoprod_inl_inr`, `AddMonoid.Coprod.map_id_id`, `AddMonoid.Coprod.clift_mk`, `ker_id`, `CentroidHom.toAddMonoidHom_id`, `graph_eq_range_prod`, `FreeAbelianGroup.toFinsupp_comp_toFreeAbelianGroup`, `AddCommGrpCat.hom_id`, `WithZero.map_id`, `AddSubmonoid.map_id`, `AddMonoid.Coprod.fst_comp_inl`, `FreeAddGroup.lift_of_eq_id`, `RingEquiv.coe_addMonoidHom_refl`, `toMultiplicative_id`, `QuotientAddGroup.homQuotientZSMulOfHom_comp_of_rightInverse`, `AddMonCat.hom_id`, `FreeAddMonoid.map_id`, `AddSubgroup.comap_id`, `negAddMonoidHom_comp_negAddMonoidHom`, `AddMonoid.Coprod.swap_comp_swap`, `AddMonoid.Coprod.lift_inl_inr`, `NonUnitalRingHom.coe_addMonoidHom_id`, `coprod_inl_inr`, `QuotientAddGroup.quotientBot_apply`, `AddCommGrpCat.ofHom_id`, `MonoidHom.toAdditive_id`, `AddSubgroup.map_id`, `comp_id`, `Finsupp.liftAddHom_singleAddHom`, `DFinsupp.liftAddHom_singleAddHom`, `AddUnits.map_id`, `snd_comp_inr`, `AddEquiv.coe_addMonoidHom_symm_comp_coe_addMonoidHom`, `Finsupp.mapDomain.addMonoidHom_id`, `DFinsupp.sumAddHom_singleAddHom`, `id_comp`, `AddCommGroup.DirectLimit.map_id`, `AddSubmonoid.LocalizationMap.map_id`, `QuotientAddGroup.quotientQuotientEquivQuotientAux_mk`, `AddSubmonoid.comap_id`, `mrange_id`, `AddCommMonCat.ofHom_id`, `AddMonoidAlgebra.mapDomainRingHom_id`, `AddGrpCat.ofHom_id`, `RingHom.coe_addMonoidHom_id`, `DFinsupp.mapRange.addMonoidHom_id`, `AddGrpCat.hom_id`, `FiniteIndexNormalAddSubgroup.comap_id`, `AddCommMonCat.hom_id`, `fst_comp_inl`, `Finsupp.mapRange.addMonoidHom_id`, `Int.castAddHom_int`, `Finsupp.toFreeAbelianGroup_comp_toFinsupp`, `QuotientAddGroup.homQuotientZSMulOfHom_id`, `AddGroupExtension.Splitting.rightHom_comp_splitting`, `DirectSum.map_id`
`instFunLike` 📖	CompOp	1814 math math: `ModuleCat.HasColimit.colimitCocone_pt_isAddCommGroup`, `AddMonoid.Coprod.swap_surjective`, `QuotientAddGroup.kerLift_mk`, `ValueDistribution.logCounting_zero`, `SkewMonoidAlgebra.mapDomain_smul`, `AddCommGrpCat.forget₂AddGroup_preservesLimitsOfShape`, `AddUnits.map_mk`, `FreeAddMonoid.lift_symm_apply`, `RCLike.conj_re`, `Finsupp.sum_toMultiset`, `coe_rangeRestrict`, `RCLike.sqrt_eq_ite`, `Finsupp.finite_of_degree_le`, `coe_ker`, `TensorAlgebra.ofDirectSum_of_tprod`, `MvPowerSeries.WithPiTopology.tendsto_trunc_atTop`, `RCLike.im_le_neg_norm_iff_eq_neg_I_mul_norm`, `DirectSum.mulHom_of_of`, `Ideal.Filtration.submodule_closure_single`, `DirectSum.mker_map`, `AddGrpCat.forget_grp_preserves_epi`, `IsAddIndecomposable.image_baseOf_neg_comp_eq`, `AddEquiv.coprodComm_apply`, `RCLike.one_re`, `DFinsupp.toMultiset_sup`, `negAddMonoidHom_apply`, `ModuleCat.forget₂_reflectsLimitsOfSize`, `Ideal.Filtration.submodule_eq_span_le_iff_stable_ge`, `CategoryTheory.opHom_apply`, `MeasureTheory.VectorMeasure.coeFnAddMonoidHom_apply`, `mgraph_eq_mrange_prod`, `RCLike.map_apply`, `LinearMap.toAddMonoidHom_coe`, `Finsupp.degree_eq_sum`, `RCLike.geometric_hahn_banach_point_open`, `isOfFinAddOrder`, `MvPowerSeries.weightedOrder_monomial`, `MvPowerSeries.eq_iff_frequently_trunc'_eq`, `taylorWithin_succ`, `AddGrpCat.forget₂_map`, `ExteriorAlgebra.GradedAlgebra.liftι_eq`, `PresentedAddMonoid.ext_iff`, `AddMonoid.Coprod.swap_inj`, `DirectSum.Gmodule.smulAddMonoidHom_apply_of_of`, `AddMonoid.Coprod.toProd_apply_inr`, `exact_of_comp_of_mem_range`, `AddCommGrpCat.biproductIsoPi_inv_comp_π_apply`, `AddCommGrpCat.comp_apply`, `PolynomialModule.comp_single`, `RingCat.FilteredColimits.instPreservesFilteredColimitsAddCommGrpCatForget₂RingHomCarrierAddMonoidHomCarrier`, `AddGrpCat.id_apply`, `RCLike.reCLM_apply`, `MvPowerSeries.trunc'_trunc'`, `RCLike.div_re_ofReal`, `FreeAddMonoid.lift_ofList`, `RCLike.separate_convex_open_set`, `MvPolynomial.degrees_monomial`, `FreeAddGroup.map.id'`, `AddMonoidAlgebra.liftNC_mul`, `FreeAddMonoid.count_apply`, `NumberField.Units.logEmbedding_fundSystem`, `coe_mk`, `Finset.singletonAddMonoidHom_apply`, `AddSubgroup.map_toAddSubmonoid`, `Function.locallyFinsuppWithin.toClosedBall_support_subset_closedBall`, `AddMonoid.Coprod.mk_of_inr`, `ModuleCat.forget₂PreservesColimitsOfSize`, `AddMonoid.Coprod.snd_apply_inr`, `SkewMonoidAlgebra.erase_apply_toFinsupp`, `Finsupp.toMultiset_strictMono`, `AddMonCat.comp_apply`, `restrictHom_apply`, `DirectSum.coe_toModule_eq_coe_toAddMonoid`, `MvPolynomial.weightedDegree_eq_zero_iff`, `MonoidHom.toAdditive_apply_apply`, `ZMod.LFunction_stdAddChar_eq_expZeta`, `FreeAddMonoid.lift_apply`, `AddSubgroup.closure_preimage_eq_top`, `RCLike.re_add_im`, `InnerProductSpace.Core.inner_mul_inner_self_le`, `qExpansionRingHom_apply`, `AddCommGrpCat.asHom_hom_apply`, `Ideal.isHomogeneous_iff_subset_iInter`, `AddSubmonoid.prod_le_iff`, `prevD_comp_left`, `DirectSum.mrange_map`, `Finsupp.singleAddHom_apply`, `Multiset.equivDFinsupp_symm_apply`, `AddCommGrpCat.FilteredColimits.forget_preservesFilteredColimits`, `ModuleCat.forget₂AddCommGroup_preservesLimitsOfSize`, `DFinsupp.liftAddHom_apply_single`, `WittVector.verschiebung_zmod`, `map_zero`, `toMultiplicativeRight_apply_apply`, `FreeAbelianGroup.toFinsupp_toFreeAbelianGroup`, `noncommPiCoprod_apply`, `Multiset.toDFinsupp_support`, `ModuleCat.mkOfSMul'_smul`, `ContinuousMultilinearMap.alternatization_apply_apply`, `Submodule.re_inner_starProjection_eq_normSq`, `NormedAddGroupHom.coe_toAddMonoidHom`, `AddMonCat.zeroHom_apply`, `AddUnits.embedProduct_injective`, `toAddEquiv_symm_apply`, `AddSubgroup.coe_addSubgroupOf`, `LinearMap.IsPositive.re_inner_nonneg_right`, `DirectSum.of_smul`, `coe_prodMap`, `LinearMap.IsSymmetric.im_inner_self_apply`, `DirectSum.fromAddMonoid_of_apply`, `AddGroupExtension.Section.exists_eq_inl_add`, `RCLike.sub_conj`, `PresentedAddMonoid.surjective_mk`, `RCLike.re_sqrt_ofReal`, `RCLike.pos_iff`, `AddMonoid.Coprod.fst_swap`, `CategoryTheory.ShortComplex.ShortExact.ab_surjective_g`, `MeasureTheory.Lp.indicatorConstLp_compMeasurePreserving`, `MeasureTheory.Lp.coeFn_const`, `CategoryTheory.Abelian.Ext.precomp_mk₀_injective_of_epi`, `WittVector.eq_iterate_verschiebung`, `MvPowerSeries.trunc'_expand_trunc'`, `AddSubmonoidClass.coe_subtype`, `SkewMonoidAlgebra.coeff_erase_same`, `AddEquiv.coprodPUnit_symm_apply`, `coe_prod`, `AddSubmonoid.prod_top`, `SemimoduleCat.instReflectsIsomorphismsAddCommMonCatForget₂LinearMapIdCarrierAddMonoidHomCarrier`, `AddChar.doubleDualEmb_injective`, `prod_map_comap_prod'`, `map_dfinsuppSumAddHom`, `inl_strictMono`, `MulEquiv.toAdditive_apply_apply`, `SkewMonoidAlgebra.mapDomain_one`, `SubAddAction.ofFixingAddSubgroup_of_inclusion_injective`, `FreeAddMonoid.count_of`, `Finsupp.weight_single_one_apply`, `CochainComplex.HomComplex.Cocycle.toCochainAddMonoidHom_apply`, `Ideal.isHomogeneous_iff_forall_subset`, `AddMonoid.Coprod.map_map`, `Finsupp.mapDomain_comapDomain_nat_add_one`, `AddMonCat.forget_reflects_isos`, `TensorAlgebra.toDirectSum_tensorPower_tprod`, `AddMonCat.forget_preservesLimitsOfSize`, `AddSubgroup.coe_comap`, `AddCommMonCat.coe_id`, `CategoryTheory.Abelian.Ext.preadditiveYoneda_homologySequenceδ_singleTriangle_apply`, `Finsupp.prod_toMultiset`, `instAddMonoidHomClass`, `iSupIndep.dfinsuppSumAddHom_injective`, `Multiset.mapAddMonoidHom_apply`, `Finsupp.toMultiset_zero`, `MonoidAlgebra.liftNC_mul`, `AddSubgroup.coe_subtype`, `AddMonoid.Coprod.swap_bijective`, `AddCommMonCat.ofHom_apply`, `toFun_eq_coe`, `RCLike.ofReal_re`, `Finsupp.le_weight_of_ne_zero'`, `DirectSum.toAddMonoid.unique`, `coe_flip_mul`, `coe_id`, `PiTensorProduct.lifts_smul`, `LinearMap.BilinMap.toQuadraticMapAddMonoidHom_apply`, `RCLike.re_add_im_ax`, `AddSubgroup.noncommPiCoprod_single`, `mulRight_bound`, `AddSubmonoid.LocalizationMap.addEquivOfLocalizations_symm_apply`, `WittVector.verschiebung_coeff_succ`, `MvPowerSeries.ext_trunc'`, `RCLike.is_real_TFAE`, `map_zsmul`, `AddGrpCat.forget_preservesLimits`, `QuotientAddGroup.mk'_eq_mk'`, `PresheafOfModules.sections_property`, `lp.coeFnAddMonoidHom_apply`, `CentroidHom.coe_toAddMonoidHom`, `RCLike.abs_im_div_norm_le_one`, `AddSubgroupClass.inclusion_self`, `norm_sub_mul_self`, `Multiset.Icc_eq`, `RingCat.forget₂AddCommGroup_preservesLimits`, `multiplesAddHom_apply`, `PresheafOfModules.toSheafify_app_apply'`, `PiTensorProduct.projectiveSeminormAux_smul`, `Matrix.IsHermitian.eigenvalues_eq`, `AddChar.directSum_apply`, `map_finsum`, `coe_of_map_add_neg`, `im_inner_eq_norm_sub_i_smul_mul_self_sub_norm_add_i_smul_mul_self_div_four`, `uliftMultiplesHom_symm_apply`, `coe_smul'`, `AddCommMonCat.FilteredColimits.forget₂AddMonPreservesFilteredColimits`, `PresheafOfModules.Derivation.d_map`, `ConvexOn.univ_sSup_of_countable_affine_eq`, `AddMonoid.End.mulRight_apply_apply`, `DomMulAct.smul_addMonoidHom_apply`, `AddAction.stabilizer_vadd_eq_stabilizer_map_conj`, `SeparationQuotient.mkAddMonoidHom_apply`, `WeierstrassCurve.Affine.Point.toClass_zero`, `AddMonCat.ext_iff`, `RCLike.ofNat_re`, `AddCommGrpCat.isColimit_iff_bijective_desc`, `groupCohomology.mem_cocycles₁_of_addMonoidHom`, `QuotientAddGroup.coe_mk'`, `AddCon.mk'_surjective`, `Set.conj_mem_fixingAddSubgroup`, `restrict_eq_zero_iff`, `SkewMonoidAlgebra.coeff_erase_ne`, `Multiset.coe_sumAddMonoidHom`, `AddSubmonoid.iSup_map_single`, `RCLike.nonpos_iff`, `RCLike.smul_re`, `HahnSeries.single.addMonoidHom_apply_coeff`, `AddSubmonoid.pi_le_iff`, `ProbabilityTheory.Kernel.coeAddHom_apply`, `DirectSum.coeAddMonoidHom_eq_dfinsuppSum`, `toZeroHom_coe`, `AddMonoid.Coprod.neg_def`, `AddCommMonCat.hom_forget₂_map`, `DirectSum.fromAddMonoid_of`, `SkewMonoidAlgebra.mapDomain_mul`, `DFinsupp.sumAddHom_single`, `TensorProduct.liftAux_tmul`, `MvPolynomial.supDegree_esymmAlgHomMonomial`, `ModuleCat.forget₂_addCommGrp_essSurj`, `MvPolynomial.le_weightedTotalDegree`, `Function.locallyFinsuppWithin.logCounting_nonneg`, `coe_toMultiplicativeRight`, `AddUnits.coeHom_apply`, `DFinsupp.coeFnAddMonoidHom_apply`, `Finsupp.toMultiset_toFinsupp`, `RCLike.abs_re_le_norm`, `DirectSum.mk_smul`, `ZMod.completedLFunction_def_even`, `Behrend.map_mod`, `MvPolynomial.pow_idealOfVars`, `MvPolynomial.idealOfVars_eq_restrictSupportIdeal`, `AddCommGrpCat.neg_hom_apply`, `AddGroupExtension.Equiv.map_inl`, `RCLike.geometric_hahn_banach_closed_point`, `Filter.pureAddMonoidHom_apply`, `AEMeasurable.im`, `AddMonoidAlgebra.divOfHom_apply_apply`, `SkewMonoidAlgebra.mapDomain_single`, `DistribMulAction.toAddMonoidHom_apply`, `Finsupp.coeFnAddHom_apply`, `Behrend.map_succ`, `AddCommGrpCat.forget_preservesLimits`, `DirectSum.mk_injective`, `Behrend.map_le_of_mem_box`, `Polynomial.trunc_toLaurent`, `MvPolynomial.totalDegree_eq`, `WittVector.iterate_verschiebung_mul`, `ker_eq_bot_iff`, `RCLike.norm_sq_eq_def_ax`, `addSubgroupOf_range_eq_of_le`, `ModuleCat.forget₂AddCommGroup_reflectsLimitOfShape`, `AddAction.map_stabilizer_le`, `Homotopy.prevD_succ_cochainComplex`, `AddCommGrpCat.forget_preservesLimitsOfShape`, `DirectSum.decompose_of_mem`, `Finsupp.mapRange.addMonoidHom_apply`, `MvPolynomial.mem_pow_idealOfVars_iff`, `WittVector.verschiebung_nonzero`, `Multiset.toDFinsupp_apply`, `fiberEquiv_symm_apply`, `RCLike.intCast_re`, `Matrix.addMonoidHomMulLeft_apply`, `DirectSum.coe_of_mul_apply_aux`, `rangeRestrict_surjective`, `RCLike.im_eq_zero_iff_isSelfAdjoint`, `Seminorm.pullback_apply`, `MvPowerSeries.trunc_C`, `FreeAddMonoid.mem_map`, `inner_mul_inner_self_le`, `commute_lmul_rmul`, `CategoryTheory.GrothendieckTopology.MayerVietorisSquare.biprodAddEquiv_symm_biprodIsoProd_hom_toBiprod_apply`, `MvPolynomial.degrees_def`, `ModN.basis_apply_eq_mkQ`, `HahnSeries.map_neg`, `RCLike.mul_im_ax`, `Function.locallyFinsuppWithin.logCounting_eval_zero`, `Polynomial.eval₂_ofFinsupp`, `AddCommGrpCat.coe_comp`, `InnerProductSpace.Core.norm_eq_sqrt_re_inner`, `DirectSum.Gmodule.of_smul_of`, `AdjoinRoot.evalEval_apply`, `MonoidAlgebra.coe_liftNCAlgHom`, `CuspForm.coeHom_apply`, `zmultiplesAddHom_symm_apply`, `Module.End.ringHomEndFinsupp_apply_apply`, `RCLike.abs_re_div_norm_le_one`, `addSubmonoidComap_apply_coe`, `norm_sub_sq`, `ValueDistribution.logCounting_top`, `ContinuousAt.compMeasurePreservingLp`, `LinearMap.BilinForm.coeFnAddMonoidHom_apply`, `RCLike.I_re_ax`, `DirectSum.sum_univ_of`, `AddGrpCat.forget_grp_preserves_mono`, `DirectSum.lie_of_of_ne`, `AddCommGrpCat.forget₂_map`, `ModuleCat.HasColimit.colimitCocone_ι_app`, `Finsupp.card_toMultiset`, `RCLike.re_nonneg_of_nonneg`, `MonoidHom.toAdditive_symm_apply_apply`, `CentroidHom.centroid_eq_centralizer_mulLeftRight`, `ZMod.LFunction_def_odd`, `LinearMap.IsSymmetric.coe_re_inner_self_apply`, `SkewMonoidAlgebra.comapSMul_def`, `AddChar.coe_compAddMonoidHom`, `id_apply`, `Sym.coe_equivNatSum_symm_apply`, `PolynomialModule.monomial_smul_single`, `Multiset.addHom_ext_iff`, `LinearMap.applyₗ'_apply_apply`, `multiplesAddHom_symm_apply`, `postcomp_extClass_surjective_of_projective_X₂`, `PresheafOfModules.Derivation.postcomp_d_apply`, `ContinuousMap.coeFnAddMonoidHom_apply`, `AddSubmonoid.closure_image_isAddIndecomposable_baseOf`, `mulRight_apply`, `ModuleCat.forget₂_addCommGroup_full`, `DirectSum.gMulHom_apply_apply`, `PresheafOfModules.Derivation.d_one`, `CategoryTheory.Idempotents.Karoubi.inclusionHom_apply`, `DoubleCentralizer.norm_def`, `FreeAddGroup.lift_mk`, `Module.DirectLimit.quotMk_of`, `PresheafOfModules.sectionsMap_coe`, `re_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two`, `AddMonoid.Coprod.mrange_lift`, `AddCommGrpCat.FilteredColimits.forget₂AddGroup_preservesFilteredColimits`, `AddMonCat.FilteredColimits.colimit_add_mk_eq`, `AddMonCat.forget_preservesLimitsOfShape`, `CategoryTheory.ShortComplex.abToCycles_apply_coe`, `CochainComplex.HomComplex.leftHomologyData_π_hom_apply`, `MvPowerSeries.weightedOrder_monomial_of_ne_zero`, `AddGrpCat.FilteredColimits.colimit_add_mk_eq`, `DirectSum.lie_of`, `rangeRestrict_injective_iff`, `AddCommMonCat.forget₂AddMonPreservesLimitsOfSize`, `re_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two`, `AddEquiv.withZeroCongr_apply`, `coe_dfinsuppSumAddHom`, `Function.locallyFinsuppWithin.logCounting_eventually_le`, `PresheafOfModules.Derivation.d_mul`, `WittVector.verschiebung_injective`, `FreeAbelianGroup.map_of_apply`, `groupCohomology.cocycles₁IsoOfIsTrivial_hom_hom_apply_apply`, `AddSubgroup.noncommPiCoprod_apply`, `DirectSum.support_of`, `Multiset.equivDFinsupp_apply`, `inner_re_zero_right`, `two_nsmul_lie_lmul_lmul_add_add_eq_zero`, `AddMonoid.Coprod.prod_mk_fst_snd`, `AddSubgroupClass.subtype_apply`, `Derivation.mapCoeffs_C`, `FGModuleCat.instFiniteCarrierSigmaObjModuleCatOfFinite`, `map_finsum_plift`, `AddMonoidAlgebra.toDirectSum_single`, `eval_apply_apply`, `qExpansion_of_mul`, `AddSubmonoid.subPairs_comap`, `flip_apply`, `AddSubgroup.mem_map_of_mem`, `DFinsupp.toMultiset_single`, `CochainComplex.HomComplex.leftHomologyData_i_hom_apply`, `MvPowerSeries.trunc'_trunc'_pow`, `ModuleCat.smul_naturality`, `Algebra.directSumGAlgebra_toFun_apply`, `LinearMap.toAddMonoidHom'_apply`, `Finset.coe_coeAddMonoidHom`, `PolynomialModule.single_smul`, `SkewMonoidAlgebra.addHom_ext_iff`, `LinearMap.compMultilinearMap_alternatization`, `CategoryTheory.Abelian.Ext.preadditiveCoyoneda_homologySequenceδ_singleTriangle_apply`, `Finsupp.toMultiset_apply`, `addSubgroupMap_apply_coe`, `DirectSum.coeLinearMap_of`, `OrderAddMonoidHom.coe_addMonoidHom`, `toAddHom_coe`, `AddMonoid.Coprod.clift_mk`, `MeasureTheory.VectorMeasure.restrictGm_apply`, `AddSubgroup.tendsto_zmultiples_subtype_cofinite`, `ModuleCat.HasColimit.colimitCocone_pt_isModule`, `DirectSum.coe_of_mul_apply_add`, `DoubleCentralizer.nnnorm_def`, `RCLike.re_monotone`, `ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_left`, `AddMonCat.forget_map`, `AddMonoidAlgebra.decomposeAux_coe`, `AddAut.conj_symm_apply`, `RCLike.im_ofReal_pow`, `CategoryTheory.preadditiveYoneda_obj`, `SemimoduleCat.forget₂_obj_moduleCat_of`, `DirectSum.ofPow`, `isClosed_range_coe`, `AddMonoidAlgebra.gradeBy.isInternal`, `MvPowerSeries.trunc_one`, `DirectSum.of_zero_smul`, `AddCommGrpCat.ext_iff`, `NumberField.Units.logEmbeddingQuot_apply`, `LowerSemicontinuousOn.isClosed_re_epigraph`, `map_mrange`, `CategoryTheory.Adjunction.compPreadditiveYonedaIso_inv_app_app_apply`, `LieRingModule.toEnd_apply_apply`, `LinearMap.IsSymmetric.hasEigenvalue_iSup_of_finiteDimensional`, `Finsupp.weight_single`, `NumberField.Units.basisOfIsMaxRank_apply`, `Module.Invertible.rTensorEquiv_apply_apply`, `toMultiplicativeRight_symm_apply_apply`, `MeasureTheory.Measure.coeAddHom_apply`, `DirectSum.coe_congr_addEquiv`, `AddMonoid.Coprod.inl_injective`, `map_add_neg`, `inner_self_im`, `WithBot.coe_addHom`, `AddEquiv.map_dfinsuppSumAddHom`, `Set.singletonAddMonoidHom_apply`, `zero_apply`, `nsmulAddMonoidHom_apply`, `groupCohomology.H1IsoOfIsTrivial_H1π_apply_apply`, `AddCommGrpCat.forget_commGrp_preserves_epi`, `map_finsum_mem'`, `AddCommMonCat.forget_preservesLimitsOfShape`, `AddCommGrpCat.coyonedaForget_inv_app_app`, `Finsupp.toFreeAbelianGroup_comp_singleAddHom`, `coe_coe`, `AddCommGrpCat.free_map_coe`, `compl₂_apply`, `mulOp_apply_apply`, `CategoryTheory.ShortComplex.ShortExact.ab_injective_f`, `inner_re_zero_left`, `inner_self_ofReal_re`, `prevD_eq_toPrev_dTo`, `FreeAddGroup.lift_apply_of`, `commute_lmul_sq_rmul`, `DirectSum.one_def`, `AddOreLocalization.numerator_isAddUnit`, `RCLike.imLm_coe`, `AddMonCat.coe_comp`, `LocallyConstant.coeFnAddMonoidHom_apply`, `fiberEquivKer_apply`, `RCLike.div_im`, `AddMonoid.Coprod.map_apply_inr`, `FreeAbelianGroup.liftAddGroupHom_apply`, `mem_graph`, `MonoidAlgebra.singleAddHom_apply`, `withTopMap_apply`, `AddCommGrpCat.coyonedaForget_hom_app_app_hom`, `ZMod.toAddCircle_inj`, `FreeAddGroup.map.comp`, `CategoryTheory.Iso.addCommGroupIsoToAddEquiv_symm_apply`, `AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff'`, `Homotopy.dNext_zero_chainComplex`, `DirectSum.of_mul_of`, `ZMod.completedLFunction_def_odd`, `RCLike.im_to_real`, `AddGroupExtension.rightHom_inl`, `PiTensorProduct.liftAux.smul`, `Fin.accumulate_invAccumulate`, `RCLike.ext_iff`, `Homotopy.prevD_chainComplex`, `AddGrpCat.forget₂AddMonPreservesLimitsOfSize`, `Finset.coe_singletonAddMonoidHom`, `AddGroupExtension.Section.exists_add_eq_inl_add_add`, `Real.Angle.coe_coeHom`, `MvPolynomial.coeff_homogeneousComponent`, `FreeAddMonoid.ofList_map`, `DFinsupp.eraseAddHom_apply`, `BoundedContinuousFunction.toContinuousMapAddMonoidHom_apply`, `AddMonoid.Coprod.snd_toProd`, `DirectSum.coe_mul_of_apply_of_not_le`, `LocallyConstant.constAddMonoidHom_apply`, `coe_graph`, `lp.singleAddMonoidHom_apply`, `sub_apply`, `RCLike.re_eq_norm_of_mul_conj`, `coe_finset_sum`, `FreeAbelianGroup.equivFinsupp_symm_apply`, `AddGrpCat.FilteredColimits.forget_preservesFilteredColimits`, `CategoryTheory.ShortComplex.exact_iff_surjective_abToCycles`, `MeasureTheory.Lp.coeFn_compMeasurePreserving`, `AddCommGrpCat.biprodIsoProd_inv_comp_snd_apply`, `AddCommMonCat.comp_apply`, `Behrend.map_injOn`, `AddEquiv.coprodAssoc_apply_inr`, `Derivation.mapCoeffs_monomial`, `AddCommMonCat.forget₂_full`, `mem_eqLocusM`, `CategoryTheory.ShortComplex.exact_iff_of_hasForget`, `RCLike.conj_eq_iff_re`, `map_nsmul`, `PolynomialModule.single_apply`, `AlgebraicGeometry.ProjIsoSpecTopComponent.FromSpec.mem_carrier_iff`, `noncommCoprod_unique`, `MultilinearMap.domCoprod_alternization_coe`, `Behrend.map_apply`, `DomMulAct.coe_smul_addMonoidHom`, `Function.locallyFinsuppWithin.logCounting_eventuallyLE`, `MvPolynomial.pow_idealOfVars_eq_span`, `prevD_nat`, `CategoryTheory.ShortComplex.ab_exact_iff_function_exact`, `AddMonCat.FilteredColimits.forget_preservesFilteredColimits`, `AddSubmonoid.mk_add_mk_neg_eq_zero`, `FreeAddMonoid.map_surjective`, `ConvexOn.sSup_of_countable_affine_eq`, `MvPolynomial.weightedHomogeneousSubmodule_eq_finsupp_supported`, `RCLike.ofNat_mul_re`, `AddChar.doubleDualEmb_apply`, `QuadraticMap.evalAddMonoidHom_apply`, `monoidEndToAdditive_apply_apply`, `AddMonoid.Coprod.swap_injective`, `AddSubmonoid.le_prod_iff`, `AddSubmonoid.closure_singleton_eq`, `AddSubmonoid.mrange_subtype`, `DFinsupp.sumAddHom_comm`, `InnerProductSpace.Core.inner_im_symm`, `AddSubgroup.addCommute_subtype_of_addCommute`, `map_closure`, `toMultiplicative_symm_apply_apply`, `AddSubmonoid.le_comap_single_pi`, `AddGrpCat.zero_apply`, `Matrix.addMonoidHomMulRight_apply`, `coe_toMultiplicativeLeft`, `GradedAlgebra.proj_recompose`, `RCLike.I_eq_zero_or_im_I_eq_one`, `MvPowerSeries.coeff_homogeneousComponent`, `MonoidHom.toAdditiveLeft_apply_apply`, `AddMonoid.Coprod.snd_surjective`, `LinearMap.tensorProductEnd_apply`, `Finsupp.degree_apply`, `DirectSum.algebraMap_apply`, `RCLike.le_iff_re_im`, `AddChar.coe_toAddMonoidHom`, `AddCommMonCat.instFullMonCatForget₂AddMonoidHomCarrierCarrier`, `ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_right`, `AddMonoidAlgebra.mapDomainBialgHom_apply`, `addSubmonoidMap_surjective`, `MeasureTheory.indicatorConstLp_univ`, `Module.End.rTensorAlgHom_apply_apply`, `ModuleCat.forget₂AddCommGroup_preservesLimits`, `Fin.accumulate_rec`, `AddCommMonCat.forget_preservesLimits`, `AddSubmonoid.iSup_map_single_le`, `inl_apply`, `Finsupp.toFreeAbelianGroup_single`, `AddGrpCat.forget_preservesLimitsOfShape`, `associator_apply`, `TensorPower.galgebra_toFun_def`, `AddSubmonoid.LocalizationMap.isAddUnit_comp`, `AddCommGrpCat.Colimits.quotUliftToQuot_ι`, `DirectSum.lof_eq_of`, `Multiset.cardHom_apply`, `Polynomial.divX_hom_toFun`, `PresheafOfModules.presheaf_map_apply_coe`, `controlled_sum_of_mem_closure_range`, `MeasureTheory.Lp.norm_const_le`, `precomp_extClass_surjective_of_projective_X₂`, `single_apply`, `LocallyConstant.comapAddMonoidHom_apply`, `DirectSum.map_injective`, `SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom_quotMk`, `QuotientAddGroup.lift_mk'`, `FreeAddGroup.map.id`, `DirectSum.of_zero_mul`, `compr₂_apply`, `Derivation.coeFnAddMonoidHom_apply`, `PresheafOfModules.Derivation.congr_d`, `Finsupp.finite_of_nat_weight_le`, `commute_lmul_rmul_sq`, `withBotMap_apply`, `RCLike.measurable_re`, `DirectSum.lmap_eq_map`, `real_inner_eq_re_inner`, `RCLike.add_conj`, `RCLike.ofReal_re_ax`, `InnerProductSpace.Core.inner_self_eq_norm_mul_norm`, `AddSubmonoid.map_inl`, `RCLike.im_eq_conj_sub`, `AddSubmonoid.map_coe_toAddMonoidHom`, `neg_apply`, `WeierstrassCurve.Affine.Point.map_injective`, `RCLike.conj_im_ax`, `finsuppSum_apply`, `AddSubmonoid.map_id`, `CategoryTheory.GrothendieckTopology.MayerVietorisSquare.mk₀_f_comp_biprodAddEquiv_symm_biprodIsoProd_hom`, `ModuleCat.forget₂_obj`, `AddCommMonCat.FilteredColimits.forget_preservesFilteredColimits`, `AddMonoid.End.mulLeft_apply_apply`, `AddAction.stabilizerEquivStabilizer_symm_apply`, `Finsupp.le_degree`, `MvPolynomial.degree_degLexDegree`, `CategoryTheory.pullbackShiftFunctorZero'_hom_app`, `AddSubmonoid.sup_eq_range`, `Matrix.singleAddMonoidHom_apply`, `MultilinearMap.domCoprod_alternization_eq`, `AddMonCat.coe_id`, `NumberField.Units.dirichletUnitTheorem.sum_logEmbedding_component`, `DirectSum.mk_apply_of_notMem`, `ContinuousWithinAt.compMeasurePreservingLp`, `AddCommMonCat.neg_hom_apply`, `MvPolynomial.DirectSum.coeAddMonoidHom_eq_support_sum`, `LinearOrderedField.inducedOrderRingHom_toFun`, `Finsupp.le_weight`, `mulRight_continuous`, `FreeAddGroup.ext_hom_iff`, `AddMonoid.Coprod.fst_toProd`, `MeasureTheory.Lp.constₗ_apply`, `RCLike.im_mul_ofReal`, `AddMonCat.FilteredColimits.M.map_mk`, `RCLike.geometric_hahn_banach_point_point`, `AddCommMonCat.forget₂Mon_preservesLimits`, `Fin.accumulate_apply`, `AddCommGrpCat.forget_commGrp_preserves_mono`, `CategoryTheory.whiskering_linearCoyoneda₂`, `DFinsupp.toMultiset_le_toMultiset`, `normedAddGroupHomCompletionHom_apply`, `AddEquiv.add_submonoid_map_symm_apply`, `range_eq_top`, `LinearMap.map_dfinsuppSumAddHom`, `ext_nat_iff`, `AddOreLocalization.numeratorHom_surjective_of_finite`, `AddMonCat.neg_hom_apply`, `MeasureTheory.Lp.norm_const'`, `Function.locallyFinsuppWithin.toClosedBall_divisor`, `SchwartzMap.coeHom_injective`, `AddGrpCat.ext_iff`, `AddCon.mrange_mk'`, `WithZero.lift_coe`, `Submodule.setLike.coe_galgebra_toFun`, `mulLeft₃_apply`, `Finsupp.mem_toMultiset`, `LinearMap.extendTo𝕜'_apply_re`, `snd_mono`, `AddSubmonoid.mem_bsupr_iff_exists_dfinsupp`, `addSubgroupComap_apply_coe`, `CategoryTheory.Pretriangulated.preadditiveYoneda_homologySequenceδ_apply`, `RCLike.re_to_real`, `ZMod.lift_castAddHom`, `AddSubgroup.injective_noncommPiCoprod_of_iSupIndep`, `prevD_eq`, `RCLike.neg_iff`, `RCLike.sqrt_eq_real_add_ite`, `AddMonoid.Coprod.toProd_apply_inl`, `RCLike.inv_im`, `AddEquiv.coprodPUnit_apply`, `restrict_mker`, `CategoryTheory.Preadditive.epi_iff_surjective`, `AddCommMonCat.ext_iff`, `PiTensorProduct.liftAux_tprodCoeff`, `FreeAddMonoid.toList_map`, `ZMod.lift_comp_coe`, `DirectSum.coeAlgHom_of`, `MulEquiv.AddMonoid.End_apply`, `MvPolynomial.monomial_mem_pow_idealOfVars_iff`, `InnerProductSpaceable.inner_.norm_sq`, `CategoryTheory.Preadditive.mono_iff_injective`, `ConvexOn.sSup_of_nat_affine_eq`, `coe_of_map_sub`, `RCLike.one_im`, `AddGroupExtension.Splitting.rightHom_splitting`, `AddMonoid.Coprod.lift_apply_inl`, `HomogeneousIdeal.mem_irrelevant_iff`, `Ideal.homogeneousHull_eq_iSup`, `AddMonCat.forget_isCorepresentable`, `RCLike.natCast_re`, `AddSubmonoid.le_pi_iff`, `Int.coe_castAddHom`, `InnerProductSpace.Core.inner_self_nonneg`, `inner_re_symm`, `AddCon.comap_eq`, `CliffordAlgebra.GradedAlgebra.ι_apply`, `FreeAbelianGroup.map_id_apply`, `AddAut.conj_inv_apply`, `NonUnitalNonAssocCommSemiring.mem_center_iff`, `Multiset.coe_countPAddMonoidHom`, `FiniteAddGrp.ofHom_apply`, `AddChar.coe_toAddMonoidHomEquiv`, `mul_apply`, `RCLike.norm_le_re_iff_eq_norm`, `Multiset.toDFinsupp_replicate`, `Ideal.mul_homogeneous_element_mem_of_mem`, `mker_fst`, `Matrix.entryAddMonoidHom_apply`, `AddMonoid.Coprod.mk_surjective`, `coe_snd`, `AddEquiv.toMultiplicativeLeft_symm_apply_symm_apply`, `CategoryTheory.Abelian.Ext.bilinearComp_apply_apply`, `DirectSum.lieAlgebraOf_apply`, `AddMonoid.Coprod.mk_of_neg_add`, `DirectSum.map_of`, `AddEquiv.coprodAssoc_symm_apply_inr_inr`, `coe_toNatLinearMap`, `map_inv₂`, `CategoryTheory.whiskering_preadditiveYoneda`, `QuotientAddGroup.sound`, `MvPowerSeries.le_weightedOrder_subst`, `Pi.constAddMonoidHom_apply`, `Subsemiring.isHomogeneous_iff_subset_iInter`, `Multiset.uIcc_eq`, `MvPowerSeries.trunc'_expand`, `multiplesHom_symm_apply`, `RCLike.ofReal_im`, `prevD_eq_zero`, `Finset.card_nsmul_quotient_add_nsmul_inter_addSubgroup_le`, `Finsupp.count_toMultiset`, `RCLike.abs_im_le_norm`, `DFinsupp.mker_mapRangeAddMonoidHom`, `map_neg`, `compHom_apply_apply`, `AddSubmonoid.fromCommLeftNeg_apply`, `AddCommGrpCat.forget₂AddGroup_preservesLimits`, `inr_mono`, `AddMonoid.Coprod.lift_apply_inr`, `RCLike.nonneg_iff`, `FreeAbelianGroup.toFinsupp_of`, `LocallyConstant.toContinuousMapAddMonoidHom_apply`, `MvPolynomial.supDegree_esymm`, `AddMonoid.Coprod.codisjoint_mrange_inl_mrange_inr`, `ZMod.toAddCircle_natCast`, `SubAddAction.fixingAddSubgroup_vadd_eq_fixingAddSubgroup_map_conj`, `SkewMonoidAlgebra.liftNC_one`, `RCLike.continuous_re`, `NumberField.mixedEmbedding.logMap_eq_logEmbedding`, `FreeAddGroup.map_eq_lift`, `AddCommMonCat.coe_comp`, `AddChar.doubleDualEmb_bijective`, `op_symm_apply_apply`, `AddCon.coe_mk'`, `AddMonoid.Coprod.swap_eq_zero`, `AddCommGrpCat.Colimits.quotToQuotUlift_ι`, `SkewMonoidAlgebra.update_eq_erase_add_single`, `AddEquiv.toMultiplicativeRight_symm_apply_apply`, `nsmul_apply`, `RCLike.zero_re`, `InnerProductSpace.Core.inner_re_symm`, `coe_mgraph`, `ModuleCat.smulNatTrans_apply_app`, `AddSubmonoid.fg_iff_exists_fin_addMonoidHom`, `RCLike.smul_im`, `AddCommGroup.DirectLimit.of_f`, `DirectSum.coe_of_mul_apply_of_not_le`, `Localization.mkAddMonoidHom_apply`, `MvPowerSeries.trunc_map`, `Finsupp.weight_single_index`, `AddMonoid.Coprod.snd_apply_inl`, `QuadraticMap.coeFnAddMonoidHom_apply`, `WittVector.iterate_verschiebung_coeff_eq_zero`, `CategoryTheory.Functor.mapExtAddHom_apply`, `SkewMonoidAlgebra.mapDomain_id`, `ENatMap_apply`, `RCLike.re_eq_add_conj`, `AddGrpCat.hasLimit_iff_small_sections`, `DivisibleHull.coeAddMonoidHom_apply`, `AddCommMonCat.coyoneda_map_app`, `LieDerivation.coeFnAddMonoidHom_apply`, `commute_lmul_lmul_sq`, `AddCommGrpCat.zero_apply`, `AddSubgroup.comap_toAddSubmonoid`, `AddGrpCat.mono_iff_injective`, `Finsupp.degree_preimage_add`, `DirectSum.mk_apply_of_mem`, `AddCommGrpCat.id_apply`, `ContinuousLinearMap.reApplyInnerSelf_apply`, `AddAction.compHom_vadd_def`, `AddSubmonoid.iSup_eq_mrange_dfinsuppSumAddHom`, `Finsupp.liftAddHom_apply`, `WithZero.mapAddHom_coe`, `PresheafOfModules.unitHomEquiv_apply_coe`, `MvPowerSeries.coeff_trunc'`, `inner_self_eq_norm_mul_norm`, `Ideal.Filtration.submodule_span_single`, `AddSubgroupClass.inclusion_mk`, `AddSubmonoid.inclusion_inj`, `Matrix.IsHermitian.sort_roots_charpoly_eq_eigenvalues₀`, `AddCommGrpCat.int_hom_ext_iff`, `AddMonoidAlgebra.mapDomainAlgHom_apply`, `Behrend.map_succ'`, `BoundedContinuousFunction.coeFnAddMonoidHom_apply`, `coe_toMultiplicative'`, `AddSubgroup.apply_coe_mem_map`, `AddEquiv.coprodAssoc_apply_inl_inr`, `AddEquiv.coprodCongr_apply`, `WittVector.verschiebung_coeff_zero`, `AddCommGrpCat.Colimits.Quot.desc_toCocone_desc_app`, `DFinsupp.toMultiset_toDFinsupp`, `Finsupp.DegLex.lt_iff`, `AddCon.kerLift_range_eq`, `mker_zero`, `AddMonoid.Coprod.swap_map`, `MvPowerSeries.coeff_weightedHomogeneousComponent`, `MvPowerSeries.order_le`, `AddSubmonoid.FG.map`, `LocallyConstant.evalAddMonoidHom_apply`, `AddOreLocalization.numeratorHom_apply`, `LinearMap.re_inner_adjoint_mul_self_nonneg`, `AddCommGrpCat.μ_forget_apply`, `MonoidAlgebra.liftNC_single`, `AddLocalization.addEquivOfQuotient_apply`, `MeasureTheory.Integrable.re_im_iff`, `AddMonoid.fg_iff_exists_freeAddMonoid_hom_surjective`, `Finsupp.degLex_def`, `uliftMultiplesHom_apply_apply`, `two_nsmul_lie_lmul_lmul_add_eq_lie_lmul_lmul_add`, `Filter.Germ.coe_coeAddHom`, `ContinuousMap.compAddMonoidHom'_apply`, `CategoryTheory.Functor.coe_mapAddHom`, `AddChar.toAddMonoidHomEquiv_symm_apply`, `AddGrpCat.forget₂_map_ofHom`, `Filter.Tendsto.compMeasurePreservingLp`, `AddCommGrpCat.epi_iff_surjective`, `WeierstrassCurve.Affine.Point.map_baseChange`, `AddAction.IsBlock.of_addSubgroup_of_conjugate`, `PresheafOfModules.freeAdjunctionUnit_app`, `AddSubmonoid.mrange_inr'`, `Matrix.entryAddMonoidHom_toAddHom`, `MvPowerSeries.coeff_trunc`, `Measurable.im`, `addSubgroupMap_surjective`, `mulLeft_continuous`, `RCLike.norm_im_le_norm`, `ModularForm.coeHom_apply`, `compHom'_apply_apply`, `MonoidHom.coe_toAdditiveLeft`, `WittVector.iterate_verschiebung_iterate_frobenius`, `ConvexOn.convex_re_epigraph`, `coe_comp_rangeRestrict`, `Int.tendsto_zmultiplesHom_cofinite`, `ValueDistribution.characteristic_sub_characteristic_inv`, `RCLike.sqrt_of_nonneg`, `FreeAbelianGroup.notMem_support_iff`, `CategoryTheory.Discrete.addMonoidalFunctor_obj`, `FreeAddGroup.freeAddGroupCongr_apply`, `AddMonoidAlgebra.GradesBy.decompose_single`, `toAddEquiv_apply`, `AddCon.kerLift_mk`, `LinearMap.isPositive_iff_complex`, `inner_self_eq_norm_sq`, `RCLike.lipschitzWith_re`, `AddCommGrpCat.forget₂_addGrp_map_ofHom`, `Finsupp.toMultiset_add`, `Measurable.re`, `WittVector.verschiebung_mul_frobenius`, `noncommPiCoprod_mrange`, `convexHullAddMonoidHom_apply`, `AddGroupExtension.Equiv.rightHom_comm`, `AddMonoidAlgebra.singleAddHom_apply`, `Matrix.diagonalAddMonoidHom_apply`, `FreeAddMonoid.lift_comp_of`, `Multiset.toFinsupp_symm_apply`, `AddMonoidAlgebra.coe_liftNCAlgHom`, `FGModuleCat.instFiniteCarrierColimitModuleCatCompForget₂LinearMapIdObjIsFG`, `FreeAbelianGroup.equivFinsupp_apply`, `QuotientAddGroup.lift_mk`, `MeasureTheory.Lp.norm_const`, `NumberField.Units.dirichletUnitTheorem.logEmbedding_eq_zero_iff`, `AddCommGrpCat.forget₂AddCommMon_preservesLimitsOfShape`, `MvPowerSeries.exists_coeff_ne_zero_and_order`, `DirectSum.coeAddMonoidHom_of`, `MeasureTheory.AEStronglyMeasurable.re`, `AddMonCat.id_apply`, `AddCon.ker_apply`, `CategoryTheory.unopHom_apply`, `RCLike.I_im`, `DFinsupp.toMultiset_inf`, `AddCommMonCat.instSmallElemForallObjCompMonCatForget₂AddMonoidHomCarrierCarrierForgetSections`, `CategoryTheory.Adjunction.compPreadditiveYonedaIso_hom_app_app_apply`, `PolynomialModule.equivPolynomial_single`, `MvPolynomial.homogeneousComponent_apply`, `op_apply_apply`, `DirectSum.GAlgebra.map_mul`, `iSupIndep.linearEquiv_apply`, `Multiset.replicateAddMonoidHom_apply`, `AddSubmonoid.mrange_inl_sup_mrange_inr`, `ext_iff`, `AddUnits.coeHom_injective`, `PresheafOfModules.toPresheaf_map_app_apply`, `ContinuousLinearMap.IsPositive.re_inner_nonneg_left`, `inl_mono`, `ZMod.LFunction_def_even`, `AddSubmonoid.subtype_injective`, `ZMod.lift_comp_castAddHom`, `AddCommGrpCat.mono_iff_injective`, `coe_toRatLinearMap`, `quotient_norm_mk_eq`, `PresheafOfModules.Derivation'.d_app`, `ModuleCat.HasColimit.instPreservesColimitAddCommGrpCatForget₂LinearMapIdCarrierAddMonoidHomCarrier`, `AddMonCat.ofHom_apply`, `coe_toMultiplicative`, `PolynomialModule.smul_single_apply`, `AddCommGrpCat.Colimits.Quot.ι_desc`, `DFinsupp.mrange_mapRangeAddMonoidHom`, `DirectSum.of_injective`, `ModuleCat.FilteredColimits.forget₂AddCommGroup_preservesFilteredColimits`, `AddEquiv.coprodAssoc_symm_apply_inl`, `CharacterModule.homEquiv_symm_apply_apply_apply`, `Matrix.diagAddMonoidHom_apply`, `ContinuousLinearMap.apply_norm_eq_sqrt_inner_adjoint_left`, `AddEquiv.punitCoprod_symm_apply`, `dNext_comp_left`, `groupCohomology.cocycles₁IsoOfIsTrivial_inv_hom_apply_coe`, `addSubmonoidMap_apply_coe`, `Summable.toCompl_tsum`, `AddCommGrpCat.Colimits.Quot.map_ι`, `WittVector.ghostComponent_verschiebung`, `RCLike.ofNat_mul_im`, `AddSubgroup.mem_comap`, `QuotientAddGroup.quotientMapAddSubgroupOfOfLe_mk`, `PiTensorProduct.liftAux_tprod`, `AddMonoid.Coprod.inr_injective`, `Nat.coe_castAddMonoidHom`, `map_mul₂`, `zsmulAddGroupHom_apply`, `AlgEquiv.algHomUnitsEquiv_apply_symm_apply`, `AddCommMonCat.hom_neg_apply`, `LinearEquiv.map_dfinsuppSumAddHom`, `AddCommGrpCat.kernelIsoKer_inv_comp_ι`, `RCLike.isCauSeq_im`, `InnerProductSpace.norm_sq_eq_re_inner`, `AddGrpCat.forget_reflects_isos`, `ModuleCat.forget₂_obj_moduleCat_of`, `ContMDiffSection.coeAddHom_apply`, `AdicCompletion.sum_of`, `WithZero.lift_zero`, `coe_toHomAddUnits`, `AddGrpCat.μ_forget_apply`, `AddCommGrpCat.forget₂AddGroup_preservesLimit`, `MonoidHom.toAdditiveRight_symm_apply_apply`, `Matrix.blockDiagonal'AddMonoidHom_apply`, `toMultiplicativeLeft_symm_apply_apply`, `AddSubmonoid.bsupr_eq_mrange_dfinsuppSumAddHom`, `DirectSum.mulHom_apply`, `DirectSum.coe_mul_of_apply_of_mem_zero`, `DirectSum.of_eq_same`, `RCLike.hasSum_re`, `FreeAbelianGroup.lift_comp_apply`, `map_zmultiples`, `AddCommGrpCat.injective_of_mono`, `CategoryTheory.ShortComplex.ShortExact.injective_f`, `coe_smul`, `restrict_apply`, `MeasureTheory.OuterMeasure.coeFnAddMonoidHom_apply`, `ModuleCat.instPreservesColimitsOfSizeAddCommGrpCatForget₂LinearMapIdCarrierAddMonoidHomCarrierOfHasColimitsOfSizeAddCommGrpMax`, `AddCommGrpCat.instFullAddGrpCatForget₂AddMonoidHomCarrierCarrier`, `AddMonoid.Coprod.fst_apply_inl`, `monoidEndToAdditive_symm_apply_apply`, `AddSubmonoid.mem_closure_image_pos_iff`, `AddEquiv.toMultiplicativeRight_symm_apply_symm_apply`, `dfinsuppSum_apply`, `mapMatrix_apply`, `AddMonoid.fg_range`, `DirectSum.sigmaCurry_apply`, `MultilinearMap.alternatization_apply`, `QuotientAddGroup.norm_lift_apply_le`, `DirectSum.coeRingHom_of`, `Isometry.inl`, `locallyFinsuppWithin.logCounting_divisor`, `AddMonoid.Coprod.map_mk_ofList`, `WeierstrassCurve.Affine.Point.map_map`, `Homotopy.dNext_succ_chainComplex`, `AddMonoid.Coprod.mk_eq_mk`, `AddMonoid.Coprod.toProd_surjective`, `AddChar.coe_doubleDualEquiv`, `Subsemiring.isHomogeneous_iff_forall_subset`, `AddCon.mk'_ker`, `WittVector.map_verschiebung`, `restrict_mrange`, `WittVector.verschiebung_frobenius`, `PresheafOfModules.Elements.fromFreeYoneda_app_apply`, `preimage_vadd_setₛₗ`, `AddMonoidAlgebra.liftNC_smul`, `AddCommGroup.DirectLimit.congr_apply_of`, `SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descMonoidHom_apply_eq`, `mker_prod_map`, `ModuleCat.HasColimit.coconePointSMul_apply`, `Complex.coe_imAddGroupHom`, `RCLike.mul_im_I_ax`, `Algebra.GrothendieckAddGroup.of_injective`, `CategoryTheory.ShortComplex.ab_exact_iff`, `AddGroupSeminorm.comp_apply`, `AddMonoidAlgebra.liftNC_single`, `AddUnits.embedProduct_apply`, `TensorProduct.SMul.aux_of`, `PrimeMultiset.coe_coePNatMonoidHom`, `MeasureTheory.VectorMeasure.mapGm_apply`, `AddCommGrpCat.coe_id`, `PreInnerProductSpace.Core.re_inner_nonneg`, `AddCommGrpCat.forget₂AddGroup_preservesLimitsOfSize`, `ModuleCat.instReflectsIsomorphismsAddCommGrpCatForget₂LinearMapIdCarrierAddMonoidHomCarrier`, `AddEquiv.toMultiplicative_symm_apply_apply`, `CochainComplex.HomComplex.CohomologyClass.toHom_bijective`, `AddCommMonCat.coe_forget₂_obj`, `WithZero.map_injective`, `ZMod.lift_coe`, `LinearMap.mul''_apply`, `Function.Exact.iff_of_ladder_addEquiv`, `FreeAddMonoid.lift_eval_of`, `RCLike.re_mul_ofReal`, `AddSubmonoid.mrange_inr`, `AlternatingMap.coe_alternatization`, `DirectSum.coe_mul_of_apply_aux`, `PiTensorProduct.mapLMultilinear_apply_apply`, `AddSubmonoid.mk_neg_add_mk_eq_zero`, `AlgebraicGeometry.Proj.basicOpen_eq_iSup_proj`, `FreeAddGroup.sum_mk`, `FreeAddMonoid.mrange_lift`, `BoxIntegral.BoxAdditiveMap.map_apply`, `SkewMonoidAlgebra.liftNC_mul`, `Matrix.blockDiag'AddMonoidHom_apply`, `AddUnits.coe_map_neg`, `AddSubmonoid.surjOn_iff_le_map`, `CategoryTheory.ShortComplex.ShortExact.surjective_g`, `AddEquiv.addSubmonoidMap_symm_apply`, `AddMonCat.forget_preservesLimits`, `AddMonoid.Coprod.fst_apply_inr`, `Finsupp.toFreeAbelianGroup_toFinsupp`, `RCLike.im_tsum`, `AddMonoid.Coprod.swap_swap`, `smulAddHom_apply`, `AddMonoidAlgebra.gradeBy.gradedMonoid`, `AddSubmonoid.one_eq_mrange`, `AddSubmonoid.top_prod`, `CategoryTheory.ShortComplex.zero_apply`, `map_list_sum`, `DirectSum.lie_of_same`, `CategoryTheory.NatTrans.appHom_apply`, `RCLike.norm_re_le_norm`, `AddAction.toPermHom_apply_symm_apply`, `AddGroupExtension.inl_injective`, `apply_nat`, `AddGroupExtension.Section.rightHom_comp_section`, `re_inner_self_pos`, `AddAut.conj_apply`, `NumberField.mixedEmbedding.logMap_unit_smul`, `zsmul_apply`, `MvPowerSeries.weightedOrder_eq_nat`, `QuotientAddGroup.quotientBot_apply`, `MvPowerSeries.totalDegree_trunc'`, `CochainComplex.HomComplex.CohomologyClass.mkAddMonoidHom_apply`, `TensorAlgebra.GradedAlgebra.ι_apply`, `WittVector.ghostComponent_zero_verschiebung`, `Function.locallyFinsuppWithin.logCounting_divisor_eq_circleAverage_sub_const`, `RCLike.geometric_hahn_banach_open`, `SkewMonoidAlgebra.sum_mapDomain_index`, `AddUnits.coe_map`, `uliftZMultiplesHom_apply_apply`, `DFinsupp.toMultiset_injective`, `FreeAddMonoid.lift_restrict`, `DMatrix.map_add`, `Finset.le_card_quotient_add_sq_inter_addSubgroup`, `LocallyFiniteOrder.orderAddMonoidHom_apply`, `SkewMonoidAlgebra.toFinsupp_mapDomain`, `MvPolynomial.monomial_mem_homogeneousSubmodule_pow_degree`, `ContinuousMultilinearMap.alternatization_apply_toContinuousMultilinearMap`, `DFinsupp.mapRange.addMonoidHom_apply`, `Multiset.toFinsupp_toMultiset`, `SkewMonoidAlgebra.update_zero_eq_erase`, `AddSubmonoid.mrange_fst`, `AddLocalization.mkHom_apply`, `InnerProductSpace.Core.inner_mul_symm_re_eq_norm`, `AddSubmonoid.mrange_snd`, `FreeAbelianGroup.mul_def`, `ContMDiffMap.coeFnAddMonoidHom_apply`, `Matrix.traceAddMonoidHom_apply`, `AddLocalization.mkHom_surjective`, `DirectSum.toAddMonoidAlgebra_of`, `CategoryTheory.Discrete.addMonoidalFunctor_δ`, `Finset.coeAddMonoidHom_apply`, `smul_apply`, `CategoryTheory.GrothendieckTopology.MayerVietorisSquare.fromBiprod_biprodIsoProd_inv_apply`, `AddMonoid.Coprod.swap_inl`, `RCLike.re_le_re`, `AddMonoidAlgebra.grade.decompose_single`, `RCLike.reLm_coe`, `sub_mem_ker_iff`, `toMultiplicative_apply_apply`, `RCLike.natCast_im`, `AddGroupExtension.Section.rightHom_section`, `map_finsum_Prop`, `AddSubmonoid.LocalizationMap.toAddMonoidHom_apply`, `AddGroupExtension.Section.exists_add_eq_add_add_inl`, `WeierstrassCurve.Affine.Point.toClass_some`, `MeasureTheory.MemLp.re`, `ModuleCat.mkOfSMul_smul`, `fiberEquiv_apply`, `AddGrpCat.instSmallElemForallObjCompMonCatForget₂AddMonoidHomCarrierCarrierForgetSections`, `RCLike.ratCast_re`, `MeasureTheory.AEStronglyMeasurable.im`, `RCLike.ofReal_eq_re_of_isSelfAdjoint`, `AddCommGrpCat.instReflectsIsomorphismsAddGrpCatForget₂AddMonoidHomCarrierCarrier`, `Matrix.IsHermitian.coe_re_apply_self`, `PresheafOfModules.sectionsMk_coe`, `MeasureTheory.Integrable.re`, `Finsupp.degree_preimage_nsmul`, `AddGroup.fg_iff_exists_freeAddGroup_hom_surjective`, `integral_coe_re_add_coe_im`, `ZMod.toAddCircle_intCast`, `RCLike.lt_iff_re_im`, `MvPowerSeries.exists_coeff_ne_zero_and_weightedOrder`, `ConvexOn.univ_sSup_affine_eq`, `AddMonoid.Coprod.snd_swap`, `HahnSeries.map_sub`, `AddCommGroup.DirectLimit.congr_symm_apply_of`, `AddCircle.continuous_mk'`, `fiberEquivKer_symm_apply`, `DistribSMul.toAddMonoidHom_apply`, `FreeAddGroup.map.mk`, `norm_add_sq`, `RCLike.norm_sq_re_conj_add`, `PresheafOfModules.Derivation.d_app`, `AddGrpCat.neg_hom_apply`, `AddChar.compAddMonoidHom_apply`, `ZMod.toAddCircle_apply`, `Finsupp.sum_hom_add_index`, `AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓUnitOpensCarrierCarrierCommRingCatRingCatSheaf`, `norm_add_mul_self`, `MultilinearMap.domCoprod_alternization`, `RCLike.conj_im`, `AddSubmonoid.mrange_inl`, `WeierstrassCurve.Affine.Point.toClass_injective`, `AddUnits.range_embedProduct`, `AddCommMonCat.coyonedaForget_inv_app_app`, `AddGrpCat.hom_neg_apply`, `MvPowerSeries.weightedOrder_le`, `WittVector.iterate_verschiebung_mul_coeff`, `InnerProductSpace.Core.inner_self_ofReal_re`, `RCLike.hasSum_iff`, `RCLike.measurable_im`, `MeasureTheory.AEEqFun.toGermAddMonoidHom_apply`, `AddMonoid.Coprod.lift_apply_mk`, `LinearMap.IsSymmetric.hasEigenvalue_iInf_of_finiteDimensional`, `AddSubgroup.inclusion_inj`, `SubAddAction.fixingAddSubgroupEquivFixingAddSubgroup_coe_apply`, `fst_mono`, `ModuleCat.reflectsColimitsOfShape`, `RCLike.re_eq_ofReal_of_isSelfAdjoint`, `coe_mul`, `AddMonoid.Coprod.mrange_mk`, `FreeAddGroup.lift_symm_apply`, `LinearMap.snd_prodOfFinsuppNat`, `Finsupp.toFinset_toMultiset`, `DomAddAct.vadd_Lp_const`, `Finsupp.degree_single`, `ModuleCat.forget₂AddCommGroup_reflectsLimitOfSize`, `AddGrpCat.forget_preservesLimitsOfSize`, `Finsupp.applyAddHom_apply`, `addMonoidHomOfForallMemZMultiples_apply_gen`, `DirectSum.of_zero_pow`, `ext_int_iff`, `AddMonoid.Coprod.mrange_inl_sup_mrange_inr`, `Matrix.PosSemidef.re_dotProduct_nonneg`, `QuotientAddGroup.map_mk'`, `DirectSum.coeFnAddMonoidHom_apply`, `PresheafOfModules.Derivation'.app_apply`, `Filter.coe_pureAddMonoidHom`, `AddMonoid.Coprod.mclosure_range_inl_union_inr`, `Module.End.baseChangeHom_apply_apply`, `MvPolynomial.weightedDecomposition.decompose'_eq`, `SchwartzMap.coe_coeHom`, `AddCommGrpCat.biprodIsoProd_inv_comp_desc_apply`, `AddSubmonoid.LocalizationMap.lift_id`, `AddEquiv.toMultiplicative_symm_apply_symm_apply`, `ProjectiveSpectrum.basicOpen_eq_union_of_projection`, `Finsupp.le_weight_of_ne_zero`, `MonoidHom.coe_toAdditive`, `CochainComplex.HomComplex.CohomologyClass.descAddMonoidHom_cohomologyClass`, `NumberField.Units.dirichletUnitTheorem.logEmbedding_component`, `DirectSum.of_natCast`, `Algebra.Extension.tensorCotangentInvFun_smul_mk`, `AddSubgroupClass.coe_inclusion`, `mem_mgraph`, `AddCommGrpCat.ofHom_apply`, `DirectSum.coe_mul_of_apply_add`, `RCLike.re_le_norm`, `Multiset.toFinsupp_eq_iff`, `ContinuousLinearMap.isPositive_iff_complex`, `AddMonoid.Coprod.mker_swap`, `Polynomial.eval₂AddMonoidHom_apply`, `AddCommGrpCat.biprodIsoProd_inv_comp_fst_apply`, `DirectSum.map_apply`, `Pi.evalAddMonoidHom_apply`, `Finsupp.mapDomain.addMonoidHom_apply`, `RCLike.ofNat_im`, `DistribMulActionHom.coe_fn_coe`, `CategoryTheory.Discrete.addMonoidalFunctor_μ`, `NumberField.Units.logEmbeddingQuot_injective`, `DirectSum.coe_of_mul_apply_of_le`, `DirectSum.support_of_subset`, `QuotientAddGroup.con_ker_eq_addConKer`, `DirectSum.coe_mul_of_apply_of_le`, `map_div₂`, `DirectSum.coe_of_mul_apply_of_mem_zero`, `ZMod.lift_injective`, `MvPolynomial.homogeneousSubmodule_eq_finsupp_supported`, `zmultiplesHom_apply`, `NonUnitalNonAssocSemiring.mem_center_iff`, `FreeAbelianGroup.liftMonoid_coe`, `CategoryTheory.Functor.mapAddHom_apply`, `coe_mulRight`, `RCLike.norm_sq_eq_def`, `AddCon.kerLift_injective`, `vaddZeroHom_apply`, `AddMonCat.hom_neg_apply`, `Matrix.IsHermitian.coe_re_diag`, `coe_comp`, `ContinuousMultilinearMap.alternatization_apply_toAlternatingMap`, `Surreal.dyadicMap_apply`, `MeasureTheory.Lp.constL_apply`, `DirectSum.of_apply`, `AddCommMonCat.instIsRightAdjointForgetAddMonoidHomCarrier`, `RCLike.ratCast_im`, `DirectSum.of_intCast`, `Function.locallyFinsuppWithin.restrictMonoidHom_apply`, `Finsupp.toMultiset_sum`, `AddSubmonoidClass.subtype_injective`, `dNext_eq_dFrom_fromNext`, `DirectSum.GAlgebra.smul_def`, `map_addCyclic`, `MvPowerSeries.coeff_mul_eq_coeff_trunc'_mul_trunc'`, `RCLike.I_mul_re`, `Multiset.toDFinsupp_injective`, `AddSubgroup.subtype_injective`, `AddGrpCat.comp_apply`, `AddSubmonoid.inclusion_injective`, `RCLike.conj_eq_iff_im`, `DirectSum.mul_eq_sum_support_ghas_mul`, `AddCon.map_apply`, `FreeAddMonoid.map_symm_apply_map_eq`, `Finsupp.range_single_one`, `SemiRingCat.forget₂AddCommMon_preservesLimits`, `Function.locallyFinsuppWithin.logCounting_mono`, `AddChar.toAddMonoidHom_apply`, `Function.ExtendByZero.hom_apply`, `map_sub`, `SkewMonoidAlgebra.liftNC_single`, `AddSubgroup.coe_map`, `WittVector.verschiebung_shift`, `SubAddAction.of_fixingAddSubgroupEmpty_mapScalars_surjective`, `DFinsupp.sumAddHom_apply`, `RCLike.conj_re_ax`, `Homotopy.prevD_zero_cochainComplex`, `MvPowerSeries.WithPiTopology.tendsto_trunc'_atTop`, `AddAction.stabilizerEquivStabilizer_apply`, `surjective_of_card_ker_le_div`, `coprod_apply`, `ZMod.toAddCircle_injective`, `SubAddAction.ofFixingAddSubgroupEmpty_equivariantMap_bijective`, `PrimeMultiset.coe_coeNatMonoidHom`, `CochainComplex.HomComplex.Cochain.shiftAddHom_apply`, `CategoryTheory.ShortComplex.ab_zero_apply`, `Function.Exact.iff_addMonoidHom_rangeRestrict`, `AddGrpCat.FilteredColimits.forget₂AddMon_preservesFilteredColimits`, `CategoryTheory.Discrete.addMonoidalFunctor_ε`, `Algebra.GrothendieckAddGroup.lift_apply`, `MultilinearMap.alternatization_coe`, `AddGroupSeminorm.coe_comp`, `CategoryTheory.Abelian.Ext.postcomp_mk₀_injective_of_mono`, `mrangeRestrict_mker`, `inner_mul_symm_re_eq_norm`, `AddCommGrpCat.HasLimit.lift_hom_apply`, `Submodule.re_inner_starProjection_nonneg`, `MeasureTheory.MemLp.im`, `AddMonoidAlgebra.liftNC_one`, `Multiset.toDFinsupp_le_toDFinsupp`, `DirectSum.liftRingHom_symm_apply_coe`, `Multiset.toDFinsupp_toMultiset`, `ModuleCat.HasColimit.colimitCocone_pt_carrier`, `ModuleCat.forget₂_addCommGrp_additive`, `MonoidAlgebra.liftNC_smul`, `closure_preimage_le`, `RCLike.realRingEquiv_apply`, `AddMonoid.Coprod.fst_surjective`, `Homotopy.comm`, `WeierstrassCurve.Affine.Point.map_some`, `Multiset.toDFinsupp_singleton`, `MulEquiv.toAdditive_apply_symm_apply`, `RingCat.forget₂AddCommGroup_preservesLimitsOfSize`, `RCLike.mul_im`, `SkewMonoidAlgebra.erase_single`, `AddMonoid.Coprod.map_apply_inl`, `AddCommMonCat.forget₂_map_ofHom`, `DirectSum.coe_mul_of_apply`, `MvPowerSeries.order_monomial`, `addCommute_inl_inr`, `RCLike.im_sq_le_normSq`, `DirectSum.mul_eq_dfinsuppSum`, `coe_single`, `MvPowerSeries.ne_zero_iff_exists_coeff_ne_zero_and_weight`, `QuotientAddGroup.map_mk`, `Finsupp.liftAddHom_apply_single`, `dNext_eq`, `addMonoidHomOfMemClosureRangeCoe_apply`, `AddSubmonoid.map_inr`, `mk'_apply`, `AddGrpCat.epi_iff_surjective`, `summable_iff_cauchySeq_finset_and_tsum_mem`, `AddCommGrpCat.forget₂_commMonCat_map_ofHom`, `LinearMap.tensorProduct_apply`, `AddCommGrpCat.forget₂AddCommMon_preservesLimitsOfSize`, `AddMonoid.Coprod.mrange_swap`, `NonUnitalRingHom.coe_toAddMonoidHom`, `Behrend.map_zero`, `RCLike.re_ofReal_pow`, `HahnSeries.map_add`, `re_inner_self_nonpos`, `RCLike.norm_sq_re_add_conj`, `LinearMap.IsSymmetric.im_inner_apply_self`, `Finsupp.embDomain.addMonoidHom_apply`, `FreeAddMonoid.comp_lift`, `Isometry.inr`, `NormedAddGroupHom.coeAddHom_apply`, `DirectSum.of_eq_of_gradedMonoid_eq`, `AddCommMonCat.forget_isCorepresentable`, `RCLike.im_to_complex`, `Multiset.coe_countAddMonoidHom`, `InnerProductSpace.Core.cauchy_schwarz_aux'`, `AddCommGrpCat.coyoneda_map_app`, `AddChar.doubleDualEquiv_symm_doubleDualEmb_apply`, `coeFn_apply`, `WittVector.verschiebung_frobenius_comm`, `CategoryTheory.Discrete.addMonoidalFunctor_η`, `TensorProduct.liftAux.smul`, `Function.locallyFinsuppWithin.logCounting_le`, `Finsupp.weight_sub_single_add`, `AddAction.toPermHom_apply_apply`, `RCLike.re_eq_complex_re`, `AddSubmonoid.mrange_inl'`, `DirectSum.ofList_dProd`, `Module.End.lTensorAlgHom_apply_apply`, `DirectSum.GAlgebra.commutes`, `RCLike.norm_to_complex`, `AddCommGrpCat.forget_map`, `FreeAbelianGroup.lift_comp`, `coe_ofMapMidpoint`, `Nonneg.coeAddMonoidHom_apply`, `RCLike.re_sq_le_normSq`, `AddSubmonoid.mem_iSup_iff_exists_dfinsupp`, `QuotientAddGroup.map_map`, `MvPolynomial.degrees_monomial_eq`, `QuotientAddGroup.lift_quot_mk`, `WittVector.iterate_verschiebung_mul_left`, `CategoryTheory.pullbackShiftFunctorZero'_inv_app`, `TwoSidedIdeal.coeAddMonoidHom_apply`, `map_multiset_sum`, `AddMonoid.Coprod.mrange_eq`, `comap_mker`, `AddSubmonoid.LocalizationMap.map_id`, `FreeAbelianGroup.lift_add_apply`, `RCLike.mul_re`, `ModuleCat.forget₂_reflectsLimits`, `AddCommGrpCat.kernelIsoKer_hom_comp_subtype`, `UniformSpace.Completion.continuous_toCompl`, `FreeAddMonoid.map_of`, `ContDiffMapSupportedIn.coe_coeHom`, `ContinuousMap.addEquivBoundedOfCompact_symm_apply`, `Matrix.blockDiagAddMonoidHom_apply`, `MeasureTheory.Lp.toLp_compMeasurePreserving`, `ModuleCat.forget₂PreservesColimitsOfShape`, `WittVector.iterate_verschiebung_coeff`, `AddAction.coe_toPermHom`, `FreeAddGroup.lift_eq_sum_map`, `coe_dfinsuppSum`, `ker_toAddSubmonoid`, `Seminorm.coeFnAddMonoidHom_apply`, `RCLike.re_le_neg_norm_iff_eq_neg_norm`, `DirectSum.Decomposition.left_inv`, `RCLike.to_complex_nonneg_iff`, `DirectSum.GAlgebra.map_one`, `Module.toAddMonoidEnd_apply_apply`, `AddMonoid.Coprod.closure_range_inl_union_inr`, `QuotientAddGroup.kerLift_mk'`, `AddCommMonCat.forget_preservesLimitsOfSize`, `ModuleCat.forget₂AddCommGroup_reflectsLimit`, `AddGrpCat.coe_comp`, `MvPolynomial.coeffAddMonoidHom_apply`, `AddGroupExtension.Equiv.inl_comm`, `Finsupp.liftAddHom_symm_apply_apply`, `NumberField.Units.logEmbeddingEquiv_apply`, `AddEquiv.toMultiplicativeLeft_symm_apply_apply`, `RCLike.im_ofReal_mul`, `FreeAbelianGroup.lift_ext_iff`, `add_apply`, `inner_self_re_eq_norm`, `prevD_comp_right`, `Circle.expHom_apply`, `LaurentPolynomial.leftInverse_trunc_toLaurent`, `DirectSum.Decomposition.right_inv`, `AddEquiv.coprodAssoc_apply_inl_inl`, `FreeAddMonoid.map_apply_map_symm_eq`, `QuotientAddGroup.quotientQuotientEquivQuotientAux_mk`, `PresheafOfModules.freeAdjunction_unit_app`, `RCLike.I_re`, `CochainComplex.HomComplex.CohomologyClass.toHom_mk_eq_zero_iff`, `DirectSum.decompose_coe`, `Multiset.toDFinsupp_inter`, `AddGroupExtension.Section.rightInverse_rightHom`, `AddSubmonoid.LocalizationMap.lift_localizationMap_mk'`, `RCLike.im_eq_complex_im`, `AddSubgroup.surjOn_iff_le_map`, `toMultiplicativeLeft_apply_apply`, `AddSubmonoid.comap_id`, `DirectSum.sigmaUncurry_apply`, `AddMonoidAlgebra.decomposeAux_single`, `SubAddAction.ofFixingAddSubgroup_equivariantMap_injective`, `AddCon.comapQuotientEquivOfSurj_symm_mk'`, `RCLike.ofReal_im_ax`, `Behrend.map_eq_iff`, `zmultiplesHom_symm_apply`, `AddGroupExtension.Equiv.rightHom_map`, `AddChar.doubleDualEmb_inj`, `Finsupp.DegLex.le_iff`, `QuotientAddGroup.mk'_surjective`, `CategoryTheory.Functor.mapExtAddHom_coe`, `InnerProductSpace.Core.inner_self_im`, `addAddMonoidHom_apply`, `mulOp_symm_apply_apply`, `AddCommMonCat.id_apply`, `coe_mulLeft`, `AddMonoid.Coprod.con_ker_mk`, `MeasureTheory.Lp.compMeasurePreserving_continuous`, `DirectSum.list_prod_ofFn_of_eq_dProd`, `Finsupp.degree_eq_zero_iff`, `coe_toIntLinearMap`, `Finsupp.toMultiset_single`, `transfer_def`, `ModuleCat.forget₂AddCommGroup_preservesLimit`, `AddCommMonCat.coyoneda_obj_map`, `Behrend.threeAPFree_image_sphere`, `WittVector.verschiebung_coeff_add_one`, `WeierstrassCurve.Affine.Point.toClass_eq_zero`, `LinearMap.evalAddMonoidHom_apply`, `AEMeasurable.re`, `mrange_id`, `QuotientAddGroup.kerLift_injective`, `FreeAddMonoid.countP_apply`, `AddChar.doubleDualEmb_eq_zero`, `MonoidHom.coe_toAdditiveRight`, `exact_of_comp_eq_zero_of_ker_le_range`, `FreeAbelianGroup.lift_neg_apply`, `AddSubgroup.mem_map`, `MonoidAlgebra.lift_def`, `ZMod.toAddCircle_eq_zero`, `Multiset.toDFinsupp_lt_toDFinsupp`, `AddSubmonoid.comap_comap`, `eq_iff`, `eq_iff_eq_on_generator`, `Homotopy.dNext_cochainComplex`, `CategoryTheory.Pretriangulated.preadditiveYoneda_shiftMap_apply`, `Multiset.toDFinsupp_union`, `AddEquiv.coprodCongr_symm_apply`, `Finsupp.sum_smul_index_addMonoidHom`, `exact_iff`, `AddSubmonoid.subtype_apply`, `AddSubmonoidClass.subtype_apply`, `dfinsuppSumAddHom_apply`, `Finsupp.comapDomain.addMonoidHom_apply`, `FreeAbelianGroup.lift_apply_of`, `MvPowerSeries.order_eq_nat`, `DirectSum.coe_of_apply`, `RCLike.re_ofReal_mul`, `norm_add_pow_two`, `zmultiplesAddHom_apply`, `coe_toZModLinearMap`, `DomMulAct.smul_Lp_const`, `AddEquiv.coprodComm_symm_apply`, `FreeAbelianGroup.liftAddEquiv_apply_apply`, `AddMonoidAlgebra.decomposeAux_eq_decompose`, `QuotientAddGroup.exists_norm_add_lt`, `norm_eq_sqrt_re_inner`, `RCLike.iInter_halfSpaces_eq`, `inner_self_nonneg`, `RCLike.continuous_im`, `LaurentPolynomial.trunc_C_mul_T`, `AlgebraicGeometry.instIsQuasicoherentOpensCarrierCarrierCommRingCatSpecTilde`, `MonoidHom.toAdditiveLeft_symm_apply_apply`, `WittVector.aeval_verschiebungPoly`, `LinearMap.IsSymmetric.re_trace_eq_sum_eigenvalues`, `dNext_comp_right`, `RCLike.normSq_add`, `RCLike.div_re`, `WittVector.verschiebung_isPoly`, `setIntegral_re_add_im`, `RCLike.geometric_hahn_banach_compact_closed`, `MvPowerSeries.ne_zero_iff_exists_coeff_ne_zero_and_degree`, `MvPolynomial.coeff_weightedHomogeneousComponent`, `mker_inr`, `FreeAddGroup.lift_of_apply`, `AddChar.doubleDualEmb_doubleDualEquiv_symm_apply`, `SemiRingCat.forget₂AddCommMon_preservesLimitsOfSize`, `DFinsupp.singleAddHom_apply`, `CategoryTheory.whiskering_linearYoneda₂`, `DirectSum.coe_congrAddEquiv`, `Finsupp.degree_mono`, `AddCommMonCat.coyonedaForget_hom_app_app_hom`, `RCLike.geometric_hahn_banach_point_closed`, `DirectSum.liftRingHom_apply`, `summable_iff_summable_compl_and_tsum_mem`, `CliffordAlgebra.GradedAlgebra.lift_ι_eq`, `RingHom.map_dfinsuppSumAddHom`, `TensorPower.toTensorAlgebra_galgebra_toFun`, `WithZero.map_map`, `re_inner_eq_norm_add_mul_self_sub_norm_sub_mul_self_div_four`, `AddMonoidAlgebra.lift_def`, `TensorAlgebra.toDirectSum_ι`, `FreeAddMonoid.hom_map_lift`, `DirectSum.lmap_of`, `PresheafOfModules.sections_ext_iff`, `Finsupp.eraseAddHom_apply`, `AddCommGrpCat.hom_neg_apply`, `MonoidHom.coe_toAdditive''`, `AddSubgroup.inclusion_injective`, `Finsupp.toMultiset_sup`, `AddSubmonoid.coe_subtype`, `AddMonoid.Coprod.swap_inr`, `DFinsupp.toMultiset_inj`, `noncommPiCoprod_single`, `norm_sub_pow_two`, `Set.coe_singletonAddMonoidHom`, `Matrix.blockDiagonalAddMonoidHom_apply`, `MvPowerSeries.trunc'_one`, `RCLike.conj_eq_re_sub_im`, `InnerProductSpace.Core.re_inner_smul_ofReal_smul_self`, `SubAddAction.fixingAddSubgroup_map_conj_eq`, `AddGrpCat.forget_isCorepresentable`, `AddChar.toAddMonoidHomEquiv_apply`, `MvPowerSeries.trunc_C_mul`, `SkewMonoidAlgebra.single_add_erase`, `AddCommGroup.DirectLimit.map_apply_of`, `UniformSpace.Completion.isDenseInducing_toCompl`, `PresheafOfModules.germ_ringCat_smul`, `FreeAddGroup.map.of`, `comp_apply`, `finset_sum_apply`, `integral_re`, `mapDMatrix_apply`, `ModuleCat.HasColimit.reflectsColimit`, `AddMonoid.Coprod.lift_swap`, `map_add`, `AlgebraicGeometry.Scheme.Modules.Hom.app_smul`, `RCLike.normSq_apply`, `DFinsupp.toMultiset_lt_toMultiset`, `FreeAddMonoid.map_map`, `Finset.imageAddMonoidHom_apply`, `Finsupp.DegLex.monotone_degree`, `coe_ofClosureMEqTopLeft`, `AddCommGrpCat.hom_add_apply`, `addMonoidEndToMultiplicative_apply_apply`, `AddSubgroup.quotientAddSubgroupOfEmbeddingOfLE_apply_mk`, `compLeft_apply`, `ConvexOn.exists_affine_le_of_lt`, `AddSubgroupClass.subtype_injective`, `inr_strictMono`, `AddUnits.continuous_embedProduct`, `ContinuousLinearMap.apply_norm_sq_eq_inner_adjoint_right`, `smulLeft_apply`, `FreeAddMonoid.countP_of`, `AddSubgroupClass.inclusion_right`, `AddEquiv.coprodAssoc_symm_apply_inr_inl`, `SheafOfModules.pushforwardSections_coe`, `WeierstrassCurve.Affine.Point.map_zero`, `DFinsupp.lsum_apply_apply`, `PresheafOfModules.fromFreeYonedaCoproduct_app_mk`, `NumberField.Units.regulator_eq_det'`, `PresheafOfModules.Sheafify.map_smul`, `MvPowerSeries.trunc'_map`, `coe_ofClosureMEqTopRight`, `re_inner_le_norm`, `SlashInvariantForm.coeHom_injective`, `OrderAddMonoidHom.coe_mk`, `DirectSum.of_zero_one`, `PresheafOfModules.germ_smul`, `MonoidAlgebra.liftNC_one`, `ValueDistribution.logCounting_coe`, `DirectSum.decompose_symm_of`, `Pi.addMonoidHom_apply`, `CharacterModule.homEquiv_apply_apply`, `ExteriorAlgebra.GradedAlgebra.ι_apply`, `AddGrpCat.instFullMonCatForget₂AddMonoidHomCarrierCarrier`, `AddSubgroup.normedMk.apply`, `RCLike.mul_re_ax`, `dNext_nat`, `LinearMap.im_inner_adjoint_mul_self_eq_zero`, `ValueDistribution.log_counting_zero_sub_logCounting_top`, `UniformSpace.Completion.toCompl_apply`, `AddSubmonoid.closure_eq_mrange`, `TestFunction.coeFnAddMonoidHom_apply`, `map_one₂`, `PresheafOfModules.toSheafify_app_apply`, `DirectSum.gsmulHom_apply_apply`, `Ideal.toIdeal_homogeneousHull_eq_iSup`, `integral_re_add_im`, `DirectSum.sum_support_of`, `AddCommGrpCat.instIsRightAdjointForgetAddMonoidHomCarrier`, `AddSubgroup.subtype_apply`, `AddAut.neg_conj_apply`, `MonoidHom.coe_toMultiplicative`, `AddCommGrpCat.forget_isCorepresentable`, `FreeAddGroup.sum.of`, `RCLike.norm_le_im_iff_eq_I_mul_norm`, `LocallyFiniteOrder.orderAddMonoidEquiv_apply`, `RCLike.re_to_complex`, `SkewMonoidAlgebra.lift_def`, `CoalgHom.coe_toAddMonoidHom`, `map_mul_iff`, `qExpansion_of_pow`, `coe_toMultiplicative''`, `coe_range`, `AffineAddMonoid.embedding_injective`, `RCLike.hasSum_im`, `ZMod.LFunction_dft`, `mrangeRestrict_surjective`, `Matrix.mulVec.addMonoidHomLeft_apply`, `AddUnits.isClosedEmbedding_embedProduct`, `PresheafOfModules.freeAdjunction_homEquiv`, `DirectSum.toAddMonoid_of`, `DirectSum.of_zero_ofNat`, `dNext_eq_zero`, `RCLike.isCauSeq_re`, `SkewMonoidAlgebra.equivMapDomain_eq_mapDomain`, `ModuleCat.forget₂AddCommGroupIsEquivalence`, `HahnSeries.coeff.addMonoidHom_apply`, `RCLike.I_im'`, `Finsupp.nsmul_single_one_image`, `inr_apply`, `Finsupp.DegLex.lt_def`, `mker_inl`, `AddEquiv.coe_toAddMonoidHom`, `mem_ker`, `Seminorm.coeFnAddMonoidHom_injective`, `MvPowerSeries.order_monomial_of_ne_zero`, `DoubleCentralizer.toProdHom_apply`, `CochainComplex.HomComplex.CohomologyClass.toHom_mk`, `QuotientAddGroup.rangeKerLift_injective`, `addOrderOf_map_dvd`, `WittVector.frobenius_verschiebung`, `Finsupp.toMultiset_sum_single`, `CategoryTheory.preadditiveCoyoneda_obj`, `NumberField.Units.regOfFamily_eq_det'`, `Continuous.compMeasurePreservingLp`, `MeasureTheory.Lp.const_val`, `DFinsupp.filterAddMonoidHom_apply`, `ConvexOn.sSup_affine_eq`, `KaehlerDifferential.quotKerTotalEquiv_apply`, `AddMonoid.Coprod.mk_of_inl`, `MeasureTheory.FiniteMeasure.toMeasureAddMonoidHom_apply`, `DirectSum.map_eq_iff`, `ContinuousOn.compMeasurePreservingLp`, `ContDiffMapSupportedIn.coeHom_injective`, `map_finsum_mem`, `AddCommMonCat.forget_map`, `ext_iff₂`, `LinearMap.IsSymmetric.coe_re_inner_apply_self`, `multiplesHom_apply`, `Finsupp.weight_eq_zero_iff_eq_zero`, `Function.locallyFinsuppWithin.logCounting_even`, `Surreal.dyadicMap_apply_pow`, `Finsupp.toMultiset_map`, `mem_range`, `WithTop.coe_addHom`, `MonoidHom.toAdditiveRight_apply_apply`, `iSupIndep_iff_dfinsuppSumAddHom_injective`, `RCLike.geometric_hahn_banach_closed_compact`, `AddSubmonoid.mem_map_equiv`, `DMatrix.map_sub`, `addMonoidHomOfTendsto_apply`, `RCLike.intCast_im`, `PresheafOfModules.freeObjDesc_app`, `DirectSum.coe_of_mul_apply`, `AddCommMonCat.forget_reflects_isos`, `flipHom_apply`, `apply_int`, `inner_im_symm`, `RCLike.re_extendTo𝕜'ₗ`, `WithZero.mapAddHom_injective`, `Finsupp.toMultiset_inf`, `prod_apply`, `Submodule.map_toAddSubmonoid`, `WeierstrassCurve.Affine.Point.toClass_apply`, `MeasureTheory.memLp_re_im_iff`, `AlgebraicGeometry.instIsIsoModulesSpecOfCarrierFromTildeΓFreeOpensCarrierCarrierCommRingCat`, `mulRight₃_apply`, `AddUnits.isEmbedding_embedProduct`, `RCLike.normSq_sub`, `CategoryTheory.ShortComplex.exact_iff_exact_map_forget₂`, `Finsupp.weight_apply`, `AddCommGrpCat.homAddEquiv_apply`, `RCLike.re_tsum`, `ContinuousLinearMap.IsPositive.re_inner_nonneg_right`, `Finsupp.degree_def`, `Finsupp.sum_mapDomain_index_addMonoidHom`, `AddSubmonoid.LocalizationMap.lift_left_inverse`, `LeftInvariantDerivation.coeFnAddMonoidHom_apply`, `AlgHom.coe_toAddMonoidHom`, `SemimoduleCat.forget₂_obj`, `FreeAbelianGroup.map_comp_apply`, `SkewMonoidAlgebra.mapDomain_apply`, `MvPolynomial.weightedHomogeneousComponent_apply`, `Finsupp.coe_orderIsoMultiset`, `MeasureTheory.Lp.isometry_compMeasurePreserving`, `ContinuousMapZero.coeFnAddMonoidHom_apply`, `QuotientAddGroup.rangeKerLift_surjective`, `QuotientAddGroup.quotientQuotientEquivQuotientAux_mk_mk`, `AddUnits.isInducing_embedProduct`, `LocallyConstant.mapAddMonoidHom_apply`, `PolynomialModule.map_single`, `MonoidHom.coe_toAdditive'`, `RCLike.imCLM_apply`, `map_finsum_of_preimage_zero`, `DirectSum.Gmodule.smul_def`, `GradedRing.proj_apply`, `SemimoduleCat.forget₂_map`, `coe_mrangeRestrict`, `ConvexOn.univ_sSup_of_nat_affine_eq`, `AddGroupExtension.rightHom_surjective`, `AddGrpCat.ofHom_apply`, `CategoryTheory.ShortComplex.ShortExact.ab_exact_iff_function_exact`, `QuotientAddGroup.mk'_apply`, `CategoryTheory.ShortComplex.instPreservesHomologyModuleCatAbForget₂LinearMapIdCarrierAddMonoidHomCarrier`, `RCLike.geometric_hahn_banach_open_open`, `HomologicalComplex.Hom.fAddMonoidHom_apply`, `RCLike.lipschitzWith_im`, `Complex.coe_reAddGroupHom`, `SkewMonoidAlgebra.singleAddHom_apply`, `WithZero.mapAddHom_mapAddHom`, `coe_fst`, `mulLeft_bound`, `commute_rmul_rmul_sq`, `FreeAddMonoid.hom_eq_iff`, `coe_finsuppSum`, `Multiset.toDFinsupp_inj`, `val_addUnitsCenterToCenterAddUnits_apply_coe`, `OrthogonalFamily.projection_directSum_coeAddHom`, `CategoryTheory.GrothendieckTopology.MayerVietorisSquare.toBiprod_apply`, `ContinuousAlternatingMap.toMultilinearAddHom_apply`, `MeasureTheory.Lp.compMeasurePreserving_val`, `addMonoidEndToMultiplicative_symm_apply_apply`, `addUnitsCenterToCenterAddUnits_injective`, `GradedRing.proj_recompose`, `MeasureTheory.Lp.norm_compMeasurePreserving`, `MultilinearMap.alternatization_def`, `integral_im`, `AddSubmonoid.map_map`, `DomMulAct.mk_smul_addMonoidHom_apply`, `SkewMonoidAlgebra.liftNC_smul`, `AddMonoidAlgebra.mapDomainRingHom_apply`, `AddEquiv.punitCoprod_apply`, `RCLike.im_le_norm`, `AddSubgroup.coe_inclusion`, `CategoryTheory.Pretriangulated.preadditiveCoyoneda_homologySequenceδ_apply`, `Matrix.IsHermitian.im_star_dotProduct_mulVec_self`, `Finsupp.toMultiset_eq_iff`, `mrange_comp`, `Fin.accumulate_injective`, `SkewMonoidAlgebra.coeff_erase_apply`, `mker_snd`, `DirectSum.map_surjective`, `CochainComplex.HomComplex_d_hom_apply`, `AlgebraicGeometry.Scheme.Modules.map_smul`, `Fin.accumulate_last`, `CochainComplex.HomComplex.CohomologyClass.homAddEquiv_apply`, `MvPowerSeries.trunc'_C`, `Multiset.coe_mapAddMonoidHom`, `RCLike.inv_re`, `AddSubgroupClass.coe_subtype`, `PiTensorProduct.mapLMultilinear_toFun_apply`, `DFinsupp.subtypeDomainAddMonoidHom_apply`, `AddCommGrpCat.hasLimit_iff_small_sections`, `Matrix.PosDef.re_dotProduct_pos`, `MvPolynomial.decomposition.decompose'_eq`, `Behrend.map_monotone`, `hasSum_iff_hasSum_compl`, `MultilinearMap.coeAddMonoidHom_apply`, `SemiRingCat.forget₂_addCommMonCat_map`, `isLinearSet_iff_exists_fin_addMonoidHom`, `MvPowerSeries.trunc'_C_mul`, `IsAddIndecomposable.baseOf_subset_pos`, `AddChar.coe_toAddMonoidHomEquiv_symm`, `AddCommGrpCat.forget_preservesLimitsOfSize`, `Function.locallyFinsuppWithin.toClosedBall_eval_within`, `coe_negAddMonoidHom`, `SkewMonoidAlgebra.support_erase`, `AddGroupExtension.Splitting.rightInverse_rightHom`, `CategoryTheory.ShortComplex.abCyclesIso_inv_apply_iCycles`, `MeasureTheory.Integrable.im`, `TensorProduct.liftAux.smulₛₗ`, `Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_hom_f_hom_apply`, `LinearMap.IsPositive.re_inner_nonneg_left`, `MeasureTheory.MemLp.toLp_const`, `AddGroupExtension.Section.exists_eq_add_inl`, `AddSubmonoid.LocalizationMap.addEquivOfLocalizations_apply`, `RCLike.zero_im`, `SheafOfModules.unitHomEquiv_apply_coe`, `PolynomialModule.eval_single`, `RCLike.im_eq_zero`, `AddGrpCat.coe_id`, `CategoryTheory.Iso.addCommGroupIsoToAddEquiv_apply`, `RCLike.geometric_hahn_banach_open_point`, `WeierstrassCurve.Affine.Point.map_id`, `Finsupp.image_pow_eq_finsuppProd_image`, `CategoryTheory.whiskering_preadditiveCoyoneda`, `ModuleCat.forget₂_map`, `AddSubgroup.map_centralizer_le_centralizer_image`, `AddGrpCat.forget₂Mon_preservesLimits`, `subAddMonoidHom_apply`, `AddSubmonoid.coe_inclusion`, `AddCommGrpCat.coyoneda_obj_map`, `AddCommGrpCat.forget_reflects_isos`, `uliftZMultiplesHom_symm_apply`, `SkewMonoidAlgebra.mapDomain_comp`, `QuotientAddGroup.map_id_apply`, `DirectSum.of_eq_of_ne`
`inverse` 📖	CompOp	1 math math: `inverse_apply`
`mk'` 📖	CompOp	4 math math: `NormedAddGroupHom.mk_toAddMonoidHom`, `CategoryTheory.Abelian.extFunctor_map_app`, `mk'_apply`, `CategoryTheory.ShiftedHom.opEquiv'_add_symm`
`toAddHom` 📖	CompOp	3 math math: `toAddHom_coe`, `toAddHom_injective`, `WithZero.lift_unique`
`toZeroHom` 📖	CompOp	30 math math: `CentroidHom.map_mul_left'`, `toFun_eq_coe`, `toZeroHom_injective`, `Matrix.diagLinearMap_apply`, `Matrix.diagonalLinearMap_apply`, `toZeroHom_coe`, `MeasureTheory.Lp.compMeasurePreservingₗ_apply`, `HahnSeries.single.linearMap_apply`, `linearMapOfMemClosureRangeCoe_apply`, `AddMonoidAlgebra.singleZeroRingHom_apply`, `Finsupp.mapRange.addMonoidHom_toZeroHom`, `ContinuousMap.coeFnLinearMap_apply`, `NonUnitalRingHom.op_symm_apply_apply`, `OrderAddMonoidHom.monotone'`, `ContinuousLinearMap.compLeftContinuous_apply`, `AddSubmonoid.leftNegEquiv_apply`, `CentroidHom.toFun_eq_coe`, `ContMDiffAddMonoidMorphism.contMDiff_toFun`, `NonUnitalRingHom.op_apply_apply`, `MonoidAlgebra.singleOneRingHom_apply`, `SeparationQuotient.normedMk_apply`, `HahnSeries.coeff.linearMap_apply`, `CentroidHom.map_mul_right'`, `map_add'`, `OrderAddMonoidHom.toFun_eq_coe`, `SkewMonoidAlgebra.DistribMulActionHom.single_toFun`, `ContinuousAddMonoidHom.continuous_toFun`, `DFinsupp.sumAddHom_toZeroHom`, `CategoryTheory.Functor.mapLinearMap_apply`, `AddGroupExtension.Splitting.rightInverse_rightHom`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`cancel_left` 📖	mathematical	`DFunLike.coe` `AddMonoidHom` `instFunLike`	`comp`	—	`ext` `comp_apply`
`cancel_right` 📖	mathematical	`DFunLike.coe` `AddMonoidHom` `instFunLike`	`comp`	—	`ext` `Function.Surjective.forall` `DFunLike.ext_iff`
`coe_coe` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `instFunLike` `AddMonoidHomClass.toAddMonoidHom`	—	—
`coe_comp` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `instFunLike` `comp`	—	—
`coe_copy` 📖	mathematical	`DFunLike.coe` `AddMonoidHom` `instFunLike`	`copy`	—	—
`coe_id` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `instFunLike` `id`	—	—
`coe_mk` 📖	mathematical	`ZeroHom.toFun` `AddZero.toZero` `AddZero.toAdd`	`DFunLike.coe` `AddMonoidHom` `instFunLike` `ZeroHom` `ZeroHom.funLike`	—	—
`comp_apply` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `instFunLike` `comp`	—	—
`comp_assoc` 📖	mathematical	—	`comp`	—	—
`comp_id` 📖	mathematical	—	`comp` `id`	—	`ext`
`comp_zero` 📖	mathematical	—	`comp` `AddZeroClass.toAddZero` `AddMonoidHom` `instZeroAddMonoidHom`	—	`ext` `map_zero` `AddMonoidHomClass.toZeroHomClass` `instAddMonoidHomClass`
`copy_eq` 📖	mathematical	`DFunLike.coe` `AddMonoidHom` `instFunLike`	`copy`	—	`DFunLike.ext'`
`ext` 📖	—	`DFunLike.coe` `AddMonoidHom` `instFunLike`	—	—	`DFunLike.ext`
`ext_iff` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `instFunLike`	—	`ext`
`id_apply` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `instFunLike` `id`	—	—
`id_comp` 📖	mathematical	—	`comp` `id`	—	`ext`
`instAddMonoidHomClass` 📖	mathematical	—	`AddMonoidHomClass` `AddMonoidHom` `instFunLike`	—	`map_add'` `ZeroHom.map_zero'`
`inverse_apply` 📖	mathematical	`DFunLike.coe` `AddMonoidHom` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `instFunLike`	`inverse`	—	—
`map_add` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `instFunLike` `AddZero.toAdd`	—	`map_add'`
`map_add'` 📖	mathematical	—	`ZeroHom.toFun` `AddZero.toZero` `toZeroHom` `AddZero.toAdd`	—	—
`map_add_neg` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `SubNegMonoid.toAddMonoid` `AddGroup.toSubNegMonoid` `SubtractionMonoid.toSubNegMonoid` `instFunLike` `AddZero.toAdd` `SubNegMonoid.toNeg`	—	`map_add` `AddMonoidHomClass.toAddHomClass` `instAddMonoidHomClass` `map_neg`
`map_exists_left_neg` 📖	mathematical	`AddZero.toAdd` `AddZero.toZero`	`DFunLike.coe`	—	`map_add_eq_zero`
`map_exists_right_neg` 📖	mathematical	`AddZero.toAdd` `AddZero.toZero`	`DFunLike.coe`	—	`map_add_eq_zero`
`map_neg` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `SubNegMonoid.toAddMonoid` `AddGroup.toSubNegMonoid` `SubtractionMonoid.toSubNegMonoid` `instFunLike` `SubNegMonoid.toNeg`	—	`map_neg` `instAddMonoidHomClass`
`map_nsmul` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `instFunLike` `AddMonoid.toNatSMul`	—	`map_nsmul` `instAddMonoidHomClass`
`map_sub` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `SubNegMonoid.toAddMonoid` `AddGroup.toSubNegMonoid` `SubtractionMonoid.toSubNegMonoid` `instFunLike` `SubNegMonoid.toSub`	—	`map_sub` `instAddMonoidHomClass`
`map_zero` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `instFunLike` `AddZero.toZero`	—	`ZeroHom.map_zero'`
`map_zsmul` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `SubNegMonoid.toAddMonoid` `AddGroup.toSubNegMonoid` `SubtractionMonoid.toSubNegMonoid` `instFunLike` `SubNegMonoid.toZSMul`	—	`map_zsmul` `instAddMonoidHomClass`
`map_zsmul'` 📖	mathematical	`DFunLike.coe` `AddMonoidHom` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `SubNegMonoid.toAddMonoid` `instFunLike` `SubNegMonoid.toNeg`	`SubNegMonoid.toZSMul`	—	`map_zsmul'` `instAddMonoidHomClass`
`mk'_apply` 📖	mathematical	`AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `SubNegMonoid.toAddMonoid` `AddGroup.toSubNegMonoid`	`DFunLike.coe` `AddMonoidHom` `instFunLike` `mk'`	—	—
`mk_coe` 📖	—	`ZeroHom.toFun` `AddZero.toZero` `ZeroHomClass.toZeroHom` `AddMonoidHom` `instFunLike` `AddMonoidHomClass.toZeroHomClass` `instAddMonoidHomClass` `AddZero.toAdd`	—	—	`AddMonoidHomClass.toZeroHomClass` `instAddMonoidHomClass` `ext`
`toAddHom_coe` 📖	mathematical	—	`AddHom.toFun` `AddZero.toAdd` `toAddHom` `DFunLike.coe` `AddMonoidHom` `instFunLike`	—	—
`toAddHom_injective` 📖	mathematical	—	`AddMonoidHom` `AddHom` `AddZero.toAdd` `toAddHom`	—	`Function.Injective.of_comp` `DFunLike.coe_injective`
`toFun_eq_coe` 📖	mathematical	—	`ZeroHom.toFun` `AddZero.toZero` `toZeroHom` `DFunLike.coe` `AddMonoidHom` `instFunLike`	—	—
`toZeroHom_coe` 📖	mathematical	—	`DFunLike.coe` `ZeroHom` `AddZero.toZero` `ZeroHom.funLike` `toZeroHom` `AddMonoidHom` `instFunLike`	—	—
`toZeroHom_injective` 📖	mathematical	—	`AddMonoidHom` `ZeroHom` `AddZero.toZero` `toZeroHom`	—	`Function.Injective.of_comp` `DFunLike.coe_injective`
`zero_apply` 📖	mathematical	—	`DFunLike.coe` `AddMonoidHom` `AddZeroClass.toAddZero` `instFunLike` `instZeroAddMonoidHom` `AddZero.toZero`	—	—
`zero_comp` 📖	mathematical	—	`comp` `AddZeroClass.toAddZero` `AddMonoidHom` `instZeroAddMonoidHom`	—	—

AddMonoidHomClass

Definitions

Name	Category	Theorems
`toAddMonoidHom` 📖	CompOp	106 math math: `CategoryTheory.Functor.mapExtLinearMap_toAddMonoidHom`, `RootPairing.linearIndepOn_root_baseOf'`, `AddMonoidAlgebra.liftNC_mul`, `CoalgHom.coe_addMonoidHom_injective`, `AddEquiv.addSubgroupMap_symm_apply`, `AlgHom.coe_addMonoidHom_injective`, `AddEquiv.toAddMonoidHom_eq_coe`, `AddMonoidHom.ker_comp_addEquiv`, `AddSubgroup.index_map_equiv`, `NonUnitalRingHom.coe_addMonoidHom_injective`, `AddSubgroup.comap_equiv_eq_map_symm`, `LinearMap.toAddMonoidHom_mulLeft`, `MonoidAlgebra.liftNC_mul`, `AddMonoidAlgebra.domCongr_toAlgHom`, `CentroidHom.coe_toAddMonoidHom`, `AddMonoidAlgebra.toRingHom_mapDomainRingEquiv`, `CharacterModule.curry_apply_apply`, `Polynomial.eval₂_ofFinsupp`, `AdjoinRoot.evalEval_apply`, `MonoidAlgebra.coe_liftNCAlgHom`, `Module.End.ringHomEndFinsupp_apply_apply`, `AddEquiv.coe_addMonoidHom_refl`, `AddGroupExtension.Splitting.coe_addMonoidHom_mk`, `RingHom.toAddMonoidHom_eq_coe`, `FreeAbelianGroup.liftMonoid_coe_addMonoidHom`, `AddEquiv.coe_addMonoidHom_comp_coe_addMonoidHom_symm`, `DirectSum.algebraMap_toAddMonoid_hom`, `OrderAddMonoidHom.coe_addMonoidHom`, `Matrix.entryAddMonoidHom_eq_comp`, `AddMonoidHom.coe_coe`, `AddEquiv.addMonoidHomCongrLeft_apply`, `CentroidHom.toAddMonoidHom_id`, `AddSubgroup.map_equiv_top`, `AddEquiv.comp_left_injective`, `AddSubmonoid.LocalizationMap.AwayMap.lift_comp`, `AddSubmonoid.map_coe_toAddMonoidHom`, `AddSubgroup.map_equiv_eq_comap_symm`, `AddEquiv.add_submonoid_map_symm_apply`, `IsUniformAddGroup.uniformContinuous_iff_isOpen_ker`, `AddEquiv.mapAddSubgroup_symm_apply`, `RingEquiv.coe_addMonoidHom_refl`, `AddEquiv.comp_right_injective`, `SkewMonoidAlgebra.liftNC_one`, `AddEquiv.comapAddSubgroup_apply`, `Submodule.natAbs_det_equiv`, `AddSubgroup.comap_toAddSubmonoid`, `AddEquiv.coprodCongr_apply`, `AddEquiv.coe_addSubgroupMap_apply`, `AddMonoidAlgebra.coe_liftNCAlgHom`, `CentroidHom.coe_toAddMonoidHom_injective`, `AddEquiv.coe_addMonoidHom_trans`, `NonUnitalRingHom.coe_addMonoidHom_id`, `AddMonoidAlgebra.liftNC_smul`, `ContinuousAddMonoidHom.toAddMonoidHom_toContinuousAddMonoidHom`, `OrderAddMonoidHom.coe_comp_addMonoidHom`, `SkewMonoidAlgebra.liftNC_mul`, `AddEquiv.addSubmonoidMap_symm_apply`, `AddSubmonoid.equivMapOfInjective_coe_addEquiv`, `AddEquiv.mapAddSubgroup_apply`, `Finsupp.mapRange.addEquiv_toAddMonoidHom`, `Ideal.natAbs_det_equiv`, `CentroidHom.toAddMonoidHom_eq_coe`, `DirectSum.ringHom_ext'_iff`, `DistribMulActionHom.coe_fn_coe`, `LinearMap.eqLocus_toAddSubmonoid`, `AddMonoidAlgebra.liftNC_one`, `DirectSum.liftRingHom_symm_apply_coe`, `MonoidAlgebra.liftNC_smul`, `AddSubgroup.equivMapOfInjective_coe_addEquiv`, `NonUnitalRingHom.coe_comp_addMonoidHom`, `AddEquiv.coe_addMonoidHom_symm_comp_coe_addMonoidHom`, `LocallyFiniteOrder.orderAddMonoidHom_toAddMonoidHom`, `CentroidHom.coe_comp_addMonoidHom`, `LinearMap.mulRight_toAddMonoidHom`, `AddSubgroup.map_symm_eq_iff_map_eq`, `AddCon.comapQuotientEquivOfSurj_symm_mk'`, `AlgHom.toRingHom_toAddMonoidHom`, `RingEquiv.coe_addMonoidHom_trans`, `DirectSum.toSemiring_coe_addMonoidHom`, `MonoidAlgebra.lift_def`, `RingHom.coe_addMonoidHom_mk`, `AddEquiv.coprodCongr_symm_apply`, `Submodule.map_toAddSubgroup`, `AddMonoidAlgebra.lift_def`, `AddMonoidHom.coe_addMonoidHom_mkRingHomOfMulSelfOfTwoNeZero`, `RingHom.coe_addMonoidHom_id`, `MonoidAlgebra.liftNC_one`, `FreeAbelianGroup.liftMonoid_symm_coe`, `ContinuousAddMonoidHom.coe_toAddMonoidHom`, `Matrix.entryLinearMap_toAddMonoidHom`, `SkewMonoidAlgebra.lift_def`, `CoalgHom.coe_toAddMonoidHom`, `AddEquiv.addMonoidHomCongrRight_apply`, `RingHom.coe_addMonoidHom_injective`, `NonUnitalRingHom.coe_addMonoidHom_mk`, `LinearMap.toAddMonoidHom_mulRight`, `Submodule.map_toAddSubmonoid`, `AddSubmonoid.topEquiv_toAddMonoidHom`, `AlgHom.coe_toAddMonoidHom`, `SemimoduleCat.forget₂_map`, `OrderAddMonoidHom.toAddMonoidHom_eq_coe`, `SkewMonoidAlgebra.liftNC_smul`, `AddEquiv.comapAddSubgroup_symm_apply`, `LinearMap.mulLeft_toAddMonoidHom`, `AddGroupExtension.Splitting.rightHom_comp_splitting`, `ModuleCat.forget₂_map`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`toAddHomClass` 📖	mathematical	—	`AddHomClass` `AddZero.toAdd`	—	—
`toZeroHomClass` 📖	mathematical	—	`ZeroHomClass` `AddZero.toZero`	—	—

Function.Injective

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`isAddTorsionFree` 📖	mathematical	`DFunLike.coe` `AddMonoidHom` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `AddMonoidHom.instFunLike`	`IsAddTorsionFree`	—	`IsAddTorsionFree.nsmul_right_injective` `map_nsmul` `AddMonoidHom.instAddMonoidHomClass`
`isMulTorsionFree` 📖	mathematical	`DFunLike.coe` `MonoidHom` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `MonoidHom.instFunLike`	`IsMulTorsionFree`	—	`IsMulTorsionFree.pow_left_injective` `map_pow` `MonoidHom.instMonoidHomClass`

Function.Surjective

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`add_comm` 📖	—	`DFunLike.coe` `AddHom` `AddHom.funLike`	—	—	`map_add` `AddHom.addHomClass`
`mul_comm` 📖	—	`DFunLike.coe` `MulHom` `MulHom.funLike`	—	—	`MulHom.mulHomClass`

IsDedekindFiniteAddMonoid

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`of_injective` 📖	mathematical	`DFunLike.coe`	`IsDedekindFiniteAddMonoid`	—	`map_add` `AddMonoidHomClass.toAddHomClass` `map_zero` `AddMonoidHomClass.toZeroHomClass` `add_eq_zero_comm`

IsDedekindFiniteMonoid

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`of_injective` 📖	mathematical	`DFunLike.coe`	`IsDedekindFiniteMonoid`	—	`map_mul` `MonoidHomClass.toMulHomClass` `map_one` `MonoidHomClass.toOneHomClass`

LibraryNote

Definitions

Name	Category	Theorems
`hom_simp_lemma_priority` 📖	CompOp	—

Monoid.End

Definitions

Name	Category	Theorems
`instFunLike` 📖	CompOp	4 math math: `coe_one`, `coe_pow`, `coe_mul`, `instMonoidHomClass`
`instInhabited` 📖	CompOp	—
`instMonoid` 📖	CompOp	3 math math: `coe_pow`, `Subgroup.pointwise_smul_def`, `MulDistribMulAction.toMonoidEnd_apply`
`instMul` 📖	CompOp	7 math math: `monoidEndToAdditive_apply_apply`, `MulEquiv.AddMonoid.End_apply`, `coe_mul`, `monoidEndToAdditive_symm_apply_apply`, `MulEquiv.Monoid.End_apply`, `addMonoidEndToMultiplicative_apply_apply`, `addMonoidEndToMultiplicative_symm_apply_apply`
`instOne` 📖	CompOp	1 math math: `coe_one`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`coe_mul` 📖	mathematical	—	`DFunLike.coe` `Monoid.End` `instFunLike` `instMul`	—	—
`coe_one` 📖	mathematical	—	`DFunLike.coe` `Monoid.End` `instFunLike` `instOne`	—	—
`coe_pow` 📖	mathematical	—	`DFunLike.coe` `Monoid.End` `instFunLike` `Monoid.toNatPow` `instMonoid` `Nat.iterate`	—	—
`instMonoidHomClass` 📖	mathematical	—	`MonoidHomClass` `Monoid.End` `instFunLike`	—	`MonoidHom.instMonoidHomClass`

MonoidHom

Definitions

Name	Category	Theorems
`coeToMulHom` 📖	CompOp	—
`coeToOneHom` 📖	CompOp	—
`comp` 📖	CompOp	267 math math: `CommGrpCat.uliftFunctor_map`, `Submonoid.LocalizationMap.lift_comp_lift_eq`, `Monoid.PushoutI.hom_ext_iff`, `cancel_left`, `Finsupp.mulHom_ext'_iff`, `GroupSeminorm.comp_assoc`, `groupHomology.mapCycles₂_comp_assoc`, `AddMonoidAlgebra.algHom_ext'_iff`, `functions_ext'_iff`, `groupHomology.coinfNatTrans_app`, `Con.hom_ext_iff`, `MulEquiv.monoidHomCongrLeft_apply`, `Subgroup.subtype_comp_inclusion`, `MulEquiv.coe_monoidHom_symm_comp_coe_monoidHom`, `SkewMonoidAlgebra.algHom_ext'_iff`, `Matrix.GeneralLinearGroup.map_comp_apply`, `comp_inv`, `Rep.coindToInd_of_support_subset_orbit`, `groupHomology.mapCycles₁_comp_apply`, `comp_div`, `Monoid.Coprod.map_comp_inr`, `FDRep.endRingEquiv_symm_comp_ρ`, `MonoidAlgebra.algHom_ext'_iff`, `Monoid.Coprod.map_comp_inl`, `DirichletCharacter.changeLevel_def`, `groupHomology.mapCycles₁_comp_assoc`, `coprod_comp_inr`, `Monoid.Coprod.fst_comp_inr`, `noncommCoprod_comp_inr`, `coe_comp`, `prod_comp_prodMap`, `groupCohomology.cochainsMap_comp`, `FiniteIndexNormalSubgroup.comap_comp`, `Monoid.Coprod.clift_comp_mk`, `Submonoid.LocalizationMap.AwayMap.lift_comp`, `Submonoid.LocalizationMap.lift_of_comp`, `Monoid.Coprod.map_map`, `CompTriple.comp`, `Submonoid.LocalizationMap.ofMulEquivOfLocalizations_comp`, `CommMonCat.hom_comp`, `Rep.resCoindAdjunction_unit_app_hom_hom`, `Representation.Coinvariants.le_comap_ker`, `Monoid.CoprodI.inv_def`, `CommMonCat.coyonedaType_obj_map`, `Abelianization.hom_ext_iff`, `Abelianization.map_map_apply`, `groupCohomology.mapShortComplexH2_comp_assoc`, `comp_mul`, `MulEquiv.monoidHomCongrLeftEquiv_apply`, `MulEquiv.comp_right_injective`, `map_range`, `one_comp`, `CategoryTheory.Discrete.monoidalFunctorComp_isMonoidal`, `SubgroupClass.subtype_comp_inclusion`, `Representation.instIsTrivialSubtypeMemSubgroupSubmoduleInvariantsCompLinearMapIdSubtypeToInvariants`, `MonoidAlgebra.mapDomainBialgHom_comp`, `compHom_apply_apply`, `groupHomology.mapCycles₁_comp`, `Monoid.Coprod.mrange_eq`, `groupHomology.map_comp`, `groupCohomology.map_comp`, `MonoidAlgebra.mapDomainBialgHom_mapDomainBialgHom`, `LinearMap.detAux_def`, `Monoid.PushoutI.of_comp_eq_base`, `CommGrpCat.coyonedaType_map_app`, `MonCat.ofHom_comp`, `range_comp`, `unitary.toUnits_comp_map`, `SemidirectProduct.congr'_apply_right`, `Submonoid.LocalizationMap.map_comp_map`, `groupCohomology.infNatTrans_app`, `groupHomology.cyclesMap_comp_assoc`, `Algebra.GrothendieckGroup.lift_symm_apply`, `WithOne.mapMulHom_comp`, `AddMonoidAlgebra.ringHom_ext'_iff`, `CommMonCat.ofHom_comp`, `Con.lift_comp_mk'`, `IsScalarTower.AlgEquiv.restrictNormalHom_comp`, `MonoidAlgebra.mapDomainAlgHom_comp`, `groupHomology.map_comp_assoc`, `QuotientGroup.lift_comp_mk'`, `mrange_comp`, `CompTriple.comp_eq`, `DirichletCharacter.changeLevel_toUnitHom`, `Abelianization.coe_lift_symm`, `ZMod.unitsMap_comp`, `fst_comp_inr`, `Matrix.GeneralLinearGroup.map_comp`, `groupCohomology.mapShortComplexH2_comp`, `subtype_comp_rangeRestrict`, `Rep.instIsTrivialCarrierVModuleCatOfCompLinearMapIdρ`, `div_comp`, `groupCohomology.cochainsMap_comp_assoc`, `Rep.indResAdjunction_counit_app_hom_hom`, `Monoid.Coprod.snd_comp_inr`, `Monoid.Coprod.swap_comp_inr`, `Rep.coindToInd_apply`, `groupHomology.mapCycles₂_comp`, `WithZero.monoidWithZeroHom_ext_iff`, `comp_assoc`, `Monoid.Coprod.fst_comp_inl`, `SemidirectProduct.rightHom_comp_inr`, `noncommCoprod_unique`, `Abelianization.lift_of_comp`, `SemidirectProduct.congr'_apply_left`, `Submonoid.map_map`, `SemidirectProduct.congr'_symm_apply_right`, `CommRing.Pic.mapAlgebra_comp_mapAlgebra`, `groupCohomology.cocyclesMap_comp`, `isLocalHom_comp`, `QuotientGroup.mk'_comp_subtype`, `snd_comp_inr`, `Monoid.Coprod.snd_comp_toProd`, `FreeMonoid.comp_lift`, `IsMulIndecomposable.image_baseOf_inv_comp_eq`, `groupHomology.mapCycles₂_comp_apply`, `Submonoid.LocalizationMap.isUnit_comp`, `Function.MulExact.monoidHom_comp_eq_zero`, `groupHomology.mapShortComplexH1_comp`, `WithZero.lift'_unique`, `Rep.coinvariantsShortComplex_f`, `invMonoidHom_comp_invMonoidHom`, `fst_comp_inl`, `prodMap_def`, `MulEquiv.monoidHomCongrRightEquiv_apply`, `groupHomology.mapShortComplexH2_comp`, `Monoid.PushoutI.homEquiv_apply_coe`, `GrpCat.ofHom_comp`, `SkewMonoidAlgebra.lift_unique'`, `Representation.instIsTrivialSubtypeMemSubgroupCoinvariantsCompLinearMapIdSubtypeToCoinvariants`, `Abelianization.map_comp`, `snd_comp_prod`, `CommGrpCat.hom_comp`, `QuotientGroup.homQuotientZPowOfHom_comp_of_rightInverse`, `Submonoid.LocalizationMap.lift_comp`, `Action.res_map_hom`, `Rep.toCoinvariantsMkQ_hom`, `OrderMonoidHom.coe_comp_monoidHom`, `groupHomology.chainsMap_comp`, `Monoid.Coprod.comp_lift`, `comp_noncommPiCoprod`, `AddMonoidAlgebra.lift_unique'`, `QuotientGroup.homQuotientZPowOfHom_comp`, `Rep.quotientToInvariantsFunctor_obj_V`, `Monoid.Coprod.lift_comp_swap`, `Monoid.PushoutI.hom_ext_nonempty_iff`, `MonCat.uliftFunctor_map`, `CategoryTheory.SingleObj.mapHom_comp`, `Unitary.toUnits_comp_map`, `Submonoid.LocalizationMap.of_mulEquivOfMulEquiv`, `map_mrange`, `GroupExtension.rightHom_comp_inl`, `HNNExtension.hom_ext_iff`, `Representation.le_comap_invariants`, `Con.comap_eq`, `SemidirectProduct.congr'_symm_apply_left`, `groupCohomology.cocyclesMap_comp_assoc`, `Monoid.Coprod.hom_ext_iff`, `Monoid.Coprod.snd_comp_inl`, `comp_toFunctor`, `Submonoid.comap_comap`, `Monoid.Coprod.toProd_comp_inr`, `groupCohomology.mapShortComplexH1_comp`, `MulEquiv.coe_monoidHom_trans`, `Subgroup.goursat_surjective`, `Action.resComp_inv_app_hom`, `id_comp`, `SemidirectProduct.rightHom_comp_inl`, `mul_comp`, `MonoidAlgebra.mapDomainRingHom_comp`, `Action.res_obj_ρ`, `Monoid.Coprod.swap_comp_swap`, `Units.map_comp`, `ker_comp_mulEquiv`, `Monoid.Coprod.lift_comp_inr`, `Rep.Action_ρ_eq_ρ`, `Monoid.Coprod.snd_comp_swap`, `GroupExtension.Splitting.rightHom_comp_splitting`, `MulEquiv.monoidHomCongrRight_apply`, `Con.lift_apply_mk'`, `Monoid.Coprod.lift_comp_inl`, `coprod_unique`, `Monoid.Coprod.toProd_comp_inl`, `fst_comp_prod`, `RegularWreathProduct.rightHom_comp_inl_eq_id`, `prod_unique`, `Representation.toCoinvariants_mk`, `snd_comp_inl`, `groupCohomology.mapShortComplexH1_comp_assoc`, `Subgroup.comap_comap`, `MonoidAlgebra.ringHom_ext'_iff`, `Rep.indMap_hom`, `Monoid.CoprodI.lift_symm_apply`, `WithOne.map_comp`, `CommMonCat.uliftFunctor_map`, `Representation.ind_mk`, `cancel_right`, `MulEquiv.comp_left_injective`, `Monoid.Coprod.map_comp_map`, `CommRing.Pic.mapRingHom_comp_mapRingHom`, `Monoid.Coprod.fst_comp_toProd`, `FDRep.of_ρ`, `Monoid.Coprod.fst_comp_swap`, `Rep.coinvariantsTensorIndHom_mk_tmul_indVMk`, `Monoid.CoprodI.ext_hom_iff`, `CommGrpCat.ofHom_comp`, `SemidirectProduct.mulEquivSubgroup_apply`, `Submonoid.subtype_comp_inclusion`, `Submonoid.LocalizationMap.map_comp`, `CommGrpCat.coyonedaType_obj_map`, `QuotientGroup.monoidHom_ext_iff`, `SemidirectProduct.mulEquivSubgroup_symm_apply`, `WithZero.map'_comp`, `Rep.res_obj_ρ`, `Monoid.Coprod.range_eq`, `SemidirectProduct.lift_unique`, `liftOfRightInverse_comp`, `SkewMonoidAlgebra.ringHom_ext'_iff`, `MonCat.hom_comp`, `Abelianization.lift_symm_apply`, `OpenSubgroup.comap_comap`, `QuotientGroup.ker_le_range_iff`, `range_le_ker_iff`, `FreeMonoid.map_comp`, `comp_id`, `RingEquiv.coe_monoidHom_trans`, `GrpCat.uliftFunctor_map`, `noncommCoprod_comp_inl`, `groupHomology.cyclesMap_comp`, `Submonoid.LocalizationMap.map_map`, `MulEquiv.monoidHomCongrRightEquiv_symm_apply`, `Monoid.CoprodI.lift_comp_of'`, `toHomUnitsMulEquiv_symm_apply`, `MulEquiv.coe_monoidHom_comp_coe_monoidHom_symm`, `comp_one`, `Representation.ind_apply`, `Rep.ihom_obj_ρ`, `Monoid.CoprodI.lift_comp_of`, `coprod_comp_inl`, `Action.resComp_hom_app_hom`, `Submonoid.LocalizationMap.ofMulEquivOfLocalizations_eq`, `Rep.coinvariantsTensorIndInv_mk_tmul_indMk`, `groupCohomology.H1InfRes_f`, `comp_coprod`, `inv_comp`, `ValuationSubring.unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk`, `Submonoid.LocalizationMap.ofMulEquivOfDom_eq`, `GrpCat.hom_comp`, `MulEquiv.monoidHomCongrLeftEquiv_symm_apply`, `comap_mker`, `cyclotomicCharacter.continuous`, `Submonoid.LocalizationMap.lift_comp_lift`, `Monoid.Coprod.swap_comp_map`, `FDRep.endRingEquiv_comp_ρ`, `ext_int_iff`, `Representation.IndV.hom_ext_iff`, `comap_ker`, `CommMonCat.coyonedaType_map_app`, `Rep.indResHomEquiv_symm_apply_hom`, `Subgroup.map_map`, `SemidirectProduct.monoidHomSubgroup_apply`, `comp_noncommCoprod`, `groupCohomology.map_comp_assoc`, `MonoidAlgebra.lift_unique'`, `WithZero.map'_map'`, `comp_apply`, `Monoid.Coprod.swap_comp_inl`
`copy` 📖	CompOp	2 math math: `coe_copy`, `copy_eq`
`id` 📖	CompOp	251 math math: `QuotientGroup.homQuotientZPowOfHom_id`, `MonoidAlgebra.liftMagma_apply_apply`, `groupHomology.mapShortComplexH2_id`, `DistribMulActionHom.one_apply`, `NonUnitalAlgHom.one_apply`, `Unitization.inrNonUnitalAlgHom_toFun`, `NonUnitalSubalgebraClass.coe_subtype`, `MulEquiv.coe_monoidHom_symm_comp_coe_monoidHom`, `groupCohomology.cocyclesMap_id_comp_assoc`, `MulSemiringActionHom.comp_id`, `groupHomology.cyclesMap_id_comp`, `Submonoid.LocalizationMap.map_id`, `NonUnitalAlgHom.fst_toFun`, `CommRing.Pic.mapAlgebra_self`, `NonUnitalAlgHom.range_comp_le_range`, `NonUnitalAlgHom.restrictScalars_injective`, `groupHomology.chainsMap_id`, `AddMonoidAlgebra.mapDomainNonUnitalAlgHom_apply`, `Unitization.algHom_ext'_iff`, `NonUnitalStarAlgHom.coe_restrictScalars`, `CategoryTheory.SingleObj.mapHom_id`, `NonUnitalAlgHom.coe_lmul_eq_mul`, `DomMulAct.smul_mulDistribActionHom_apply`, `groupHomology.comap_coinvariantsKer_pOpcycles_range_subtype_pOpcycles_eq_top`, `WithOne.map_id`, `ZMod.unitsMap_self`, `MonoidAlgebra.mapDomainRingHom_id`, `FreeGroup.lift_of_eq_id`, `NonUnitalSubalgebra.inclusion_mk`, `groupHomology.map_id`, `RingHom.coe_monoidHom_id`, `NonUnitalSubalgebra.inclusion_right`, `FreeNonUnitalNonAssocAlgebra.of_comp_lift`, `id_toFunctor`, `groupHomology.chainsMap_id_f_map_mono`, `MonoidAlgebra.coe_liftNCAlgHom`, `CStarMatrix.norm_def'`, `NonUnitalStarAlgHomClass.instNonUnitalStarRingHomClassOfStarHomClass`, `CompTriple.IsId.eq_id`, `FreeNonUnitalNonAssocAlgebra.lift_unique`, `GrpCat.hom_id`, `MonoidAlgebra.distribMulActionHom_ext'_iff`, `NonUnitalAlgHom.prod_apply`, `groupHomology.mapCycles₁_id_comp_assoc`, `Unitization.lift_symm_apply_apply`, `MulSemiringActionHom.id_apply`, `CStarMatrix.toCLMNonUnitalAlgHom_eq_toCLM`, `graph_eq_range_prod`, `CommGrpCat.hom_id`, `groupHomology.map_id_comp`, `FreeLieAlgebra.liftAux_map_smul`, `Subgroup.comap_id`, `CompTriple.instIsId`, `NonUnitalAlgebra.range_id`, `SkewMonoidAlgebra.nonUnitalAlgHom_ext'_iff`, `NonUnitalAlgHom.subsingleton`, `SMulCommClass.toDistribMulActionHom_toFun`, `Subgroup.smul_diff_smul'`, `groupHomology.mapCycles₂_id_comp`, `MonoidAlgebra.mapDomainAlgHom_id`, `NonUnitalAlgHom.coe_restrictScalars`, `NonUnitalAlgHom.restrictScalars_apply`, `Action.resId_inv_app_hom`, `NonUnitalAlgHom.toAlgHom_zero`, `QuotientGroup.quotientQuotientEquivQuotientAux_mk`, `NonUnitalAlgHom.prod_fst_snd`, `NonUnitalSubalgebra.inclusion_injective`, `NonUnitalAlgHom.prodEquiv_apply`, `MulSemiringActionHom.map_smul`, `Algebra.norm_self`, `groupHomology.mapCycles₁_id_comp_apply`, `NonUnitalAlgHom.prodEquiv_symm_apply`, `groupCohomology.cochainsMap_id_comp`, `DistribMulActionHom.id_apply`, `groupCohomology.map_id`, `Subgroup.map_id`, `WithOne.mapMulHom_id`, `Monoid.Coprod.snd_comp_inr`, `NonUnitalAlgHom.injective_codRestrict`, `NonUnitalAlgHom.coe_inr`, `NonUnitalAlgHom.coe_Lmul`, `NonUnitalAlgHom.coe_one`, `Monoid.Coprod.fst_comp_inl`, `groupCohomology.mono_map_0_of_mono`, `DomMulAct.instSMulCommClassDistribMulActionHomId`, `NonUnitalSubalgebra.toSubring_subtype`, `SemidirectProduct.rightHom_comp_inr`, `FreeLieAlgebra.liftAux_map_add`, `Ideal.Quotient.stabilizerHomSurjectiveAuxFunctor_aux`, `FreeNonUnitalNonAssocAlgebra.lift_symm_apply`, `WeakDual.CharacterSpace.coe_toNonUnitalAlgHom`, `NonUnitalAlgHom.coe_prod`, `NonUnitalAlgHom.coe_restrictScalars'`, `NonUnitalSubalgebraClass.subtype_injective`, `LieHom.toNonUnitalAlgHom_toFun`, `WeakDual.CharacterSpace.toAlgHom_apply`, `Submonoid.map_id`, `snd_comp_inr`, `FreeLieAlgebra.liftAux_map_mul`, `AddMonoidHom.toMultiplicative_id`, `groupHomology.chainsMap_id_f_map_epi`, `mrange_id`, `CommRing.Pic.mapRingHom_id`, `groupCohomology.cochainsMap_id_f_map_mono`, `groupHomology.chainsMap_id_comp`, `transfer_center_eq_pow`, `groupCohomology.mapShortComplexH1_id`, `NonUnitalSubalgebra.inclusion_inclusion`, `NonUnitalStarAlgHom.coe_toNonUnitalAlgHom`, `groupHomology.mapShortComplexH1_id_comp`, `NonUnitalAlgHom.inl_apply`, `invMonoidHom_comp_invMonoidHom`, `NonUnitalSubalgebra.map_toNonUnitalSubsemiring`, `fst_comp_inl`, `Units.map_id`, `DistribMulActionHom.coe_one`, `Unitization.lift_range`, `NonUnitalAlgHom.snd_prod`, `DistribMulActionHom.instLinearMapClassId`, `PositiveLinearMap.gnsStarAlgHom_apply`, `MulSemiringActionHom.id_comp`, `GroupSeminorm.comp_id`, `groupHomology.map_id_comp_H0Iso_hom_assoc`, `AddMonoidAlgebra.coe_liftNCAlgHom`, `FreeLieAlgebra.liftAux_spec`, `groupCohomology.mapShortComplexH2_id_comp_assoc`, `groupHomology.mapCycles₂_id_comp_assoc`, `CommMonCat.hom_id`, `groupCohomology.map_id_comp_H0Iso_hom_apply`, `Unitization.lift_symm_apply`, `groupHomology.chainsMap_id_f_hom_eq_mapRange`, `groupCohomology.map_id_comp_H0Iso_hom`, `groupHomology.mapCycles₁_id_comp`, `Monoid.Coprod.lift_inl_inr`, `mgraph_eq_mrange_prod`, `DomMulAct.mk_smul_mulDistribActionHom_apply`, `MonoidAlgebra.nonUnitalAlgHom_ext'_iff`, `QuotientGroup.homQuotientZPowOfHom_comp_of_rightInverse`, `AddMonoidAlgebra.nonUnitalAlgHom_ext'_iff`, `FiniteIndexNormalSubgroup.comap_id`, `Monoid.CoprodI.lift_of'`, `AddMonoidAlgebra.liftMagma_apply_apply`, `IsInvariantSubring.coe_subtypeHom'`, `groupCohomology.map_id_comp_assoc`, `noncommCoprod_inl_inr`, `MulSemiringActionHom.coe_polynomial`, `MonoidAlgebra.liftMagma_symm_apply`, `AddMonoidAlgebra.liftMagma_symm_apply`, `coe_id`, `coprod_inl_inr`, `groupCohomology.mapShortComplexH1_id_comp`, `toAdditive_id`, `LieHom.toNonUnitalAlgHom_apply`, `AlgEquiv.restrictNormalHom_id`, `NonUnitalAlgHom.fst_prod`, `LinearMap.nonUnitalAlgHom_comp_convMul_distrib`, `NonUnitalAlgHom.comp_mul'`, `id_comp`, `NonUnitalAlgHom.snd_apply`, `groupHomology.mapShortComplexH1_id`, `groupCohomology.map_id_comp`, `Monoid.Coprod.clift_mk`, `GrpCat.ofHom_id`, `Monoid.Coprod.swap_comp_swap`, `NonUnitalAlgHom.coe_inl`, `groupHomology.chainsFunctor_map`, `groupCohomology.cochainsMap_id_f_map_epi`, `MonoidAlgebra.mapDomainNonUnitalAlgHom_apply`, `NonUnitalAlgHom.prod_toFun`, `NonUnitalStarSubalgebra.toSubring_subtype`, `NonUnitalAlgebra.map_top`, `FreeMonoid.map_id`, `groupHomology.map_id_comp_H0Iso_hom_apply`, `NonUnitalAlgHom.toAlgHom_apply`, `Abelianization.lift_of`, `NonUnitalAlgHom.coe_id`, `id_apply`, `groupHomology.functor_map`, `RingEquiv.toMonoidHom_refl`, `MonCat.ofHom_id`, `groupHomology.mapCycles₂_id_comp_apply`, `GroupExtension.Splitting.rightHom_comp_splitting`, `Finsupp.distribMulActionHom_ext'_iff`, `groupCohomology.H1Map_id`, `RegularWreathProduct.rightHom_comp_inl_eq_id`, `groupCohomology.mapShortComplexH1_id_comp_assoc`, `Unitization.inrNonUnitalAlgHom_apply`, `NonUnitalAlgebra.comap_top`, `IsModuleTopology.continuous_of_distribMulActionHom`, `groupHomology.map_id_comp_H0Iso_hom`, `Abelianization.map_id`, `NonUnitalAlgHom.range_comp`, `groupCohomology.cocyclesMap_id`, `Unitization.lift_range_le`, `groupCohomology.mapShortComplexH2_id_comp`, `NonUnitalSubalgebra.toNonUnitalSubsemiring_subtype`, `CommGrpCat.ofHom_id`, `SkewMonoidAlgebra.distribMulActionHom_ext'_iff`, `Submonoid.comap_id`, `NonUnitalAlgebra.map_bot`, `Action.resId_hom_app_hom`, `groupHomology.mapShortComplexH2_id_comp`, `Monoid.CoprodI.of_leftInverse`, `NonUnitalAlgHom.inr_apply`, `QuotientGroup.quotientBot_apply`, `NonUnitalSubalgebra.coe_inclusion`, `NonUnitalSubalgebraClass.subtype_apply`, `NonUnitalAlgebra.range_eq_top`, `groupCohomology.cocyclesMap_id_comp`, `groupCohomology.mapShortComplexH2_id`, `Unitization.lift_apply_apply`, `NonUnitalSubalgebra.map_map`, `groupCohomology.cochainsFunctor_map`, `Pi.evalStarAlgHom_apply`, `comp_id`, `NonUnitalAlgHom.fst_apply`, `CommMonCat.ofHom_id`, `Monoid.Coprod.map_id_id`, `groupHomology.epi_map_0_of_epi`, `Matrix.GeneralLinearGroup.map_id`, `MulEquiv.coe_monoidHom_comp_coe_monoidHom_symm`, `AddMonoidAlgebra.distribMulActionHom_ext'_iff`, `groupCohomology.cochainsMap_id_f_hom_eq_compLeft`, `FreeNonUnitalNonAssocAlgebra.lift_of_apply`, `ker_id`, `MulEquiv.coe_monoidHom_refl`, `RingEquiv.coe_monoidHom_refl`, `groupCohomology.cochainsMap_id_comp_assoc`, `NonUnitalAlgHom.coe_codRestrict`, `NonUnitalStarAlgHom.map_star'`, `groupHomology.cyclesMap_id`, `SkewMonoidAlgebra.DistribMulActionHom.single_toFun`, `DistribMulActionHom.comp_id`, `Unitization.lift_apply`, `WithZero.map'_id`, `groupCohomology.map_id_comp_H0Iso_hom_assoc`, `FreeNonUnitalNonAssocAlgebra.hom_ext_iff`, `NonUnitalSubalgebra.map_id`, `groupCohomology.functor_map`, `Subgroup.smul_diff'`, `MonoidAlgebra.mapDomainBialgHom_id`, `MonCat.hom_id`, `NonUnitalSubalgebra.range_val`, `NonUnitalAlgHom.subtype_comp_codRestrict`, `IsInvariantSubring.coe_subtypeHom`, `Abelianization.equivOfComm_symm_apply`, `NonUnitalAlgHom.snd_toFun`, `DistribMulActionHom.id_comp`, `LieHom.toNonUnitalAlgHom_injective`, `FreeNonUnitalNonAssocAlgebra.lift_comp_of`, `groupCohomology.cochainsMap_id`
`instFunLike` 📖	CompOp	2000 math math: `LinearMap.det_toContinuousLinearMap`, `DistribMulAction.toModuleAut_apply`, `AddMonoidAlgebra.singleHom_apply`, `Polynomial.Gal.galActionHom_restrict`, `Monoid.Coprod.inl_injective`, `CompTriple.comp_apply`, `MulAction.compHom_smul_def`, `Rep.resCoindHomEquiv_apply_hom`, `DomMulAct.smul_monoidHom_apply`, `Fin.sign_cycleIcc_of_eq`, `IsLocalization.mul_add_inv_left`, `Submonoid.mrange_inr'`, `DilationEquiv.toPerm_apply`, `SemidirectProduct.rightHom_inl`, `Submonoid.mem_map_equiv`, `FreeMonoid.map_apply_map_symm_eq`, `Filter.Germ.coe_coeMulHom`, `Matrix.GeneralLinearGroup.det_scalar`, `NonUnitalAlgHom.map_smul`, `Module.Basis.det_comp_basis`, `DirichletCharacter.changeLevel_injective`, `MulChar.equivToUnitHom_mul_apply`, `OreLocalization.numeratorHom_surjective_of_finite`, `Submonoid.map_inr`, `orderOf_map_dvd`, `Rep.coe_linearization_obj_ρ`, `ContMDiffMap.coeFnMonoidHom_apply`, `AddMonoidAlgebra.lift_apply'`, `AlternatingMap.domCoprod.summand_mk''`, `Unitary.conjStarAlgAut_trans_conjStarAlgAut`, `CommMonCat.instFullMonCatForget₂MonoidHomCarrierCarrier`, `Monoid.Coprod.fst_apply_inl`, `LinearMap.detAux_def''`, `Matrix.det_apply`, `mker_snd`, `Algebra.norm_algebraMap_of_basis`, `Monoid.PushoutI.NormalWord.base_smul_def`, `coprod_apply`, `ContinuousAlgHom.toAlgHomMonoidHom_apply`, `MeasureTheory.Measure.addHaar_image_continuousLinearMap`, `Action.leftRegularTensorIso_inv_hom`, `Monoid.Coprod.swap_inr`, `Matrix.SpecialLinearGroup.map_mapGL`, `CompTriple.instRootCompTriple`, `Complex.linearEquiv_det_conjLIE`, `QuotientGroup.kerLift_mk`, `groupCohomology.mem_cocycles₂_def`, `Pi.constMonoidHom_apply`, `CircleDeg1Lift.translate_inv_apply`, `QuotientGroup.lift_mk`, `ClassGroup.mk0_integralRep`, `TannakaDuality.FiniteGroup.toRightFDRepComp_in_rightRegular`, `Monoid.Coprod.lift_apply_inr`, `ZMod.unitsMap_surjective`, `map_finprod_mem'`, `RatFunc.coe_mapAlgHom_eq_coe_map`, `groupCohomology.d₂₃_hom_apply`, `Function.ExtendByOne.hom_apply`, `Monoid.Coprod.map_apply_inl`, `ContinuousMap.compMonoidHom'_apply`, `zpowersMulHom_symm_apply`, `Matrix.GeneralLinearGroup.map_det`, `Polynomial.Gal.galActionHom_bijective_of_prime_degree'`, `monoidHomOfForallMemZpowers_apply_gen`, `HNNExtension.NormalWord.prod_group_smul`, `ClassGroup.mk0_eq_mk0_iff`, `PiTensorProduct.mapLMonoidHom_apply`, `AddEquiv.toMultiplicative_apply_symm_apply`, `NNReal.rpowMonoidHom_apply`, `Matrix.SpecialLinearGroup.continuous_toGL`, `transferSylow_eq_pow`, `Ideal.absNorm_span_insert`, `CommMonCat.forget₂_full`, `FreeMonoid.hom_eq_iff`, `ModularForm.mul_slash`, `map_mul₂`, `QuotientGroup.coe_mk'`, `CongruenceSubgroup.Gamma_mem'`, `toAdditive_apply_apply`, `FreeGroup.prod.of`, `preimage_smul_setₛₗ`, `instMonoidHomClass`, `Algebra.norm_eq_zero_iff_of_basis`, `Equiv.Perm.Basis.toCentralizer_equivariant`, `Monoid.Coprod.con_ker_mk`, `restrictHomKerEquiv_apply_coe`, `ClassGroup.mk_eq_mk_of_coe_ideal`, `Matrix.SpecialLinearGroup.coeToGL_det`, `toMulHom_coe`, `WittVector.teichmuller_mul_pow_coeff_of_ne`, `CommMonCat.forget_preservesLimitsOfShape`, `groupHomology.chainsMap_f_3_comp_chainsIso₃_apply`, `Nat.divisorsHom_apply`, `NumberField.norm_norm_le_norm_mul_house_pow`, `AddMonoidHom.toMultiplicativeRight_apply_apply`, `Matrix.GeneralLinearGroup.map_comp_apply`, `LocallyConstant.comapMonoidHom_apply`, `Polynomial.Gal.restrict_smul`, `Orientation.linearEquiv_det_rotation`, `SubMulAction.ofFixingSubgroup_equivariantMap_injective`, `PresentedGroup.ext_iff`, `MonCat.forget_map`, `Monoid.Foldl.ofFreeMonoid_apply`, `Unitary.conjStarAlgAut_symm`, `SkewMonoidAlgebra.lift_apply'`, `groupHomology.d₃₂_single`, `MonCat.forget_reflects_isos`, `LinearEquiv.transvection.det_eq_one`, `finrank_quotient_span_eq_natDegree_norm`, `ContinuousMultilinearMap.alternatization_apply_apply`, `Action.ofMulAction_apply`, `CommGrpCat.forget₂CommMon_preservesLimitsOfShape`, `Representation.finsupp_apply`, `GrpCat.surjective_of_epi`, `AffineEquiv.coe_homothetyUnitsMulHom_eq_homothetyHom_coe`, `LinearEquiv.det_trans`, `OnePoint.isBoundedAt_iff_exists_SL2Z`, `Submonoid.mrange_inl'`, `Rep.leftRegularHom_hom`, `MulAut.conjNormal_symm_apply`, `SubMulAction.fixingSubgroup_smul_eq_fixingSubgroup_map_conj`, `ClassGroup.Quot_mk_eq_mk`, `AlternatingMap.map_perm`, `ZLattice.covolume_eq_det_inv`, `WithZero.map'_coe`, `Unitary.toUnits_injective`, `Representation.Coinvariants.mk_inv_tmul`, `Matrix.SpecialLinearGroup.toGLPos_injective`, `Equiv.Perm.sign_of_cycleType`, `Polynomial.Gal.galActionHom_injective`, `MeasureTheory.addHaarScalarFactor_smul_inv_eq_distribHaarChar`, `Matrix.det_permute`, `LinearMap.det_mulLeft`, `Subgroup.coe_inclusion`, `Algebra.PowerBasis.norm_gen_eq_coeff_zero_minpoly`, `Algebra.norm_eq_matrix_det`, `WithZero.lift'_symm_apply_apply`, `Representation.asAlgebraHom_single`, `Submonoid.LocalizationMap.map_id`, `PresentedGroup.mk_eq_mk_of_mul_inv_mem`, `Submonoid.prod_top`, `CategoryTheory.PreGaloisCategory.autEmbedding_injective`, `Monoid.CoprodI.NeWord.mulHead_prod`, `FreeMonoid.lift_restrict`, `CategoryTheory.Iso.commGroupIsoToMulEquiv_symm_apply`, `Representation.asGroupHom_apply`, `ProfiniteGrp.ProfiniteCompletion.lift_eta`, `LinearMap.det_pi`, `SpecialLinearGroup.coe_zpow`, `MulAut.conjNormal_val`, `Ideal.intNorm_mem_spanNorm`, `CongruenceSubgroup.exists_Gamma_le_conj'`, `SpecialLinearGroup.toLinearEquiv_symm_to_linearMap`, `Unitary.conjStarAlgEquiv_unitaryLinearIsometryEquiv`, `sum_hom_units`, `SkewMonoidAlgebra.single_algebraMap_eq_algebraMap_mul_of`, `CommMonCat.coe_forget₂_obj`, `Equiv.Perm.subtypePerm_ofSubtype`, `DirichletCharacter.changeLevel_def`, `CommGrpCat.epi_iff_surjective`, `Submonoid.le_pi_iff`, `Rep.coe_res_obj_ρ`, `Subgroup.conj_smul_subgroupOf`, `AddEquiv.toMultiplicativeRight_apply_apply`, `Representation.ofCoinvariantsTprodLeftRegular_mk_tmul_single`, `LinearMap.det_zero`, `Equiv.Perm.Basis.toCentralizer_apply`, `MonCat.ext_iff`, `SkewMonoidAlgebra.smul_of`, `Traversable.foldMap_hom_free`, `MulAction.stabilizerEquivStabilizer_symm_apply`, `NumberField.mixedEmbedding.norm_eq_norm`, `NonUnitalAlgSemiHomClass.toDistribMulActionSemiHomClass`, `ModularForm.SL_slash`, `CongruenceSubgroup.conjGL_coe`, `SpecialLinearGroup.mem_center_iff_spec`, `Rep.diagonalHomEquiv_symm_apply`, `MulSemiringActionHom.instMulSemiringActionSemiHomClassCoeMonoidHom`, `SemidirectProduct.inl_aut`, `GrpCat.forget_reflects_isos`, `rotation_injective`, `CoxeterSystem.lift_apply_simple`, `IsPrimitiveRoot.norm_toInteger_sub_one_of_eq_two_pow`, `SemidirectProduct.inr_injective`, `cyclotomicCharacter.toZModPow`, `IsFractionRing.fieldEquivOfAlgEquivHom_apply`, `UpperHalfPlane.im_smul_eq_div_normSq`, `DirichletCharacter.conductor_le_conductor_mem_conductorSet`, `Equiv.Perm.subtypeCongrHom_injective`, `Submonoid.LocalizationMap.mulEquivOfLocalizations_apply`, `QuotientGroup.rangeKerLift_injective`, `Subgroup.comap_toSubmonoid`, `InfiniteGalois.normalAutEquivQuotient_apply`, `RatFunc.map_apply_ofFractionRing_mk`, `Monoid.Coprod.fst_toProd`, `CommGrpCat.hasLimit_iff_small_sections`, `map_div₂`, `groupHomology.mem_cycles₂_iff`, `CategoryTheory.yonedaMonObjIsoOfRepresentableBy_hom_app_hom_apply`, `Sylow.coe_smul`, `Equiv.Perm.disjoint_ofSubtype_noncommPiCoprod`, `pinGroup.val_inv_toUnits_apply`, `WithOne.lift_coe`, `QuadraticAlgebra.norm_algebraMap`, `Matrix.GeneralLinearGroup.map_inv_mul_map`, `AddConstMap.coe_addLeftHom_apply`, `finset_prod_apply`, `NumberField.FinitePlace.prod_eq_inv_abs_norm`, `CommGrpCat.forget_map`, `GroupExtension.rightHom_inl`, `groupCohomology.d₁₂_hom_apply`, `map_list_prod`, `GroupExtension.Splitting.rightInverse_rightHom`, `SubMulAction.ofFixingSubgroup_of_inclusion_injective`, `WittVector.map_teichmuller`, `MonCat.forget_preservesLimitsOfSize`, `FreeGroup.ext_hom_iff`, `SpecialLinearGroup.mem_center_iff`, `Representation.directSum_apply`, `CategoryTheory.yonedaMonObjIsoOfRepresentableBy_inv_app_hom_apply`, `Traversable.foldlm.ofFreeMonoid_comp_of`, `Ideal.torsionMapQuot_apply`, `CommGrpCat.forget_preservesLimits`, `Finset.coeMonoidHom_apply`, `Monoid.CoprodI.lift_of`, `Monoid.Coprod.swap_surjective`, `UpperHalfPlane.moebius_im`, `ModularForm.prod_slash`, `DirichletCharacter.changeLevel_trans`, `MeasureTheory.addHaarScalarFactor_smul_eq_distribHaarChar_inv`, `GrpCat.forget_isCorepresentable`, `zpowersHom_symm_apply`, `SemidirectProduct.inr_inj`, `MonCat.forget_preservesLimitsOfShape`, `Equiv.Perm.sign_subtypePerm`, `FractionalIdeal.absNorm_eq'`, `Function.sSup_div_semiconj`, `coe_comp`, `FreeMonoid.countP_of`, `FiniteGaloisIntermediateField.mem_fixingSubgroup_iff`, `Ideal.map_spanIntNorm`, `Action.rightDual_ρ`, `Subgroup.apply_coe_mem_map`, `SpecialLinearGroup.coe_one`, `StarMonoidHom.coe_toMonoidHom`, `Monoid.Coprod.toProd_surjective`, `groupCohomology.d₀₁_hom_apply`, `Con.mk'_surjective`, `Representation.ofMulAction_apply`, `Rep.linearization_single`, `Equiv.Perm.disjoint_ofSubtype_of_memFixedPoints_self`, `groupHomology.d₃₂_single_one_thd`, `rotation_apply`, `toMatrix_rotation`, `Matrix.SpecialLinearGroup.map_intCast_injective`, `QuotientGroup.kerLift_mk'`, `Equiv.Perm.OnCycleFactors.kerParam_injective`, `SetSemiring.imageHom_def`, `ModularForm.prod_slash_sum_weights`, `CommGrpCat.FilteredColimits.forget₂Group_preservesFilteredColimits`, `toHomPerm_apply_symm_apply`, `MulSemiringActionHom.map_smulₛₗ`, `MonoidAlgebra.mem_adjoin_support`, `BoundedContinuousFunction.toContinuousMapMonoidHom_apply`, `Monoid.Coprod.map_map`, `Finset.imageMonoidHom_apply`, `Algebra.dvd_algebraMap_intNorm_self`, `AddMonoidHom.coe_toMultiplicativeRight`, `Representation.apply_sub_id_partialSum_eq`, `HNNExtension.equiv_eq_conj`, `AlgEquiv.restrictNormalHom_surjective`, `WittVector.teichmuller_zero`, `QuadraticAlgebra.norm_zero`, `Equiv.Perm.sign_trans_trans_symm`, `subgroupOf_range_eq_of_le`, `AddMonoidAlgebra.divOfHom_apply_apply`, `uliftZPowersHom_symm_apply`, `Orientation.map_eq_neg_iff_det_neg`, `MulSemiringActionHom.coe_fn_coe'`, `ProfiniteGrp.ProfiniteCompletion.lift_eta_assoc`, `IsLocalization.lift_spec_mul_add`, `InfiniteGalois.toAlgEquivAux_eq_proj_of_mem`, `CommGrpCat.ext_iff`, `ModularGroup.sl_moeb`, `ClassGroup.mk0_eq_one_iff`, `DirichletCharacter.LFunction_changeLevel`, `Algebra.algebraMap_intNorm`, `fiberEquivKer_symm_apply`, `Representation.Coinvariants.mk_tmul_inv`, `CategoryTheory.MonObj.ofRepresentableBy_one`, `Finset.singletonMonoidHom_apply`, `FreeGroup.lift_apply_of`, `SemidirectProduct.inl_left_mul_inr_right`, `CommGrpCat.coe_comp`, `SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descMonoidHom_quotMk`, `SpecialLinearGroup.mem_range_toGeneralLinearGroup_iff`, `Matrix.SpecialLinearGroup.coe_int_neg`, `MonoidAlgebra.of_mem_span_of_iff`, `Equiv.Perm.viaFintypeEmbedding_sign`, `CategoryTheory.PreGaloisCategory.autMapHom_apply`, `OnePoint.isZeroAt_iff_exists_SL2Z`, `Submonoid.top_prod`, `Rep.resCoindAdjunction_unit_app_hom_hom`, `Equiv.Perm.ofSubtype_apply_mem_iff_mem`, `groupCohomology.mem_cocycles₁_def`, `RootPairing.Equiv.weightHom_toLinearMap`, `Equiv.Perm.sign_symm`, `Matrix.permMatrixHom_apply`, `CommGrpCat.forget₂Group_preservesLimit`, `Representation.Coinvariants.le_comap_ker`, `CommGrpCat.hom_inv_apply`, `NumberField.Units.dirichletUnitTheorem.seq_next`, `Monoid.CoprodI.inv_def`, `Ideal.rootsOfUnityMapQuot_injective`, `Con.ker_apply`, `Abelianization.mk_eq_of`, `Abelianization.map_map_apply`, `Filter.pureMonoidHom_apply`, `LinearMap.det_baseChange`, `RatFunc.map_injective`, `MonoidAlgebra.lift_symm_apply`, `IsPrimitiveRoot.norm_toInteger_sub_one_of_eq_two`, `Polynomial.eval₂_ofFinsupp`, `LocallyConstant.evalMonoidHom_apply`, `LinearMap.isUnit_iff_isUnit_det`, `GroupExtension.Equiv.map_inl`, `NumberField.FinitePlace.prod_eq_inv_abs_norm_int`, `map_frobeniusEquiv_symm`, `map_iterate_frobenius`, `CongruenceSubgroup.exists_Gamma_le_conj`, `groupCohomology.cochainsMap_f_1_comp_cochainsIso₁_apply`, `groupCohomology.mapCocycles₂_comp_i_apply`, `LinearEquiv.det_baseChange`, `MonoidAlgebra.support_gen_of_gen'`, `modularCyclotomicCharacter.spec`, `CommMonCat.coe_comp`, `Representation.ofMulAction_single`, `Submonoid.mrange_inl_sup_mrange_inr`, `AddConstMap.toEnd_apply`, `Fin.sign_cycleRange`, `groupHomology.single_one_snd_sub_single_one_fst_mem_boundaries₂`, `Equiv.Perm.sign_eq_prod_prod_Ioi`, `groupCohomology.cochainsMap_f_3_comp_cochainsIso₃_apply`, `RatFunc.coe_mapRingHom_eq_coe_map`, `Equiv.Perm.prod_Ioi_comp_eq_sign_mul_prod`, `Monoid.Coprod.swap_injective`, `Representation.ofDistribMulAction_apply_apply`, `Finset.coe_coeMonoidHom`, `Ideal.Quotient.stabilizerHom_surjective`, `toAdditive_symm_apply_apply`, `Matrix.SpecialLinearGroup.isClosedEmbedding_toGL`, `HNNExtension.NormalWord.prod_cons`, `isOfFinOrder`, `Valuation.unit_map_eq`, `linearIndependent_monoidHom`, `alternatingGroup.normalClosure_swap_mul_swap_five`, `SubgroupClass.subtype_injective`, `Subgroup.coe_map`, `MonoidWithZeroHom.coe_copy`, `groupHomology.d₂₁_single_inv_mul_ρ_add_single`, `ValuativeExtension.mapValueGroupWithZero_mk`, `FundamentalGroup.map_apply`, `inr_apply`, `inr_mono`, `Con.kerLift_mk`, `ZMod.AddAutEquivUnits_symm_apply`, `MonCat.oneHom_apply`, `Monoid.Coprod.mrange_swap`, `PreTilt.mk_comp_untilt_eq_coeff_zero`, `HNNExtension.inv_t_mul_of`, `compLeft_apply`, `IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_pow_ne_two`, `val_unitsCenterToCenterUnits_apply_coe`, `zpowersHom_apply`, `powersHom_symm_apply`, `IsFreeGroupoid.SpanningTree.functorOfMonoidHom_map`, `Rep.ρ_hom`, `Monoid.Coprod.closure_range_inl_union_inr`, `Monoid.CoprodI.mrange_eq_iSup`, `coe_snd`, `CongruenceSubgroup.Gamma1_mem'`, `AddChar.toMonoidHomEquiv_symm_apply`, `CommGrpCat.FilteredColimits.forget_preservesFilteredColimits`, `compHom_apply_apply`, `Equiv.Perm.sign_of_cycleType_eq_replicate`, `FreeAbelianGroup.liftMonoid_coe_addMonoidHom`, `Units.coe_map`, `Subgroup.Normal.conj_smul_eq_self`, `Monoid.PushoutI.NormalWord.base_smul_eq_smul`, `Units.map_mk`, `pinGroup.val_toUnits_apply`, `QuadraticAlgebra.norm_eq_one_iff_mem_unitary`, `Submonoid.iSup_map_mulSingle_le`, `LinearMap.det_mulRight`, `Zsqrtd.mker_norm_eq_unitary`, `Matrix.SpecialLinearGroup.toLin'_symm_to_linearMap`, `Orientation.det_rotation`, `IsLocalization.surj''`, `isLocalHom_toMonoidHom`, `RatFunc.map_apply`, `AddEquiv.toMultiplicativeLeft_apply_symm_apply`, `Monoid.Coprod.mrange_eq`, `DistribMulAction.toAddMonoidEnd_apply`, `CommGrpCat.μ_forget_apply`, `Representation.Coinvariants.sub_mem_ker`, `Action.FunctorCategoryEquivalence.functor_map_app`, `ComplexShape.TensorSigns.ε'_succ`, `Algebra.discr_powerBasis_eq_norm`, `FreeMonoid.lift_ofList`, `Pi.evalMonoidHom_apply`, `LinearMap.associated_det_of_eq_comp`, `spinGroup.val_inv_toUnits_apply`, `CommMonCat.coyoneda_obj_map`, `closure_preimage_le`, `MonCat.instIsRightAdjointForgetMonoidHomCarrier`, `Orientation.map_eq_iff_det_pos`, `IsPrimitiveRoot.norm_eq_neg_one_pow`, `IsCyclotomicExtension.norm_zeta_pow_sub_one_of_prime_ne_two`, `transferSylow_restrict_eq_pow`, `Polynomial.leadingCoeffHom_apply`, `isCusp_SL2Z_iff'`, `Subgroup.pointwise_smul_def`, `ContinuousCohomology.I_obj_ρ_apply`, `MulEquiv.coprodPUnit_symm_apply`, `Equiv.Perm.decomposeFin.symm_sign`, `FreeMonoid.mem_map`, `IsPrimitiveRoot.autToPow_eq_modularCyclotomicCharacter`, `Subgroup.quotientSubgroupOfEmbeddingOfLE_apply_mk`, `ModularGroup.denom_apply`, `MeasureTheory.addHaarScalarFactor_smul_eq_distribHaarChar`, `CommMonCat.forget_preservesLimitsOfSize`, `CoxeterSystem.getElem_succ_leftInvSeq_alternatingWord`, `SkewMonoidAlgebra.lift_of`, `OreLocalization.numeratorHom_inj`, `CommGrpCat.forget₂CommMon_preservesLimitsOfSize`, `Ideal.relNorm_singleton`, `Submonoid.inclusion_inj`, `CommGrpCat.forget_commGrp_preserves_mono`, `Representation.single_smul`, `SkewMonoidAlgebra.lift_single`, `CategoryTheory.Aut.toEnd_apply`, `AffineEquiv.linear_equivUnitsAffineMap_symm_apply`, `GrpCat.FilteredColimits.forget₂Mon_preservesFilteredColimits`, `Submonoid.LocalizationMap.lift_localizationMap_mk'`, `uliftPowersHom_symm_apply`, `MulEquiv.coprodAssoc_symm_apply_inr_inr`, `Ideal.map_relNorm`, `SkewMonoidAlgebra.lift_apply`, `Equiv.Perm.cycleType_ofSubtype`, `AddChar.coe_toMonoidHomEquiv`, `CircleDeg1Lift.coe_toOrderIso_symm`, `RootPairing.weylGroup_apply_root`, `MonCat.FilteredColimits.M.map_mk`, `Rack.envelAction_prop`, `CircleDeg1Lift.translate_iterate`, `CategoryTheory.IsCommMonObj.ofRepresentableBy`, `GroupExtension.Section.exists_mul_eq_mul_mul_inl`, `CommRing.Pic.mapRingHom_id_apply`, `CommGrpCat.forget₂Group_preservesLimits`, `AddEquiv.toMultiplicativeRight_apply_symm_apply`, `FreeGroup.map.mk`, `Matrix.GeneralLinearGroup.continuous_det`, `Algebra.algebraMap_intNorm_fractionRing`, `AddMonoidAlgebra.of_apply`, `LinearMap.det_eq_one_of_not_module_finite`, `Matrix.SpecialLinearGroup.map_apply_coe`, `CategoryTheory.PreGaloisCategory.toAut_hom_app_apply`, `IsPrimitiveRoot.coe_autToPow_apply`, `IntermediateField.restrictRestrictAlgEquivMapHom_apply`, `Algebra.norm_eq_prod_embeddings`, `zpowersMulHom_apply`, `NumberField.mixedEmbedding.exists_ne_zero_mem_ringOfIntegers_of_norm_le`, `Representation.prod_apply_apply`, `Matrix.coe_detMonoidHom`, `Representation.tprod_apply`, `invMonoidHom_apply`, `MonoidAlgebra.mvPolynomial_aeval_of_surjective_of_closure`, `AddMonoidHom.toMultiplicativeRight_symm_apply_apply`, `SemiRingCat.forget₂Mon_preservesLimitsOfSize`, `WithOne.mapMulHom_coe`, `AlternatingMap.map_congr_perm`, `MeasureTheory.Measure.map_linearMap_addHaar_eq_smul_addHaar`, `Monoid.Coprod.map_apply_inr`, `Associates.mkMonoidHom_apply`, `Monoid.PushoutI.NormalWord.prod_cons`, `FreeMonoid.lift_comp_of`, `Monoid.Coprod.prod_mk_fst_snd`, `Units.map_neg`, `CommRingCat.instIsRightAdjointCommMonCatForget₂RingHomCarrierMonoidHomCarrier`, `RingOfIntegers.isUnit_norm_of_isGalois`, `FreeAbelianGroup.liftAddEquiv_symm_apply`, `Representation.IntertwiningMap.isIntertwiningMap_of_mem_center`, `smulOneHom_apply`, `Abelianization.map_of`, `Monoid.fg_iff_exists_freeMonoid_hom_surjective`, `Monoid.CoprodI.mclosure_iUnion_range_of`, `CircleDeg1Lift.coe_toOrderIso_inv`, `map_finprod_of_preimage_one`, `Subgroup.subtype_apply`, `WithOne.mapMulHom_injective`, `mker_inl`, `Submonoid.closure_singleton_eq`, `LocallyConstant.coeFnMonoidHom_apply`, `unitSphereToUnits_apply_coe`, `apply_mnat`, `MonoidAlgebra.single_algebraMap_eq_algebraMap_mul_of`, `Subgroup.coe_subtype`, `FreeGroupBasis.lift_apply_apply`, `groupHomology.single_one_fst_sub_single_one_snd_mem_boundaries₂`, `Equiv.Perm.sign_prodExtendRight`, `coe_toPrincipalIdeal`, `WithOne.map_injective`, `CategoryTheory.IsCommMon.ofRepresentableBy`, `RootPairing.Equiv.coweightHom_injective`, `toOneHom_coe`, `submonoidMap_apply_coe`, `Algebra.map_intNormAux`, `Subgroup.subtype_injective`, `Monoid.PushoutI.of_apply_eq_base`, `Algebra.norm_localization`, `inv_apply`, `CategoryTheory.essImage_yonedaMon`, `DirichletCharacter.LSeries_changeLevel`, `IsBaseChange.det_endHom`, `SpecialLinearGroup.coe_inv`, `Con.comapQuotientEquivOfSurj_symm_mk'`, `MeasureTheory.Measure.addHaar_preimage_continuousLinearMap`, `Algebra.coe_norm_int`, `Monoid.PushoutI.lift_base`, `LinearEquiv.det_symm_mul_det`, `Equiv.Perm.subtypeEquivSubtypePerm_apply_coe`, `Representation.Coinvariants.mk_self_apply`, `CommGrpCat.instFullGrpCatForget₂MonoidHomCarrierCarrier`, `Equiv.Perm.sign_abs`, `Submonoid.mrange_inr`, `WeierstrassCurve.Affine.CoordinateRing.mk_XYIdeal'_neg_mul`, `Fin.sign_cycleIcc_of_ge`, `IsPrimitiveRoot.sub_one_norm_isPrimePow`, `MeasureTheory.distribHaarChar_apply`, `groupHomology.d₂₁_single_inv_self_ρ_sub_self_inv`, `transfer_def`, `groupHomology.chainsMap_f_single`, `CategoryTheory.yonedaMon_naturality_assoc`, `CommMonCat.instSmallElemForallObjCompMonCatForget₂MonoidHomCarrierCarrierForgetSections`, `GrpCat.SurjectiveOfEpiAuxs.g_apply_infinity`, `MvPolynomial.monomialOneHom_apply`, `ContinuousCohomology.Iobj_ρ_apply`, `coeFn_apply`, `MulChar.equivToUnitHom_symm_coe`, `WithZero.lift'_coe`, `Subgroup.mem_map_of_mem`, `MonoidAlgebra.mem_span_support`, `SkewMonoidAlgebra.of_apply`, `Units.coe_map_inv`, `IntermediateField.restrictRestrictAlgEquivMapHom_surjective`, `IntermediateField.AdjoinSimple.norm_gen_eq_one`, `Equiv.Perm.Basis.card_ofPermHom_support`, `fiberEquiv_symm_apply`, `coe_of_map_div`, `DirichletCharacter.changeLevel_one`, `Algebra.norm_eq_of_equiv_equiv`, `QuadraticAlgebra.norm_eq_one`, `groupCohomology.cocycles₂_map_one_snd`, `Monoid.PushoutI.NormalWord.prod_summand_smul`, `CategoryTheory.PreGaloisCategory.autIsoFibers_inv_app`, `AbsoluteValue.coe_toMonoidHom`, `LinearMap.det_prodMap`, `noncommPiCoprod_apply`, `Set.singletonMonoidHom_apply`, `TannakaDuality.FiniteGroup.sumSMulInv_apply`, `SubgroupClass.subtype_apply`, `Representation.mem_invariants_iff_of_forall_mem_zpowers`, `Dilation.ratioHom_apply`, `Equiv.Perm.Basis.ofPermHom_apply`, `CommMonCat.val_units_map_hom_apply`, `CommMonCat.coyonedaForget_inv_app_app`, `MultilinearMap.domCoprod_alternization_coe`, `unitSphereToUnits_injective`, `Matrix.SpecialLinearGroup.mapGL_injective`, `AddMonoidAlgebra.lift_apply`, `Representation.ofQuotient_coe_apply`, `MulEquiv.coprodPUnit_apply`, `RootPairing.Hom.coweightHom_injective`, `isIntegral_localization'`, `OnePoint.map_smul`, `NumberField.Units.complexEmbedding_injective`, `QuadraticAlgebra.norm_neg`, `CategoryTheory.PreGaloisCategory.continuous_mapAut_whiskeringRight`, `NumberField.Units.norm`, `IsPrimitiveRoot.norm_eq_one`, `Localization.mkHom_surjective`, `Representation.mem_invariants`, `SpecialLinearGroup.congr_linearEquiv_coe_apply`, `restrictHomKerEquiv_symm_coe_apply`, `Matrix.SpecialLinearGroup.mapGL_inj`, `monoidEndToAdditive_apply_apply`, `pow_apply`, `CategoryTheory.yonedaMon_naturality`, `Algebra.norm_eq_iff`, `GroupExtension.Section.exists_eq_inl_mul`, `IsPrimitiveRoot.autToPow_injective`, `map_cyclic`, `IsPrimitiveRoot.norm_toInteger_pow_sub_one_of_two`, `QuotientGroup.quotientQuotientEquivQuotientAux_mk`, `Equiv.Perm.sign_trans`, `iterate_map_frobenius`, `IsFreeGroup.lift_of`, `AlgHom.coe_toMonoidHom`, `Subgroup.noncommPiCoprod_apply`, `AffineEquiv.coe_homothetyUnitsMulHom_apply_symm`, `DirichletCharacter.Even.toUnitHom_eval_neg_one`, `AddMonoidHom.toMultiplicative_symm_apply_apply`, `DistribMulActionHom.coe_fn_coe'`, `MulChar.coe_mk`, `AddMonoidHom.coe_toMultiplicativeLeft`, `mrange_comp`, `RegularWreathProduct.left_inl`, `Monoid.CoprodI.lift_mrange_le`, `CommMonCat.val_inv_units_map_hom_apply`, `mem_eqLocusM`, `SpecialLinearGroup.det_eq_one`, `Submonoid.mrange_subtype`, `CommMonCat.forget₂_map_ofHom`, `IsPrimitiveRoot.norm_pow_sub_one_of_prime_ne_two`, `groupCohomology.cocycles₂_ρ_map_inv_sub_map_inv`, `toAdditiveLeft_apply_apply`, `CongruenceSubgroup.IsArithmetic.conj`, `noncommPiCoprod_mulSingle`, `DirichletCharacter.toUnitHom_eq_char'`, `Equiv.Perm.sign_sumCongr`, `DirichletCharacter.changeLevel_toUnitHom`, `Equiv.Perm.sign_eq_prod_prod_Iio`, `IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_ne_two'`, `Unitary.conjStarAlgAut_ext_iff'`, `MulEquiv.coprodAssoc_apply_inl_inl`, `GroupLike.valMonoidHom_apply`, `Monoid.PushoutI.NormalWord.cons_head`, `LinearMap.det_eq_det_toMatrix_of_finset`, `FiniteField.norm_surjective`, `PiTensorProduct.tprodMonoidHom_apply`, `FractionalIdeal.canonicalEquiv_mk0`, `map_finprod_mem`, `Abelianization.lift_apply_of`, `FreeMonoid.map_surjective`, `CategoryTheory.PreGaloisCategory.toAutHomeo_apply`, `toFun_eq_coe`, `CommRing.Pic.mapAlgebra_apply`, `Matrix.det_permutation`, `FreeGroup.injective_lift_of_ping_pong`, `Action.tensorUnit_ρ`, `Submonoid.iSup_map_mulSingle`, `CommGrpCat.mono_iff_injective`, `MulAction.stabilizerEquivStabilizer_apply`, `eq_iff_eq_on_generator`, `Monoid.Coprod.toProd_apply_inr`, `Rep.ofMulDistribMulAction_ρ_apply_apply`, `WithOne.mapMulHom_mapMulHom`, `SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom_quotMk`, `map_one₂`, `coe_prodMap`, `CommGrpCat.forget_isCorepresentable`, `CategoryTheory.essImage_yonedaGrp`, `NumberField.InfinitePlace.ComplexEmbedding.conjugate_sign`, `Ideal.Quotient.stabilizerHom_surjective_of_profinite`, `LinearMap.det_toLin'`, `coe_of_map_mul_inv`, `Submodule.linearEquiv_det_reflection`, `FreeGroup.map.id'`, `Monoid.CoprodI.Word.mem_smul_iff_of_ne`, `Matrix.SpecialLinearGroup.isClosedEmbedding_mapGLInt`, `Monoid.fg_range`, `RingOfIntegers.norm_algebraMap`, `GroupExtension.Section.rightHom_section`, `FreeGroup.map.comp`, `Monoid.CoprodI.Word.of_smul_def`, `Matrix.GeneralLinearGroup.map_inv`, `coe_toHomUnits`, `FreeGroup.lift_symm_apply`, `Submonoid.pi_le_iff`, `toMulEquiv_symm_apply`, `CategoryTheory.PreGaloisCategory.toAutMulEquiv_apply`, `Rep.indResAdjunction_counit_app_hom_hom`, `FDRep.hom_hom_action_ρ`, `CategoryTheory.Functor.mapAction_obj_ρ_apply`, `Action.FintypeCat.toEndHom_apply`, `QuotientGroup.mk'_apply`, `SemiRingCat.FilteredColimits.forget₂Mon_preservesFilteredColimits`, `IsGalois.normalAutEquivQuotient_apply`, `CategoryTheory.SingleObj.functor_map`, `QuotientGroup.rangeKerLift_surjective`, `Representation.FiniteCyclicGroup.coinvariantsKer_eq_range`, `groupHomology.mapCycles₁_comp_i_apply`, `prod_map_comap_prod'`, `Complex.det_conjAe`, `WeierstrassCurve.Affine.CoordinateRing.norm_smul_basis`, `Subgroup.HasDetOne.det_eq`, `Monoid.Coprod.lift_apply_inl`, `CategoryTheory.PreGaloisCategory.toAut_isHomeomorph`, `Algebra.norm_inv`, `TannakaDuality.FiniteGroup.leftRegular_apply`, `IsLocalization.lift_mk'`, `FiniteField.algebraMap_norm_eq_pow`, `GrpCat.forget₂Mon_preservesLimits`, `Submonoid.mk_mul_mk_inv_eq_one`, `Representation.apply_bijective`, `Equiv.Perm.viaEmbeddingHom_apply`, `CongruenceSubgroup.isArithmetic_conj_SL2Z`, `WeierstrassCurve.Affine.CoordinateRing.degree_norm_smul_basis`, `Submonoid.mk_inv_mul_mk_eq_one`, `Orientation.map_eq_det_inv_smul`, `CommMonCat.forget_map`, `Representation.linHom_apply`, `FreeMonoid.map_map`, `GrpCat.SurjectiveOfEpiAuxs.agree`, `noncommCoprod_unique`, `LocallyConstant.toContinuousMapMonoidHom_apply`, `MulAction.stabilizer_smul_eq_stabilizer_map_conj`, `Monoid.PushoutI.NormalWord.prod_base_smul`, `MonoidAlgebra.lift_apply'`, `unitary.toUnits_injective`, `MulChar.coe_toMonoidHom`, `ModularGroup.denom_S`, `CommMonCat.comp_apply`, `Ideal.Quotient.stabilizerHomSurjectiveAuxFunctor_aux`, `div_apply`, `fiberEquivKer_apply`, `CommMonCat.hom_forget₂_map`, `FiniteField.algebraMap_norm_eq_prod_pow`, `DistribMulAction.toModuleEnd_apply`, `MulAction.IsBlock.of_subgroup_of_conjugate`, `MonoidAlgebra.mapDomainBialgHom_apply`, `NumberField.IsCMField.unitsMulComplexConjInv_apply_torsion`, `Monoid.CoprodI.Word.cons_eq_smul`, `RegularWreathProduct.toPermInj`, `MulEquiv.coprodAssoc_apply_inl_inr`, `GroupSeminorm.coe_comp`, `Algebra.PreSubmersivePresentation.jacobian_eq_det_aevalDifferential`, `Polynomial.det_taylorLinearEquiv_toLinearMap`, `LinearEquiv.det_symm`, `Matrix.SpecialLinearGroup.isInducing_mapGL`, `Equiv.Perm.Basis.ofPermHom_mem_centralizer`, `Monoid.Coprod.fst_apply_inr`, `CoxeterSystem.rightInvSeq_concat`, `IsFractionRing.stabilizerHom_surjective`, `InfiniteGalois.proj_adjoin_singleton_val`, `MulAction.toPermHom_apply`, `Submonoid.le_comap_mulSingle_pi`, `MulEquiv.withOneCongr_apply`, `Units.unitary_eq`, `SpecialLinearGroup.centerEquivRootsOfUnity_symm_apply`, `Monoid.Coprod.swap_map`, `MulEquiv.coe_toMonoidHom`, `CommGrpCat.ofHom_apply`, `WithOne.lift_one`, `groupCohomology.cocycles₁_map_inv`, `Submonoid.coe_subtype`, `Submonoid.map_map`, `Equiv.Perm.sign_bij`, `mem_ker`, `MonoidAlgebra.freeAlgebra_lift_of_surjective_of_closure`, `WeakDual.CharacterSpace.toAlgHom_apply`, `GrpCat.forget_preservesLimits`, `GrpCat.hom_inv_apply`, `MulSemiringAction.toRingAut_apply`, `Ideal.Quotient.stabilizerHom_apply`, `coe_coe`, `CommMonCat.forget₂Mon_preservesLimitsOfSize`, `Submonoid.map_id`, `Rep.indToCoindAux_mul_fst`, `isLocalHom_comp`, `restrict_apply`, `CoxeterSystem.lengthParity_comp_simple`, `Action.FunctorCategoryEquivalence.inverse_obj_ρ_apply`, `Representation.asAlgebraHom_single_one`, `LinearMap.isUnit_det`, `RegularWreathProduct.fun_id`, `Fin.sign_cycleIcc_of_le`, `Rep.ihom_obj_ρ_apply`, `FreeMonoid.count_of`, `SpecialLinearGroup.det_coe`, `Representation.finsupp_single`, `Polynomial.Gal.restrictComp_surjective`, `map_inv₂`, `AddConstEquiv.toAddConstMapHom_apply`, `MeasureTheory.distribHaarChar_eq_of_measure_smul_eq_mul`, `SemidirectProduct.mul_def`, `coe_finsuppProd`, `QuotientGroup.mk'_eq_mk'`, `Equiv.Perm.support_ofSubtype`, `QuotientGroup.map_mk'`, `SemidirectProduct.rightHom_surjective`, `QuadraticAlgebra.norm_natCast`, `MulEquiv.coprodCongr_apply`, `TannakaDuality.FiniteGroup.equivApp_inv`, `mulAutArrow_apply_apply`, `Equiv.Perm.mem_alternatingGroup`, `Matrix.det_reindex`, `Subgroup.injective_noncommPiCoprod_of_iSupIndep`, `groupCohomology.norm_ofAlgebraAutOnUnits_eq`, `AddChar.toMonoidHom_apply`, `Equiv.Perm.sign_of_pow_two_eq_one`, `PresentedGroup.mk_eq_one_iff`, `QuadraticAlgebra.norm_coe`, `Matrix.IsHermitian.cfcAux_apply`, `CategoryTheory.PreGaloisCategory.toAut_surjective_of_isPretransitive`, `IsPrimitiveRoot.norm_pow_sub_one_eq_prime_pow_of_ne_zero`, `groupCohomology.mem_cocycles₂_iff`, `linear_isometry_complex`, `GroupExtension.rightHom_surjective`, `FreeMonoid.comp_lift`, `QuotientGroup.map_mk`, `CategoryTheory.PreGaloisCategory.autMulEquivAutGalois_symm_app`, `QuotientGroup.quotientMapSubgroupOfOfLe_mk`, `QuadraticAlgebra.isUnit_iff_norm_isUnit`, `unitsCenterToCenterUnits_injective`, `coe_invMonoidHom`, `HNNExtension.of_mul_t`, `IsMulIndecomposable.image_baseOf_inv_comp_eq`, `RatFunc.map_apply_div`, `Subgroup.normalizerMonoidHom_apply_apply_coe`, `mrange_id`, `IntermediateField.restrictRestrictAlgEquivMapHom_injective`, `RingOfIntegers.norm_norm`, `Monoid.PushoutI.ofCoprodI_of`, `map_neg`, `LocallyConstant.mapMonoidHom_apply`, `SubgroupClass.inclusion_mk`, `CuspForm.coe_translate`, `Rep.ofDistribMulAction_ρ_apply_apply`, `Equiv.Perm.sign_refl`, `CommMonCat.inv_hom_apply`, `PiTensorProduct.singleAlgHom_apply`, `HNNExtension.of_mul_inv_t`, `Units.isEmbedding_embedProduct`, `LinearMap.det_eq_zero_iff_ker_ne_bot`, `rotationOf_rotation`, `CommRingCat.commMon_forget₂_obj_coe`, `Submodule.natAbs_det_equiv`, `IsPrimitiveRoot.norm_pow_sub_one_of_prime_pow_ne_two`, `Finsupp.prod_hom_add_index`, `MonoidAlgebra.exists_finset_adjoin_eq_top`, `Equiv.Perm.ofSubtype_subtypePerm_of_mem`, `MonCat.comp_apply`, `TannakaDuality.FiniteGroup.equivApp_hom`, `Submonoid.LocalizationMap.isUnit_comp`, `Ideal.Quotient.stabilizerQuotientInertiaEquiv_mk`, `CommGrpCat.forget_reflects_isos`, `NumberField.Units.dirichletUnitTheorem.seq_norm_le`, `unitary.val_toUnits_apply`, `FreeMonoid.lift_eval_of`, `div_mem_ker_iff`, `MonoidAlgebra.smul_of`, `transfer_center_eq_pow`, `mrangeRestrict_surjective`, `CategoryTheory.yonedaGrp_naturality_assoc`, `QuotientGroup.lift_mk'`, `LinearEquiv.isUnit_det'`, `mulSingle_apply`, `Submonoid.subtype_injective`, `WittVector.dvd_sub_sum_teichmuller_iterateFrobeniusEquiv_coeff`, `controlled_prod_of_mem_closure_range`, `SemidirectProduct.left_inl`, `IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_eq`, `CategoryTheory.Groupoid.CategoryTheory.Functor.mapVertexGroup_apply`, `Subgroup.noncommPiCoprod_mulSingle`, `mulAutArrow_apply_symm_apply`, `AlgEquiv.toLinearMapHom_apply_apply`, `Module.Basis.orientation_comp_linearEquiv_eq_iff_det_pos`, `groupHomology.H1AddEquivOfIsTrivial_single`, `Pi.monoidHom_apply`, `Subgroup.map_centralizer_le_centralizer_image`, `GroupExtension.Section.exists_mul_eq_inl_mul_mul`, `groupCohomology.mem_cocycles₁_iff`, `Monoid.CoprodI.of_apply`, `PresentedGroup.mk_eq_mk_of_inv_mul_mem`, `coe_comp_rangeRestrict`, `AddMonoidAlgebra.lift_single`, `SemidirectProduct.inl_aut_inv`, `QuadraticAlgebra.coe_norm_eq_mul_star`, `SpecialLinearGroup.coe_mul`, `Monoid.Coprod.snd_swap`, `PowerSeries.coe_divXPowOrderHom`, `Algebra.intNorm_eq_of_isLocalization`, `MonCat.coe_comp`, `AddMonoidHom.coe_toMultiplicative'`, `WittVector.fontaineThetaModPPow_teichmuller`, `op_symm_apply_apply`, `GrpCat.coe_comp`, `Subgroup.mem_comap`, `GrpCat.hasLimit_iff_small_sections`, `CommMonCat.coe_id`, `CircleDeg1Lift.translate_apply`, `Traversable.foldl.unop_ofFreeMonoid`, `CommRing.Pic.mapAlgebra_self_apply`, `Matrix.SpecialLinearGroup.toGL_injective`, `AddMonoidAlgebra.lift_symm_apply`, `MulAut.conj_apply`, `CategoryTheory.Discrete.monoidalFunctor_ε`, `Algebra.norm_eq_norm_adjoin`, `Equiv.Perm.Basis.toPermHom_apply_toCentralizer`, `Unitary.conjStarAlgAut_apply`, `Monoid.Coprod.mrange_lift`, `IsPrimitiveRoot.norm_toInteger_sub_one_eq_one`, `restrict_eq_one_iff`, `Submodule.unitsToPic_apply`, `Subgroup.map_toSubmonoid`, `IntermediateField.LinearDisjoint.norm_algebraMap`, `FractionalIdeal.absNorm_eq`, `QuotientGroup.kerLift_injective`, `Action.leftRegularTensorIso_hom_hom`, `Representation.norm_comp_self`, `ext_iff`, `MonCat.forget_isCorepresentable`, `IsPrimitiveRoot.norm_sub_one_of_prime_ne_two'`, `Con.kerLift_range_eq`, `Module.Basis.orientation_comp_linearEquiv_eq_neg_iff_det_neg`, `EisensteinSeries.vecMul_SL2_mem_gammaSet`, `Equiv.Perm.ofSubtype_apply_of_mem`, `FundamentalGroup.mapOfEq_apply`, `MeasureTheory.AEEqFun.toGermMonoidHom_apply`, `AddEquiv.toMultiplicative_apply_apply`, `LinearIsometryEquiv.toLinearEquiv_smul`, `Unitary.toAlgEquiv_conjStarAlgAut`, `coe_ofClosureMEqTopLeft`, `Equiv.Perm.sign_mul`, `groupHomology.inhomogeneousChains.d_single`, `GroupExtension.Equiv.rightHom_comm`, `LinearEquiv.det_conj`, `RootPairing.Equiv.weightHom_apply`, `IsMulIndecomposable.baseOf_subset_one_lt`, `Subrepresentation.apply_mem_toSubmodule`, `LinearMap.det_map`, `LinearMap.hasEigenvalue_zero_tfae`, `FreeMonoid.toList_map`, `submonoidComap_apply_coe`, `Representation.coind'_apply_apply`, `WittVector.ghostComponentModPPow_teichmuller_coeff`, `PositiveLinearMap.gnsStarAlgHom_apply`, `map_closure`, `RingOfIntegers.isUnit_norm`, `coe_toAdditiveLeft`, `LinearMap.detAux_def'`, `inl_mono`, `SpecialLinearGroup.coe_pow`, `Equiv.Perm.Basis.ofPermHom_support`, `freeGroupEquivCoprodI_symm_apply`, `Action.FintypeCat.quotientToEndHom_mk`, `GroupExtension.Equiv.inl_comm`, `MeasureTheory.Measure.addHaar_image_linearMap`, `Matrix.SpecialLinearGroup.toLin_equiv.symm_toLinearMap_eq`, `QuotientGroup.quotientQuotientEquivQuotientAux_mk_mk`, `map_div`, `zpowGroupHom_apply`, `compHom'_apply_apply`, `QuotientGroup.congr_apply`, `groupCohomology.cochainsMap_f_2_comp_cochainsIso₂_apply`, `MulSemiringActionHom.map_one'`, `MulSemiringActionHom.coe_fn_coe`, `modularCyclotomicCharacter.unique`, `Set.up_image`, `Representation.asModuleEquiv_symm_map_rho`, `RegularWreathProduct.right_inl`, `QuotientGroup.con_ker_eq_conKer`, `Representation.ofModule_asModule_act`, `CommGrpCat.inv_hom_apply`, `LinearMap.eval_charpoly`, `Subgroup.conj_smul_le_of_le`, `IsCyclotomicExtension.Rat.galEquivZMod_restrictNormal_apply`, `NonUnitalAlgHom.addHomMk_coe`, `SubmonoidClass.coe_subtype`, `Monoid.foldlM.ofFreeMonoid_apply`, `FractionalIdeal.absNorm_span_singleton`, `ContinuousLinearEquiv.smulLeft_apply_apply`, `divMonoidHom_apply`, `Equiv.Perm.sign_symm_trans_trans`, `mulExact_of_comp_of_mem_range`, `IsCyclotomicExtension.norm_zeta_sub_one_of_prime_ne_two`, `AddMonoidAlgebra.lift_of`, `Algebra.norm_eq_prod_automorphisms`, `powersHom_apply`, `CongruenceSubgroup.finiteIndex_conjGL`, `CommRing.Pic.mapRingHom_mapRingHom`, `MulEquiv.coprodAssoc_apply_inr`, `AddMonoidHom.coe_toMultiplicative`, `IsPrimitiveRoot.sub_one_norm_eq_eval_cyclotomic`, `range_eq_top`, `AddMonoid.End.natCast_def`, `SetSemiring.down_imageHom`, `LinearMap.det_dualMap`, `Equiv.Perm.mem_ofSign`, `Rep.indToCoindAux_mul_snd`, `Matrix.SpecialLinearGroup.range_toGL`, `AffineEquiv.linearHom_apply`, `HNNExtension.t_mul_of`, `Equiv.Perm.ofSubtype_subtypePerm`, `finsuppProd_apply`, `Set.coe_singletonMonoidHom`, `IsFreeGroup.lift_symm_apply`, `Monoid.CoprodI.NeWord.prod_singleton`, `Matrix.SpecialLinearGroup.isEmbedding_toGL`, `mgraph_eq_mrange_prod`, `Representation.asAlgebraHom_of`, `Equiv.Perm.sigmaCongrRightHom_injective`, `MulAut.conj_inv_apply`, `LinearMap.detAux_comp`, `powersMulHom_apply`, `Matrix.SpecialLinearGroup.coe_GLPos_neg`, `Submonoid.le_prod_iff`, `mem_range`, `LinearMap.det_id`, `inl_apply`, `CircleDeg1Lift.coe_toOrderIso`, `CircleDeg1Lift.translate_zpow`, `sum_hom_units_eq_zero`, `toAdditiveRight_symm_apply_apply`, `IsLocalization.toInvSubmonoid_mul`, `AddMonoidHom.toMultiplicativeLeft_symm_apply_apply`, `IsCyclotomicExtension.norm_zeta_sub_one_of_isPrimePow`, `Unitary.conjStarAlgAut_symm_apply`, `Equiv.Perm.ofSubtype_apply_coe`, `Submodule.det_reflection`, `ClassGroup.mk_mk0`, `NumberField.mixedEmbedding.exists_ne_zero_mem_ideal_of_norm_le`, `LocallyConstant.constMonoidHom_apply`, `Rep.linearization_obj_ρ`, `Ideal.pointwise_smul_eq_comap`, `CategoryTheory.yonedaGrpObjIsoOfRepresentableBy_hom`, `MonoidAlgebra.singleHom_apply`, `MulEquiv.submonoidMap_symm_apply`, `GrpCat.forget₂_map`, `Unitary.toRingEquiv_conjStarAlgAut`, `Module.Basis.det_comp`, `OreLocalization.numeratorHom_apply`, `CategoryTheory.yonedaGrpObjIsoOfRepresentableBy_inv`, `rotation_trans`, `Unitary.val_toUnits_apply`, `DirichletCharacter.changeLevel_factorsThrough`, `IsPrimitiveRoot.norm_sub_one_of_prime_ne_two`, `ClassGroup.mk0_eq_mk0_iff_exists_fraction_ring`, `Complex.linearEquiv_det_conjAe`, `Set.conj_mem_fixingSubgroup`, `mulMonoidHom_apply`, `groupHomology.lsingle_comp_chainsMap_f`, `LinearMap.det_restrictScalars`, `MeasureTheory.Measure.addHaar_image_continuousLinearEquiv`, `map_sub_swap`, `Submonoid.LocalizationMap.mulEquivOfLocalizations_symm_apply`, `AddChar.toMonoidHomEquiv_apply`, `monoidEndToAdditive_symm_apply_apply`, `apply_mint`, `ModularForm.SL_slash_def`, `NumberField.Units.complexEmbedding_apply`, `GrpCat.SurjectiveOfEpiAuxs.h_apply_fromCoset'`, `ClassGroup.mk_eq_one_iff`, `MultilinearMap.alternatization_apply`, `pinGroup.toUnits_injective`, `groupCohomology.cochainsMap_f`, `inhomogeneousCochains.d_hom_apply`, `PresentedGroup.mk_surjective`, `Equiv.Perm.OnCycleFactors.cycleType_kerParam_apply_apply`, `rangeRestrict_surjective`, `monoidHomOfTendsto_apply`, `coe_graph`, `MulAction.map_stabilizer_le`, `MulSemiringActionHom.map_mul'`, `CommGrpCat.coe_id`, `IsPrimitiveRoot.norm_eq_one_of_linearly_ordered`, `CommGrpCat.id_apply`, `MulEquiv.punitCoprod_symm_apply`, `MulEquiv.coprodComm_symm_apply`, `FixedPoints.toAlgAut_surjective`, `coe_mulSingle`, `Matrix.IsHermitian.spectral_theorem`, `AddMonoidAlgebra.liftNC_smul`, `IsInvariantSubring.coe_subtypeHom'`, `SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descMonoidHom_apply_eq`, `mker_inr`, `toPrincipalIdeal_eq_iff`, `groupHomology.d₂₁_single_self_inv_ρ_sub_inv_self`, `ClassGroup.norm_lt`, `Complex.expMonoidHom_apply`, `LinearMap.det_eq_sign_charpoly_coeff`, `RingOfIntegers.coe_norm`, `Function.MulExact.iff_monoidHom_rangeRestrict`, `groupHomology.single_ρ_self_add_single_inv_mem_boundaries₁`, `PowerSeries.map_invUnitsSub`, `Algebra.smul_units_def`, `SemidirectProduct.right_inr`, `groupHomology.H1ToTensorOfIsTrivial_H1π_single`, `GrpCat.SurjectiveOfEpiAuxs.h_apply_infinity`, `SpecialLinearGroup.centerEquivRootsOfUnity_apply`, `noncommPiCoprod_mrange`, `Equiv.Perm.decomposeOption_symm_sign`, `WeierstrassCurve.Affine.CoordinateRing.mk_XYIdeal'_mul_mk_XYIdeal'`, `Action.ρ_self_inv_apply`, `Monoid.Coprod.mk_of_inr`, `Monoid.PushoutI.Reduced.exists_normalWord_prod_eq`, `Matrix.GeneralLinearGroup.map_swap`, `CommGrpCat.comp_apply`, `NumberField.IsCMField.unitsMulComplexConjInv_apply`, `AddConstEquiv.toPerm_apply`, `Representation.isTrivial_def`, `NonUnitalAlgHom.toFun_eq_coe`, `Equiv.Perm.subtypeCongrHom_apply`, `coe_compAddChar`, `IsLocalization.toInvSubmonoid_surjective`, `IsPrimitiveRoot.prime_norm_toInteger_sub_one_of_prime_ne_two`, `QuadraticAlgebra.norm_intCast`, `QuadraticAlgebra.mker_norm_eq_unitary`, `FreeMonoid.mrange_lift`, `Ideal.norm_mem_spanNorm`, `MulEquiv.toAdditive_symm_apply_apply`, `IsPrimitiveRoot.norm_pow_sub_one_two`, `Algebra.norm_complex_apply`, `RootPairing.Equiv.indexHom_apply`, `GroupLike.val_toUnits_apply`, `uliftPowersHom_apply_apply`, `associated_norm_prod_smith`, `ClassGroup.mk_eq_mk`, `restrict_mker`, `CategoryTheory.ActionCategory.curry_apply_left`, `CommMonCat.instIsRightAdjointForgetMonoidHomCarrier`, `Action.ρ_one`, `GrpCat.ofHom_apply`, `Unitary.val_inv_toUnits_apply`, `MonCat.FilteredColimits.forget_preservesFilteredColimits`, `commute_inl_inr`, `GroupExtension.Section.exists_eq_mul_inl`, `WittVector.fontaineTheta_teichmuller`, `MulSemiringActionHom.coe_polynomial`, `SpecialLinearGroup.toLinearEquiv_symm_apply`, `CategoryTheory.PresheafOfGroups.OneCochain.ev_precomp`, `GrpCat.mono_iff_injective`, `pinGroup.mem_lipschitzGroup`, `CommMonCat.forget_reflects_isos`, `Action.ρAut_apply_hom`, `Real.map_linearMap_volume_pi_eq_smul_volume_pi`, `CoxeterSystem.lengthParity_simple`, `coe_mk`, `CategoryTheory.MonObj.ofRepresentableBy_mul`, `MonoidAlgebra.support_gen_of_gen`, `GrpCat.instSmallElemForallObjCompMonCatForget₂MonoidHomCarrierCarrierForgetSections`, `coe_id`, `Equiv.Perm.OnCycleFactors.coe_toPermHom`, `SemidirectProduct.rightHom_eq_right`, `CommGrpCat.coyonedaForget_hom_app_app_hom`, `DistribMulActionHom.instDistribMulActionSemiHomClassCoeMonoidHom`, `TannakaDuality.FiniteGroup.sumSMulInv_single_id`, `transferCenterPow_apply`, `StarMonoidHom.coe_mk`, `Equiv.Perm.sign_permCongr`, `Rep.leftRegularHom_hom_single`, `MeasureTheory.Measure.map_linearMap_addHaar_pi_eq_smul_addHaar`, `WithOne.map_map`, `map_mrange`, `Representation.free_single_single`, `IsPrimitiveRoot.norm_toInteger_sub_one_of_prime_ne_two`, `coe_rangeRestrict`, `IsLocalRing.surjective_units_map_of_local_ringHom`, `Monoid.Coprod.mker_swap`, `Representation.IntertwiningMap.isIntertwining'`, `Finset.le_card_quotient_mul_sq_inter_subgroup`, `mker_fst`, `SlashInvariantForm.slash_action_eqn_SL''`, `Action.Hom.comm_assoc`, `HNNExtension.hom_ext_iff`, `QuotientGroup.congr_mk'`, `GrpCat.FilteredColimits.colimit_mul_mk_eq`, `LinearMap.det_eq_one_of_finrank_eq_zero`, `ContinuousMultilinearMap.alternatization_apply_toContinuousMultilinearMap`, `ZMod.unitsMap_val`, `AddMonoidAlgebra.lift_of'`, `RootPairing.Equiv.coweightHom_op`, `Representation.le_comap_invariants`, `Con.comap_eq`, `Equiv.Perm.sumCongrHom_injective`, `mk'_apply`, `Representation.invariants_eq_inter`, `Abelianization.equivOfComm_apply`, `BoundedContinuousFunction.coeFnMonoidHom_apply`, `AffineMap.linearHom_apply`, `DistribMulActionHom.map_zero'`, `Algebra.norm_eq_one_of_not_exists_basis`, `Traversable.foldr.ofFreeMonoid_comp_of`, `ValuationSubring.surjective_unitGroupToResidueFieldUnits`, `isClosed_range_coe`, `FreeGroup.prod_mk`, `CategoryTheory.PreGaloisCategory.autMulEquivAutGalois_π`, `Algebra.norm_eq_of_algEquiv`, `AddMonoidHom.toMultiplicative_apply_apply`, `DistribMulActionHom.map_add'`, `GrpCat.inv_hom_apply`, `PresentedGroup.one_of_mem`, `GroupExtension.inl_conjAct_comm`, `WeierstrassCurve.Affine.Point.toClass_some`, `Module.Basis.det_basis`, `Monoid.CoprodI.iSup_mrange_of`, `Representation.norm_self_apply`, `Monoid.CoprodI.Word.equivPair_head_smul_equivPair_tail`, `MonCat.val_units_map_hom_apply`, `MulAut.conj_symm_apply`, `MonoidAlgebra.of_commute`, `Subgroup.HasDetPlusMinusOne.det_eq`, `map_iterateFrobenius`, `Matrix.GeneralLinearGroup.map_mul_map_inv`, `SubgroupClass.inclusion_inclusion`, `NumberField.exists_ideal_in_class_of_norm_le`, `CommGrpCat.forget₂Group_preservesLimitsOfSize`, `linearEquiv_det_rotation`, `Matrix.GeneralLinearGroup.val_inv_det_apply`, `Submonoid.comap_comap`, `SemidirectProduct.rightHom_inr`, `SpecialLinearGroup.toGeneralLinearGroup_injective`, `Monoid.CoprodI.Word.equivPair_smul_same`, `IsLocalization.toInvSubmonoid_eq_mk'`, `MonoidAlgebra.mem_ideal_span_of_image`, `SeparationQuotient.mkMonoidHom_apply`, `fst_mono`, `prod_apply`, `MonoidAlgebra.lift_apply`, `SubgroupClass.coe_subtype`, `Algebra.intNorm_zero`, `CommMonCat.forget_preservesLimits`, `IsPrimitiveRoot.card_quotient_toInteger_sub_one`, `InfiniteGalois.toAlgEquivAux_eq_liftNormal`, `Projectivization.smul_def`, `ModularGroup.im_smul_eq_div_normSq`, `Con.coe_mk'`, `Monoid.CoprodI.Word.prod_cons`, `Matrix.SpecialLinearGroup.coe_GL_coe_matrix`, `restrictHom_apply`, `FreeGroup.map.of`, `IntermediateField.AdjoinSimple.norm_gen_eq_prod_roots`, `LinearMap.det_smul`, `MonoidAlgebra.of_injective`, `Ideal.rootsOfUnityMapQuot_apply`, `Equiv.Perm.sigmaCongrRightHom_apply`, `ext_mint_iff`, `Representation.self_inv_apply`, `eq_iff`, `GrpCat.instFullMonCatForget₂MonoidHomCarrierCarrier`, `CategoryTheory.Functor.mapEnd_apply`, `GroupSeminorm.comp_apply`, `CongruenceSubgroup.mem_conjGL`, `Traversable.foldrm.ofFreeMonoid_comp_of`, `MeasureTheory.Measure.addHaar_preimage_linearMap`, `SubMulAction.fixingSubgroup_map_conj_eq`, `Matrix.SpecialLinearGroup.map_intCast_inj`, `AffineMap.coe_homothetyHom`, `Representation.isTrivial_apply`, `Monoid.Coprod.clift_mk`, `Ideal.natAbs_det_equiv`, `Matrix.SpecialLinearGroup.coe_to_GLPos_to_GL_det`, `MonCat.Colimits.cocone_naturality_components`, `map_inv`, `coe_ofClosureMEqTopRight`, `QuotientGroup.exists_norm_mul_lt`, `FreeMonoid.hom_map_lift`, `Subgroup.inclusion_inj`, `Matrix.SpecialLinearGroup.toLin'_to_linearMap`, `Representation.IsIntertwiningMap.isIntertwining`, `IsFractionRing.fieldEquivOfAlgEquivHom_injective`, `CommMonCat.FilteredColimits.forget₂Mon_preservesFilteredColimits`, `MeasureTheory.Measure.addHaar_preimage_continuousLinearEquiv`, `Module.Basis.map_orientation_eq_det_inv_smul`, `GrpCat.forget_grp_preserves_epi`, `ker_eq_bot_iff`, `IsDedekindDomain.HeightOneSpectrum.valuationOfNeZero_eq`, `Monoid.Coprod.mrange_mk`, `GrpCat.SurjectiveOfEpiAuxs.h_apply_fromCoset`, `groupHomology.single_one_snd_sub_single_one_snd_mem_boundaries₂`, `NonUnitalAlgHom.map_mul'`, `Matrix.det_mul_aux`, `Rep.unit_iso_comm`, `SkewMonoidAlgebra.of_injective`, `Rep.leftRegularHomEquiv_symm_single`, `CommMonCat.coyoneda_map_app`, `rangeRestrict_injective_iff`, `Ideal.spanNorm_singleton`, `coe_toAdditive`, `ModularGroup.tendsto_lcRow0`, `cosetToCuspOrbit_apply_mk`, `MulEquiv.coprodComm_apply`, `NumberField.exists_ne_zero_mem_ringOfIntegers_of_norm_le_mul_sqrt_discr`, `SubgroupClass.coe_inclusion`, `coe_dfinsuppProd`, `FreeGroupBasis.lift_symm_apply`, `FreeMonoid.lift_symm_apply`, `Monoid.Coprod.inv_def`, `Unitary.conjStarAlgAut_mul_apply`, `Matrix.GeneralLinearGroup.val_inv_scalar_apply`, `smulMonoidHom_apply`, `map_one`, `ValuationSubring.unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk_apply`, `SubmonoidClass.subtype_injective`, `OnePoint.exists_mem_SL2`, `Monoid.PushoutI.NormalWord.cons_eq_smul`, `IsDedekindDomain.selmerGroup.fromUnitLift_injective`, `DistribMulActionHom.coe_fn_coe`, `Polynomial.Gal.restrict_surjective`, `LinearMap.det_comp`, `AffineEquiv.coe_homothetyUnitsMulHom_apply`, `groupHomology.chainsMap_f_hom`, `CategoryTheory.PreGaloisCategory.autEmbedding_apply`, `QuotientGroup.map_id_apply`, `Matrix.det_permute'`, `DistribMulAction.toAddAut_apply`, `cyclotomicCharacter.toZModPow_toFun`, `MulDistribMulAction.toMulAut_apply`, `Localization.mulEquivOfQuotient_apply`, `LinearMap.transvection.det`, `Equiv.Perm.ofSubtype_eq_iff`, `RootPairing.Equiv.weightHom_injective`, `Monoid.Coprod.swap_eq_one`, `Matrix.mem_glpos`, `Polynomial.Gal.galActionHom_bijective_of_prime_degree`, `Rep.ofHom_ρ`, `FreeAbelianGroup.liftMonoid_coe`, `map_finprod_Prop`, `HNNExtension.of_injective`, `SL_reduction_mod_hom_val`, `RingHom.coe_mk`, `Ideal.torsionMapQuot_injective`, `Equiv.Perm.ofSubtype_swap_eq`, `Monoid.Coprod.snd_apply_inl`, `pinGroup.mem_iff`, `Pi.coe_evalMonoidHom`, `LinearEquiv.det_mul_det_symm`, `Monoid.Coprod.lift_swap`, `Function.MulExact.iff_of_ladder_mulEquiv`, `DirichletCharacter.changeLevel_eq_one_iff`, `id_apply`, `Units.isClosedEmbedding_embedProduct`, `Monoid.Coprod.fst_swap`, `Matrix.GeneralLinearGroup.val_map_apply`, `RegularWreathProduct.rightHom_eq_right`, `Algebra.norm_one_add_smul`, `groupCohomology.mapCocycles₁_comp_i_apply`, `inl_strictMono`, `FreeGroup.map.id`, `Submonoid.FG.map`, `InfiniteGalois.finGaloisGroupFunctor_map_proj_eq_proj`, `Algebra.norm_apply`, `Algebra.norm_eq_zero_iff'`, `Rep.trivial_def`, `Polynomial.det_taylorLinearEquiv`, `WeierstrassCurve.Affine.CoordinateRing.coe_norm_smul_basis`, `UpperHalfPlane.det_J`, `Monoid.Coprod.mk_of_inl`, `Submonoid.closure_image_isMulIndecomposable_baseOf`, `DirichletCharacter.FactorsThrough.existsUnique`, `Sylow.coe_subgroup_smul`, `IsLocalRing.ResidueField.mapAut_apply`, `Monoid.Coprod.snd_toProd`, `QuadraticAlgebra.norm_def`, `Traversable.Free.map_eq_map`, `Finset.card_pow_quotient_mul_pow_inter_subgroup_le`, `InfiniteGalois.restrictNormalHom_continuous`, `MultilinearMap.alternatization_coe`, `Submonoid.mrange_snd`, `MulEquiv.toAdditive_symm_apply_symm_apply`, `MonCat.coe_id`, `ClassGroup.exists_mk0_eq_mk0`, `LinearMap.det_conj`, `Rep.hom_comm_apply`, `CongruenceSubgroup.mem_conjGL'`, `GrpCat.FilteredColimits.forget_preservesFilteredColimits`, `PreTilt.untilt_iterate_frobeniusEquiv_symm_pow`, `Equiv.Perm.ofSubtype_apply_of_not_mem`, `MulEquiv.punitCoprod_apply`, `SubgroupClass.inclusion_right`, `AddMonoid.End.intCast_def`, `GrpCat.forget_preservesLimitsOfShape`, `MonCat.FilteredColimits.colimit_mul_mk_eq`, `Equiv.Perm.OnCycleFactors.toPermHom_apply`, `IsPrimitiveRoot.norm_toInteger_pow_sub_one_of_prime_pow_ne_two`, `FiniteField.unitsMap_norm_surjective`, `CommGrpCat.forget_commGrp_preserves_epi`, `FiniteGrp.ofHom_apply`, `Monoid.PushoutI.NormalWord.of_smul_eq_smul`, `Submonoid.LocalizationMap.lift_left_inverse`, `MonoidAlgebra.liftNC_smul`, `Algebra.algebraMap_intNorm_of_isGalois`, `iteratedWreathToPermHomInj`, `AddMonoidAlgebra.of_injective`, `Rep.FiniteCyclicGroup.homResolutionIso_inv_f_hom_apply_hom_hom_apply`, `ClassGroup.mk0_surjective`, `AddAut.mulLeft_apply_apply`, `Localization.mkHom_apply`, `groupCohomology.cochainsMap_f_hom`, `Equiv.Perm.ofSubtype_injective`, `Units.coeHom_apply`, `AlgEquiv.smul_units_def`, `GrpCat.ext_iff`, `lipschitzGroup.coe_mem_iff_mem`, `eval_apply_apply`, `SubMulAction.ofFixingSubgroupEmpty_equivariantMap_bijective`, `ModularGroup.exists_one_half_le_im_smul_and_norm_denom_le`, `cyclotomicCharacter.spec`, `CommMonCat.id_apply`, `abelianizationCongr_of`, `Monoid.CoprodI.Word.prod_rcons`, `MulSemiringAction.toAlgAut_apply`, `MeasureTheory.distribHaarChar_eq_div`, `inr_strictMono`, `SubMulAction.fixingSubgroupEquivFixingSubgroup_coe_apply`, `Representation.toCoinvariants_mk`, `Subgroup.mem_map`, `GroupLike.val_inv_toUnits_apply`, `Submonoid.mrange_inl`, `CommGrpCat.coyonedaForget_inv_app_app`, `ext_mnat_iff`, `MonCat.val_inv_units_map_hom_apply`, `Representation.IsTrivial.out`, `QuadraticAlgebra.norm_one`, `Matrix.SpecialLinearGroup.toLin_equiv.toLinearMap_eq`, `DirichletCharacter.changeLevel_self_toUnitHom`, `CategoryTheory.PreGaloisCategory.autEmbedding_range`, `IsGalois.map_fixingSubgroup`, `Units.continuous_embedProduct`, `NumberField.isUnit_iff_norm`, `CommMonCat.forget₂Mon_preservesLimits`, `SpecialLinearGroup.toGeneralLinearGroup_toLinearEquiv_apply`, `ModularForm.prod_fintype_slash`, `coe_mgraph`, `PowerSeries.coe_orderHom`, `one_apply`, `Matrix.SpecialLinearGroup.coe_GLPos_coe_GL_coe_matrix`, `GrpCat.SurjectiveOfEpiAuxs.h_apply_fromCoset_nin_range`, `coe_fst`, `MonoidAlgebra.single_eq_algebraMap_mul_of`, `LinearMap.det_zero''`, `Representation.coind_apply`, `CommMonCat.coyonedaForget_hom_app_app_hom`, `Submonoid.mrange_fst`, `Con.mk'_ker`, `SubmonoidClass.subtype_apply`, `Matrix.GeneralLinearGroup.map_apply`, `Algebra.norm_eq_prod_roots`, `MeasureTheory.distribHaarChar_mul`, `MeasureTheory.Measure.addHaar_preimage_linearEquiv`, `map_mul_inv`, `OreLocalization.numerator_isUnit`, `Real.expMonoidHom_apply`, `Algebra.norm_zero`, `Equiv.Perm.sign_extendDomain`, `SubMulAction.of_fixingSubgroupEmpty_mapScalars_surjective`, `Traversable.foldl.ofFreeMonoid_comp_of`, `SemidirectProduct.inl_inj`, `Equiv.Perm.cycleFactorsFinset_conj`, `MulEquiv.coprodCongr_symm_apply`, `Monoid.CoprodI.Word.mem_smul_iff`, `WittVector.teichmuller_coeff_zero`, `Monoid.Coprod.swap_bijective`, `Representation.ind_mk`, `Action.tensor_ρ`, `Monoid.CoprodI.Word.smul_eq_of_smul`, `coe_mrangeRestrict`, `SkewMonoidAlgebra.single_eq_algebraMap_mul_of`, `WeierstrassCurve.map_Δ'`, `LinearMap.det_zero'`, `groupHomology.d₂₁_single_one_snd`, `GroupExtension.Equiv.rightHom_map`, `LinearEquiv.coe_inv_det`, `IsGaloisGroup.mulEquivAlgEquiv_symm_apply`, `CategoryTheory.Discrete.monoidalFunctor_obj`, `Algebra.norm_eq_one_of_not_module_finite`, `AlgEquiv.restrictNormalHom_apply`, `InfiniteGalois.mk_toAlgEquivAux`, `ModularForm.slash_def`, `Algebra.norm_norm`, `FractionalIdeal.absNorm_div_norm_eq_absNorm_div_norm`, `CommGrpCat.one_apply`, `WithZero.map'_surjective_iff`, `CategoryTheory.PreGaloisCategory.AutGalois.π_apply`, `map_dvd`, `CategoryTheory.Functor.homMonoidHom_apply`, `groupHomology.d₃₂_single_one_snd`, `NumberField.IsCMField.indexRealUnits_eq_two_iff`, `Circle.coeHom_apply`, `DistribMulActionHom.map_smulₑ`, `Polynomial.Splits.toPermHom_apply_eq_one_or_isSwap_of_ncard_le_of_mem_inertia`, `SemidirectProduct.inv_left`, `FDRep.Iso.conj_ρ`, `map_isConj`, `MulEquiv.coprodAssoc_symm_apply_inl`, `FractionalIdeal.map_canonicalEquiv_mk0`, `DistribMulActionHom.toFun_eq_coe`, `Submonoid.comap_id`, `ModularForm.is_invariant_one'`, `Traversable.foldMap_hom`, `GroupExtension.Section.rightInverse_rightHom`, `IsCyclotomicExtension.norm_zeta_eq_one`, `det_rotation`, `sign_finRotate`, `Monoid.Coprod.mclosure_range_inl_union_inr`, `Rep.coinvariantsTensorIndHom_mk_tmul_indVMk`, `Units.embed_product_star`, `uliftZPowersHom_apply_apply`, `Equiv.Perm.IsCycle.sign`, `IsPrimitiveRoot.autToPow_spec`, `Equiv.Perm.sign_inv`, `MulChar.coe_toUnitHom`, `coe_range`, `InfiniteGalois.proj_of_le`, `ClassGroup.mk_eq_one_of_coe_ideal`, `GrpCat.forget₂_map_ofHom`, `Monoid.Foldr.ofFreeMonoid_apply`, `map_finprod`, `QuadraticAlgebra.norm_star`, `FreeAbelianGroup.ofMulHom_coe`, `Polynomial.Gal.card_complex_roots_eq_card_real_add_card_not_gal_inv`, `Matrix.IsHermitian.star_mul_self_mul_eq_diagonal`, `GrpCat.one_apply`, `ext_iff₂`, `Matrix.SpecialLinearGroup.toLin'_injective`, `Equiv.Perm.IsThreeCycle.sign`, `ValuativeExtension.mapPosSubmonoid_apply_coe`, `map_zpowers`, `Con.map_apply`, `SpecialLinearGroup.coe_toGeneralLinearGroup_apply`, `IsCyclotomicExtension.Rat.associated_norm_zeta_sub_one`, `LinearMap.det_ring`, `AlternatingMap.domDomCongr_perm`, `LinearMap.coe_det`, `Subgroup.closure_preimage_eq_top`, `CategoryTheory.Discrete.monoidalFunctor_μ`, `GrpCat.forget_map`, `Equiv.Perm.IsCycle.forall_commute_iff`, `Matrix.GeneralLinearGroup.map_one`, `MulAut.conjNormal_inv_apply`, `Equiv.Perm.IsSwap.sign_eq`, `Submodule.mulExact_unitsMap_spanSingleton_unitsToPic`, `Con.kerLift_injective`, `Monoid.CoprodI.of_leftInverse`, `subgroupMap_surjective`, `IsCyclotomicExtension.norm_zeta_pow_sub_one_two`, `op_apply_apply`, `Representation.linHom.mem_invariants_iff_comm`, `Matrix.GeneralLinearGroup.map_mul`, `restrict_surjective`, `rotation_symm`, `modularCyclotomicCharacter'.spec'`, `Subgroup.conj_smul_eq_self_of_mem`, `monoidHomOfMemClosureRangeCoe_apply`, `Representation.IntertwiningMap.isIntertwining`, `WeierstrassCurve.inv_map_Δ'`, `groupHomology.mapCycles₂_comp_i_apply`, `PiTensorProduct.mapMonoidHom_apply`, `CoxeterSystem.lengthParity_eq_ofAdd_length`, `OrderMonoidHom.coe_monoidHom`, `Submonoid.prod_le_iff`, `QuotientGroup.quotientBot_apply`, `GroupExtension.inl_injective`, `Polynomial.Splits.surjective_toPermHom_of_iSup_inertia_eq_top`, `Matrix.SpecialLinearGroup.coe_matrix_coe`, `CommGrpCat.instReflectsIsomorphismsGrpCatForget₂MonoidHomCarrierCarrier`, `QuadraticAlgebra.algebraMap_norm_eq_mul_star`, `Matrix.SpecialLinearGroup.mapGL_coe_matrix`, `CategoryTheory.PreGaloisCategory.endEquivAutGalois_π`, `HNNExtension.NormalWord.of_smul_eq_smul`, `AddMonoidHom.toMultiplicativeLeft_apply_apply`, `RingOfIntegers.algebraMap_norm_algebraMap`, `LinearMap.det_eq_det_mul_det`, `RestrictedProduct.evalMonoidHom_apply`, `CommGrpCat.forget_preservesLimitsOfSize`, `IsLocalization.mul_toInvSubmonoid`, `Rep.freeLift_hom`, `FreeMonoid.lift_apply`, `ProfiniteGrp.toLimitFun_continuous`, `Equiv.Perm.sign_prodCongrRight`, `AlgHom.toEnd_apply`, `Action.isContinuous_def`, `SemidirectProduct.mulEquivSubgroup_symm_apply`, `CommGrpCat.forget₂_grp_map_ofHom`, `LinearEquiv.det_coe_symm`, `CategoryTheory.Functor.mapAction_map_hom`, `Subgroup.coe_subgroupOf`, `zpow_apply`, `groupHomology.d₁₀_single_inv`, `groupHomology.mkH1OfIsTrivial_apply`, `freeGroupEquivCoprodI_apply`, `mker_one`, `Representation.ofMulDistribMulAction_apply_apply`, `restrictRootsOfUnity_coe_apply`, `FixedPoints.toAlgAut_bijective`, `Unitary.conjStarAlgAut_symm_unitaryLinearIsometryEquiv`, `ProfiniteGrp.diagram_map`, `map_multiset_prod`, `CircleDeg1Lift.translationNumber_translate`, `image_commutatorSet_closureCommutatorRepresentatives`, `Equiv.Perm.IsCycle.commute_iff`, `dfinsuppProd_apply`, `InfiniteGalois.algEquivToLimit_continuous`, `AddAut.mulLeft_apply_symm_apply`, `LinearMap.det_toMatrix'`, `Equiv.Perm.sign_eq_sign_of_equiv`, `Algebra.PowerBasis.norm_gen_eq_prod_roots`, `Equiv.Perm.sign_one`, `MulChar.ofUnitHom_coe`, `Matrix.SpecialLinearGroup.det_mapGL`, `coe_toAdditiveRight`, `IsScalarTower.AlgEquiv.restrictNormalHom_comp_apply`, `MonoidAlgebra.lift_def`, `CommGrpCat.coyoneda_obj_map`, `LinearMap.det_cases`, `Subgroup.HasDetPlusMinusOne.abs_det`, `CommRingCat.commMon_forget₂_map`, `ClassGroup.equiv_mk0`, `GroupExtension.Splitting.rightHom_splitting`, `ModularForm.SL_slash_apply`, `Rep.freeLift_hom_single_single`, `Subgroup.inclusion_injective`, `Subalgebra.LinearDisjoint.norm_algebraMap`, `Ideal.relNorm_apply`, `Action.leftDual_ρ`, `Polynomial.Gal.restrictProd_injective`, `NumberField.exists_ne_zero_mem_ideal_of_norm_le_mul_sqrt_discr`, `Monoid.PushoutI.NormalWord.Transversal.injective`, `Algebra.norm_algebraMap`, `EisensteinSeries.eisSummand_SL2_apply`, `MulEquiv.Monoid.End_apply`, `NumberField.Ideal.tendsto_norm_le_and_mk_eq_div_atTop`, `Equiv.Perm.sumCongrHom_apply`, `Submonoid.divPairs_comap`, `FreeAbelianGroup.liftAddEquiv_apply_apply`, `CommGrpCat.forget₂_commMonCat_map_ofHom`, `GrpCat.forget_preservesLimitsOfSize`, `GroupExtension.Section.rightHom_comp_section`, `Submonoid.fromCommLeftInv_apply`, `Monoid.Coprod.mk_surjective`, `WittVector.teichmuller_mul_pow_coeff`, `CommMonCat.FilteredColimits.forget_preservesFilteredColimits`, `toAdditiveLeft_symm_apply_apply`, `toHomPerm_apply_apply`, `QuotientGroup.mk'_surjective`, `Submonoid.closure_image_one_lt_and_isMulIndecomposable`, `groupHomology.d₂₁_single`, `BoundedContinuousFunction.charMonoidHom_apply`, `Submonoid.map_inl`, `CategoryTheory.ActionCategory.curry_apply_right`, `Monoid.CoprodI.Word.rcons_eq_smul`, `Equiv.Perm.extendDomainHom_apply`, `NumberField.mixedEmbedding.fundamentalCone.sum_expMap_symm_apply`, `Equiv.Perm.ofSubtype_support_disjoint`, `Group.fg_iff_exists_freeGroup_hom_surjective`, `DirichletCharacter.Odd.toUnitHom_eval_neg_one`, `Pi.evalStarAlgHom_apply`, `MulEquiv.coprodAssoc_symm_apply_inr_inl`, `subgroupComap_apply_coe`, `Monoid.Coprod.lift_apply_mk`, `Matrix.det_apply'`, `LinearMap.not_hasEigenvalue_zero_tfae`, `Algebra.intNorm_map_algEquiv`, `AddMonoidAlgebra.lift_def`, `Ideal.norm_mem_relNorm`, `Rep.applyAsHom_hom`, `OrderMonoidHom.coe_mk`, `Algebra.intNorm_intNorm`, `map_neg_one`, `mem_graph`, `Equiv.Perm.sign_prod_list_swap`, `MonCat.forget_preservesLimits`, `DirichletCharacter.changeLevel_primitiveCharacter`, `Matrix.SpecialLinearGroup.toLin'_symm_apply`, `ClassGroup.mk0_eq_mk0_inv_iff`, `QuadraticAlgebra.det_toLinearMap_eq_norm`, `Algebra.GrothendieckGroup.lift_apply`, `AlgEquiv.toAlgHomHom_apply`, `SemidirectProduct.right_inl`, `coe_toAdditive''`, `HNNExtension.equiv_symm_eq_conj`, `FreeGroup.freeGroupCongr_apply`, `Action.Iso.conj_ρ`, `groupHomology.single_inv_ρ_self_add_single_mem_boundaries₁`, `FreeGroup.lift_mk`, `Submonoid.LocalizationMap.lift₀_apply`, `Representation.dual_apply`, `restrict_mrange`, `Circle.toUnits_apply`, `Monoid.CoprodI.lift_word_ping_pong`, `Equiv.Perm.IsSwap.of_subtype_isSwap`, `Algebra.norm_eq_zero_iff`, `Profinite.NobelingProof.list_prod_apply`, `SemidirectProduct.mul_left`, `LinearEquiv.automorphismGroup.toLinearMapMonoidHom_apply`, `ClassGroup.mk_canonicalEquiv`, `NumberField.InfinitePlace.prod_eq_abs_norm`, `MonoidAlgebra.mapDomainRingHom_apply`, `CategoryTheory.PreGaloisCategory.toAut_bijective`, `transfer_eq_pow`, `Representation.dualTensorHom_comm`, `Matrix.GeneralLinearGroup.val_scalar_apply`, `Equiv.Perm.sign_prodCongrLeft`, `Subgroup.commute_subtype_of_commute`, `mulExact_iff`, `FreeMonoid.countP_apply`, `SemiRingCat.forget₂_monCat_map`, `groupHomology.lsingle_comp_chainsMap_f_assoc`, `compr₂_apply`, `compAddChar_apply`, `MulAction.IwasawaStructure.is_conj`, `QuotientGroup.map_map`, `ClassGroup.equivPic_apply`, `IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_mod_eq`, `groupHomology.single_mem_cycles₂_iff`, `fiberEquiv_apply`, `Algebra.norm_eq_of_ringEquiv`, `Ideal.absNorm_span_singleton`, `CategoryTheory.Discrete.monoidalFunctor_η`, `Monoid.Coprod.map_mk_ofList`, `Monoid.Coprod.swap_swap`, `PreTilt.mk_untilt_eq_coeff_zero`, `Submonoid.sup_eq_range`, `ClassGroup.norm_le`, `Algebra.GrothendieckGroup.of_injective`, `FreeMonoid.count_apply`, `CategoryTheory.PreGaloisCategory.AutGalois.π_surjective`, `addMonoidEndToMultiplicative_apply_apply`, `Representation.ind_apply`, `map_finprod_plift`, `FreeMonoid.map_symm_apply_map_eq`, `Representation.isIntertwiningMap_iff`, `Monoid.Coprod.fst_surjective`, `MulChar.coe_equivToUnitHom`, `Subgroup.normalizerMonoidHom_apply_symm_apply_coe`, `LinearEquiv.det_refl`, `SemidirectProduct.mk_eq_inl_mul_inr`, `flip_apply`, `RingOfIntegers.coe_algebraMap_norm`, `WittVector.teichmuller_coeff_pos`, `Equiv.Perm.sign_of_cycleType'`, `powMonoidHom_apply`, `Matrix.IsHermitian.conjStarAlgAut_star_eigenvectorUnitary`, `Representation.self_comp_norm`, `Equiv.Perm.sign_subtypeCongr`, `GrpCat.SurjectiveOfEpiAuxs.g_apply_fromCoset`, `EisensteinSeries.eisensteinSeries_slash_apply`, `MonoidAlgebra.mapDomainAlgHom_apply`, `SpecialLinearGroup.baseChange_apply_coe`, `CongruenceSubgroup.IsCongruenceSubgroup.conjGL`, `IsPrimitiveRoot.norm_of_cyclotomic_irreducible`, `Submonoid.surjOn_iff_le_map`, `GrpCat.epi_iff_surjective`, `FreeMonoid.map_of`, `FreeGroup.lift_eq_prod_map`, `map_frobenius`, `Subgroup.coe_comap`, `autAdjoinRootXPowSubC_root`, `Submonoid.inclusion_injective`, `AddSubgroup.pointwise_smul_def`, `FreeAbelianGroup.liftMonoid_symm_coe`, `Representation.inv_self_apply`, `map_iterate_frobeniusEquiv_symm`, `Equiv.Perm.ofSubtype_mem_stabilizer`, `LinearMap.detAux_id`, `mrangeRestrict_mker`, `Algebra.isIntegral_norm`, `map_mul`, `QuadraticAlgebra.norm_mem_nonZeroDivisors_iff`, `QuotientGroup.lift_quot_mk`, `Con.mrange_mk'`, `SkewMonoidAlgebra.mem_span_support`, `MonCat.id_apply`, `Submonoid.coe_inclusion`, `Representation.apply_eq_of_coe_eq`, `NonUnitalStarAlgHom.map_star'`, `MeasureTheory.distribHaarChar_pos`, `mker_prod_map`, `coe_finset_prod`, `coe_toMultiplicative`, `Algebra.norm_eq_prod_embeddings_gen`, `Sylow.smul_def`, `GrpCat.id_apply`, `LinearMap.det_toMatrix`, `exteriorPower.toTensorPower_apply_ιMulti`, `Submonoid.LocalizationMap.toMonoidHom_apply`, `ContinuousLinearEquiv.smulLeft_apply_toLinearEquiv`, `Monoid.Coprod.toProd_apply_inl`, `subgroupMap_apply_coe`, `SkewMonoidAlgebra.lift_def`, `Equiv.Perm.prod_Iio_comp_eq_sign_mul_prod`, `LinearMap.det_toLin`, `SemidirectProduct.left_inr`, `Subgroup.surjOn_iff_le_map`, `LinearMap.det_eq_one_of_subsingleton`, `AddMonoidHom.coe_toMultiplicative''`, `DomMulAct.mk_smul_monoidHom_apply`, `map_zpow`, `flipHom_apply`, `DirichletCharacter.changeLevel_self`, `Equiv.optionCongr_sign`, `CommMonCat.ofHom_apply`, `MulAction.coe_toPermHom`, `Action.FunctorCategoryEquivalence.functor_obj_map`, `ModularGroup.coeHom_apply`, `Monoid.CoprodI.lift_injective_of_ping_pong`, `powersMulHom_symm_apply`, `CommGrpCat.coyoneda_map_app`, `RootPairing.Equiv.coweightHom_toLinearMap`, `Monoid.PushoutI.NormalWord.summand_smul_def`, `SpecialLinearGroup.coe_div`, `NumberField.Units.complexEmbedding_inj`, `surjective_of_card_ker_le_div`, `AddMonoidAlgebra.of'_eq_of`, `snd_mono`, `FreeMonoid.ofList_map`, `Units.range_embedProduct`, `Unitary.conjStarAlgAut_star_apply`, `groupHomology.H1AddEquivOfIsTrivial_symm_tmul`, `groupHomology.single_mem_cycles₁_iff`, `Monoid.foldrM.ofFreeMonoid_apply`, `Action.ρAut_apply_inv`, `groupHomology.d₂₁_single_ρ_add_single_inv_mul`, `CircleDeg1Lift.translate_pow`, `Monoid.Coprod.mrange_inl_sup_mrange_inr`, `Matrix.SpecialLinearGroup.isInducing_toGL`, `GrpCat.μ_forget_apply`, `Equiv.Perm.ofSubtype_subtypePerm_of_not_mem`, `SpecialLinearGroup.toLinearEquiv_injective`, `IsPrimitiveRoot.norm_toInteger_sub_one_of_prime_ne_two'`, `Action.Hom.comm`, `Action.ρ_inv_self_apply`, `Polynomial.Gal.restrictDvd_surjective`, `RootPairing.Equiv.coweightHom_apply`, `Units.embedProduct_injective`, `PresentedMonoid.surjective_mk`, `LinearIsometryEquiv.toContinuousLinearEquiv_smul`, `Monoid.PushoutI.lift_of`, `FiniteField.algebraMap_norm_eq_pow_sum`, `Representation.mem_invtSubmodule`, `TannakaDuality.FiniteGroup.equivHom_surjective`, `Submodule.mulExact_unitsToPic_mapAlgebra`, `SubgroupClass.inclusion_self`, `IsPrimitiveRoot.norm_sub_one_two`, `CategoryTheory.PreGaloisCategory.toAut_injective_of_non_trivial`, `transfer_eq_prod_quotient_orbitRel_zpowers_quot`, `TannakaDuality.FiniteGroup.equivHom_injective`, `MulChar.apply_mem_rootsOfUnity`, `toAdditiveRight_apply_apply`, `SemidirectProduct.inl_injective`, `Monoid.Coprod.mk_of_inv_mul`, `LinearMap.associated_det_comp_equiv`, `CommMonCat.forget_isCorepresentable`, `FDRep.hom_action_ρ`, `Units.isInducing_embedProduct`, `DirichletCharacter.changeLevel_eq_cast_of_dvd`, `CategoryTheory.PreGaloisCategory.endMulEquivAutGalois_pi`, `ker_toSubmonoid`, `Units.coeHom_injective`, `Units.mapContinuousMulEquiv_apply`, `MonCat.ofHom_apply`, `GrpCat.forget₂Mon_preservesLimitsOfSize`, `Monoid.Coprod.mk_eq_mk`, `UpperHalfPlane.smulAux'_im`, `CommGrpCat.forget₂Group_preservesLimitsOfShape`, `Monoid.Coprod.inr_injective`, `FractionalIdeal.coe_mk0`, `ClassGroup.mk_def`, `Equiv.Perm.sign_swap'`, `Algebra.intNorm_eq_zero`, `SpecialLinearGroup.coe_dualMap`, `WittVector.ghostComponent_teichmuller`, `GrpCat.comp_apply`, `CommGrpCat.forget₂_map`, `ValuationSubring.coe_unitGroupToResidueFieldUnits_apply`, `Monoid.Coprod.codisjoint_mrange_inl_mrange_inr`, `QuotientGroup.sound`, `QuadraticAlgebra.norm_eq_zero_iff_eq_zero`, `groupHomology.chainsMap_f_1_comp_chainsIso₁_apply`, `SemiRingCat.forget₂Mon_preservesLimits`, `Equiv.Perm.OnCycleFactors.kerParam_apply`, `MulDistribMulAction.toMonoidHom_apply`, `modularCyclotomicCharacter'.unique'`, `CommRing.Pic.mapAlgebra_mapAlgebra`, `comap_mker`, `WeierstrassCurve.Affine.Point.toClass_apply`, `Equiv.Perm.sign_swap`, `RingOfIntegers.dvd_norm`, `CategoryTheory.Discrete.monoidalFunctor_δ`, `RatFunc.map_apply_div_ne_zero`, `Subgroup.IsArithmetic.conj`, `WeierstrassCurve.VariableChange.map_u`, `RootPairing.Hom.weightHom_injective`, `CommMonCat.ext_iff`, `Complex.det_conjLIE`, `Monoid.Coprod.snd_apply_inr`, `Submonoid.LocalizationMap.lift₀_def`, `cyclotomicCharacter.continuous`, `Unitary.conjStarAlgAut_ext_iff`, `Monoid.CoprodI.Word.equivPair_tail_eq_inv_smul`, `MonoidAlgebra.lift_single`, `coe_ker`, `Equiv.Perm.sign_ofSubtype`, `Rep.coinvariantsTensorFreeToFinsupp_mk_tmul_single`, `mulExact_of_comp_eq_one_of_ker_le_range`, `MonCat.inv_hom_apply`, `MulDistribMulAction.toMonoidEnd_apply`, `TannakaDuality.FiniteGroup.rightRegular_apply`, `spinGroup.toUnits_injective`, `GrpCat.forget_grp_preserves_mono`, `WithZero.coeMonoidHom_apply`, `Submonoid.mem_closure_image_one_lt_iff`, `isLocalHom_of_comp`, `coe_toAdditive'`, `RootPairing.Equiv.indexEquiv_inv`, `IsLocalization.smul_toInvSubmonoid`, `Equiv.Perm.extendDomainHom_injective`, `PresentedMonoid.ext_iff`, `SpecialLinearGroup.toLinearEquiv_to_linearMap`, `groupHomology.mem_cycles₁_iff`, `FreeGroup.map_eq_lift`, `mul_apply`, `FreeGroup.lift_of_apply`, `CategoryTheory.PreGaloisCategory.autGaloisSystem_map_surjective`, `coe_prod`, `Units.map_neg_one`, `Matrix.SpecialLinearGroup.toGL_inj`, `ClassGroup.equiv_mk`, `IsPrimitiveRoot.norm_toInteger_pow_sub_one_of_prime_ne_two`, `Representation.trivial_apply`, `MonoidAlgebra.of_apply`, `groupHomology.single_mem_cycles₂_iff_inv`, `ModularGroup.smul_eq_lcRow0_add`, `groupHomology.d₁₀_single`, `Matrix.SpecialLinearGroup.toLin'_apply`, `ConjAct.smul_eq_mulAut_conj`, `addMonoidEndToMultiplicative_symm_apply_apply`, `Submonoid.subtype_apply`, `DirichletCharacter.FactorsThrough.eq_changeLevel`, `TannakaDuality.FiniteGroup.equivHom_apply`, `Submonoid.closure_eq_mrange`, `Monoid.PushoutI.NormalWord.cons_toList`, `ext_int_iff`, `AddEquiv.toMultiplicativeLeft_apply_apply`, `Equiv.Perm.zpow_eq_ofSubtype_subtypePerm_iff`, `Representation.self_norm_apply`, `MultilinearMap.alternatization_def`, `Monoid.Coprod.swap_inj`, `AddChar.coe_toMonoidHomEquiv_symm`, `Subgroup.hasDetPlusMinusOne_iff_abs_det`, `SkewMonoidAlgebra.liftNC_smul`, `Rep.indResHomEquiv_symm_apply_hom`, `toMulEquiv_apply`, `ClassGroup.exists_mem_finset_approx'`, `CategoryTheory.PreGaloisCategory.toAut_continuous`, `Matrix.SpecialLinearGroup.isEmbedding_mapGL`, `ContinuousMap.coeFnMonoidHom_apply`, `Units.embedProduct_apply`, `Abelianization.equivOfComm_symm_apply`, `mem_mgraph`, `CategoryTheory.PreGaloisCategory.autEmbedding_isClosedEmbedding`, `LinearEquiv.coe_det`, `CommMonCat.hom_inv_apply`, `SemidirectProduct.monoidHomSubgroup_apply`, `Action.FintypeCat.toEndHom_trivial_of_mem`, `ClassGroup.exists_mem_finsetApprox`, `GrpCat.coe_id`, `CommGrpCat.forget_preservesLimitsOfShape`, `Submodule.unitsQuotEquivRelPic_apply_coe`, `IsCyclotomicExtension.norm_zeta_pow_sub_one_of_prime_pow_ne_two`, `MonCat.hom_inv_apply`, `CategoryTheory.yonedaGrp_naturality`, `spinGroup.val_toUnits_apply`, `Equiv.Perm.sign_surjective`, `CategoryTheory.Iso.commGroupIsoToMulEquiv_apply`, `submonoidMap_surjective`, `Equiv.Perm.commute_ofSubtype_noncommPiCoprod`, `MonoidAlgebra.lift_of`, `Monoid.Coprod.swap_inl`, `ModularForm.slash_apply`, `Filter.coe_pureMonoidHom`, `Action.FintypeCat.ofMulAction_apply`, `Monoid.Coprod.snd_surjective`, `Rep.FiniteCyclicGroup.coinvariantsTensorResolutionIso_hom_f_hom_apply`, `GrpCat.instIsRightAdjointForgetMonoidHomCarrier`, `Equiv.Perm.OnCycleFactors.sign_kerParam_apply_apply`, `compl₂_apply`, `prod_galRestrict_eq_norm`, `WithZero.map'_injective_iff`, `groupHomology.chainsMap_f`, `ProfiniteGrp.diagram_obj`, `Ideal.absNorm_dvd_norm_of_mem`, `Monoid.CoprodI.of_injective`, `CategoryTheory.PreGaloisCategory.autEmbedding_range_isClosed`, `groupHomology.chainsMap_f_2_comp_chainsIso₂_apply`, `SpecialLinearGroup.toLinearEquiv_apply`, `Representation.ofMulAction_def`, `SkewMonoidAlgebra.lift_symm_apply`, `Finset.coe_singletonMonoidHom`, `AffineEquiv.constVAddHom_apply`, `map_pow`, `Matrix.GeneralLinearGroup.val_det_apply`, `UpperHalfPlane.petersson_slash`, `Equiv.Perm.viaEmbeddingHom_injective`, `MulAut.conjNormal_apply`, `Submonoid.LocalizationMap.lift_id`, `PNat.coe_coeMonoidHom`, `comp_apply`, `Submonoid.map_coe_toMonoidHom`
`inverse` 📖	CompOp	1 math math: `inverse_apply`
`mk'` 📖	CompOp	1 math math: `mk'_apply`
`toMulHom` 📖	CompOp	5 math math: `toMulHom_coe`, `SkewMonoidAlgebra.nonUnitalAlgHom_ext'_iff`, `Matrix.subsemigroupCenter_eq_scalar_map`, `toMulHom_injective`, `WithOne.lift_unique`
`toOneHom` 📖	CompOp	38 math math: `RingHom.map_add'`, `LinearEquiv.algEquivOfRing_apply`, `MulChar.toMonoidWithZeroHom_apply`, `unitsNonZeroDivisorsEquiv_apply`, `smulMonoidWithZeroHom_apply`, `GroupExtension.Splitting.rightInverse_rightHom`, `ContMDiffMonoidMorphism.contMDiff_toFun`, `AlgHom.commutes'`, `Algebra.directSumGAlgebra_toFun_apply`, `toOneHom_coe`, `Submonoid.leftInvEquiv_apply`, `toFun_eq_coe`, `mulMonoidWithZeroHom_apply`, `AddMonoidAlgebra.mapRangeAlgEquiv_apply`, `OrderMonoidHom.toFun_eq_coe`, `MonoidAlgebra.mapRangeAlgEquiv_apply`, `OrderRingHom.toFun_eq_coe`, `RingHom.toFun_eq_coe`, `ContinuousMonoidHom.continuous_toFun`, `MonoidWithZeroHom.toMonoidHom_coe`, `Ideal.quotientEquiv_apply`, `OrderRingHom.monotone'`, `OrderMonoidHom.monotone'`, `RingHom.toSemilinearMap_apply`, `Complex.uniformContinuous_ringHom_eq_id_or_conj`, `RingHom.map_zero'`, `map_mul'`, `divMonoidWithZeroHom_apply`, `LaurentSeries.comparePkg_eq_extension`, `ContinuousAlgHom.cont`, `DividedPowers.DPMorphism.ext_iff`, `MulChar.map_nonunit'`, `StarMonoidHom.map_star'`, `StarAlgHom.map_star'`, `AlgHom.toFun_eq_coe`, `toOneHom_injective`, `AlgEquiv.algHomUnitsEquiv_apply_apply`, `IsCyclotomicExtension.autEquivPow_apply`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`cancel_left` 📖	mathematical	`DFunLike.coe` `MonoidHom` `instFunLike`	`comp`	—	`ext` `comp_apply`
`cancel_right` 📖	mathematical	`DFunLike.coe` `MonoidHom` `instFunLike`	`comp`	—	`ext` `Function.Surjective.forall` `DFunLike.ext_iff`
`coe_coe` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `instFunLike` `MonoidHomClass.toMonoidHom`	—	—
`coe_comp` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `instFunLike` `comp`	—	—
`coe_copy` 📖	mathematical	`DFunLike.coe` `MonoidHom` `instFunLike`	`copy`	—	—
`coe_id` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `instFunLike` `id`	—	—
`coe_mk` 📖	mathematical	`OneHom.toFun` `MulOne.toOne` `MulOne.toMul`	`DFunLike.coe` `MonoidHom` `instFunLike` `OneHom` `OneHom.funLike`	—	—
`comp_apply` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `instFunLike` `comp`	—	—
`comp_assoc` 📖	mathematical	—	`comp`	—	—
`comp_id` 📖	mathematical	—	`comp` `id`	—	`ext`
`comp_one` 📖	mathematical	—	`comp` `MulOneClass.toMulOne` `MonoidHom` `instOneMonoidHom`	—	`ext` `map_one` `MonoidHomClass.toOneHomClass` `instMonoidHomClass`
`copy_eq` 📖	mathematical	`DFunLike.coe` `MonoidHom` `instFunLike`	`copy`	—	`DFunLike.ext'`
`ext` 📖	—	`DFunLike.coe` `MonoidHom` `instFunLike`	—	—	`DFunLike.ext`
`ext_iff` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `instFunLike`	—	`ext`
`id_apply` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `instFunLike` `id`	—	—
`id_comp` 📖	mathematical	—	`comp` `id`	—	`ext`
`instMonoidHomClass` 📖	mathematical	—	`MonoidHomClass` `MonoidHom` `instFunLike`	—	`map_mul'` `OneHom.map_one'`
`inverse_apply` 📖	mathematical	`DFunLike.coe` `MonoidHom` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `instFunLike`	`inverse`	—	—
`isDedekindFiniteMonoid_of_injective` 📖	mathematical	`DFunLike.coe`	`IsDedekindFiniteMonoid`	—	`IsDedekindFiniteMonoid.of_injective`
`map_div` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `DivInvMonoid.toMonoid` `Group.toDivInvMonoid` `DivisionMonoid.toDivInvMonoid` `instFunLike` `DivInvMonoid.toDiv`	—	`map_div` `instMonoidHomClass`
`map_exists_left_inv` 📖	mathematical	`MulOne.toMul` `MulOne.toOne`	`DFunLike.coe`	—	`map_mul_eq_one`
`map_exists_right_inv` 📖	mathematical	`MulOne.toMul` `MulOne.toOne`	`DFunLike.coe`	—	`map_mul_eq_one`
`map_inv` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `DivInvMonoid.toMonoid` `Group.toDivInvMonoid` `DivisionMonoid.toDivInvMonoid` `instFunLike` `DivInvMonoid.toInv`	—	`map_inv` `instMonoidHomClass`
`map_mul` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `instFunLike` `MulOne.toMul`	—	`map_mul'`
`map_mul'` 📖	mathematical	—	`OneHom.toFun` `MulOne.toOne` `toOneHom` `MulOne.toMul`	—	—
`map_mul_inv` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `DivInvMonoid.toMonoid` `Group.toDivInvMonoid` `DivisionMonoid.toDivInvMonoid` `instFunLike` `MulOne.toMul` `DivInvMonoid.toInv`	—	`map_mul` `MonoidHomClass.toMulHomClass` `instMonoidHomClass` `map_inv`
`map_one` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `instFunLike` `MulOne.toOne`	—	`OneHom.map_one'`
`map_pow` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `instFunLike` `Monoid.toNatPow`	—	`map_pow` `instMonoidHomClass`
`map_zpow` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `DivInvMonoid.toMonoid` `Group.toDivInvMonoid` `DivisionMonoid.toDivInvMonoid` `instFunLike` `DivInvMonoid.toZPow`	—	`map_zpow` `instMonoidHomClass`
`map_zpow'` 📖	mathematical	`DFunLike.coe` `MonoidHom` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `DivInvMonoid.toMonoid` `instFunLike` `DivInvMonoid.toInv`	`DivInvMonoid.toZPow`	—	`map_zpow'` `instMonoidHomClass`
`mk'_apply` 📖	mathematical	`MulOne.toMul` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `DivInvMonoid.toMonoid` `Group.toDivInvMonoid`	`DFunLike.coe` `MonoidHom` `instFunLike` `mk'`	—	—
`mk_coe` 📖	—	`OneHom.toFun` `MulOne.toOne` `OneHomClass.toOneHom` `MonoidHom` `instFunLike` `MonoidHomClass.toOneHomClass` `instMonoidHomClass` `MulOne.toMul`	—	—	`MonoidHomClass.toOneHomClass` `instMonoidHomClass` `ext`
`one_apply` 📖	mathematical	—	`DFunLike.coe` `MonoidHom` `MulOneClass.toMulOne` `instFunLike` `instOneMonoidHom` `MulOne.toOne`	—	—
`one_comp` 📖	mathematical	—	`comp` `MulOneClass.toMulOne` `MonoidHom` `instOneMonoidHom`	—	—
`toFun_eq_coe` 📖	mathematical	—	`OneHom.toFun` `MulOne.toOne` `toOneHom` `DFunLike.coe` `MonoidHom` `instFunLike`	—	—
`toMulHom_coe` 📖	mathematical	—	`MulHom.toFun` `MulOne.toMul` `toMulHom` `DFunLike.coe` `MonoidHom` `instFunLike`	—	—
`toMulHom_injective` 📖	mathematical	—	`MonoidHom` `MulHom` `MulOne.toMul` `toMulHom`	—	`Function.Injective.of_comp` `DFunLike.coe_injective`
`toOneHom_coe` 📖	mathematical	—	`DFunLike.coe` `OneHom` `MulOne.toOne` `OneHom.funLike` `toOneHom` `MonoidHom` `instFunLike`	—	—
`toOneHom_injective` 📖	mathematical	—	`MonoidHom` `OneHom` `MulOne.toOne` `toOneHom`	—	`Function.Injective.of_comp` `DFunLike.coe_injective`

MonoidHomClass

Definitions

Name	Category	Theorems
`toMonoidHom` 📖	CompOp	122 math math: `AddMonoidAlgebra.algHom_ext'_iff`, `Matrix.GeneralLinearGroup.map_det`, `MulEquiv.monoidHomCongrLeft_apply`, `rootsOfUnityEquivOfPrimitiveRoots_apply_coe_inv_val`, `MulEquiv.coe_monoidHom_symm_comp_coe_monoidHom`, `Subgroup.equivMapOfInjective_coe_mulEquiv`, `Subgroup.comap_equiv_eq_map_symm`, `SkewMonoidAlgebra.algHom_ext'_iff`, `FDRep.endRingEquiv_symm_comp_ρ`, `IsUniformGroup.uniformContinuous_iff_isOpen_ker`, `MonoidAlgebra.algHom_ext'_iff`, `Subgroup.isCoatom_map`, `Subgroup.comap_toSubmonoid`, `MonoidAlgebra.domCongr_toAlgHom`, `MulEquiv.comapSubgroup_symm_apply`, `RingHom.coe_monoidHom_id`, `MulEquiv.toMonoidHom_eq_coe`, `Submonoid.LocalizationMap.AwayMap.lift_comp`, `Submonoid.equivMapOfInjective_coe_mulEquiv`, `LinearEquiv.det_baseChange`, `MulEquiv.mapSubgroup_apply`, `MulEquiv.comp_right_injective`, `Valuation.unit_map_eq`, `isLocalHom_toMonoidHom`, `RingHom.toMonoidHom_eq_coe`, `LinearMap.detAux_def`, `Subgroup.map_equiv_top`, `Units.map_neg`, `MulEquiv.withZero_apply_symm_apply`, `SemidirectProduct.congr'_apply_right`, `Con.comapQuotientEquivOfSurj_symm_mk'`, `Subgroup.map_symm_eq_iff_map_eq`, `comap_upperCentralSeries`, `AddMonoidAlgebra.ringHom_ext'_iff`, `OrderMonoidHom.toMonoidHom_eq_coe`, `MulEquiv.toSingleObjEquiv_unitIso_hom`, `isIntegral_localization'`, `AlgHom.coe_toMonoidHom`, `MulEquiv.withZero_apply_apply`, `OrderMonoidIso.withZero_apply_symm_apply`, `ContAction.resCongr_hom`, `SemidirectProduct.congr'_apply_left`, `MulEquiv.toSingleObjEquiv_counitIso_hom`, `SemidirectProduct.congr'_symm_apply_right`, `MonoidHom.coe_coe`, `ContAction.res_obj_obj`, `MulEquiv.coprodCongr_apply`, `MonoidAlgebra.toRingHom_mapDomainRingEquiv`, `ContinuousMonoidHom.coe_toMonoidHom`, `OrderMonoidIso.withZero_apply_apply`, `MulEquiv.toSingleObjEquiv_unitIso_inv`, `IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_eq`, `MulEquiv.mapSubgroup_symm_apply`, `MulEquiv.toSingleObjEquiv_counitIso_inv`, `Submodule.unitsQuotEquivRelPic_symm_apply`, `SkewMonoidAlgebra.lift_unique'`, `MulEquiv.comapSubgroup_apply`, `CategoryTheory.yonedaGrpObjIsoOfRepresentableBy_hom`, `MulEquiv.submonoidMap_symm_apply`, `CategoryTheory.yonedaGrpObjIsoOfRepresentableBy_inv`, `IsLocalization.toLocalizationMap_toMonoidHom`, `OrderMonoidHom.coe_comp_monoidHom`, `PowerSeries.map_invUnitsSub`, `Algebra.smul_units_def`, `AddMonoidAlgebra.lift_unique'`, `ClassGroup.equivPic_symm_apply`, `Subgroup.index_map_equiv`, `IsScalarTower.of_compHom`, `SemidirectProduct.congr'_symm_apply_left`, `MulEquiv.coe_monoidHom_trans`, `MonoidHom.ker_comp_mulEquiv`, `LinearOrderedCommGroupWithZero.fst_apply`, `ContAction.resCongr_inv`, `MonoidWithZeroHom.copy_eq`, `GroupExtension.Splitting.rightHom_comp_splitting`, `MulEquiv.monoidHomCongrRight_apply`, `RingHom.coe_monoidHom_injective`, `AlgEquiv.smul_units_def`, `MonoidAlgebra.ringHom_ext'_iff`, `MulEquiv.coe_subgroupMap_apply`, `MulEquiv.coprodCongr_symm_apply`, `MulEquiv.comp_left_injective`, `WeierstrassCurve.map_Δ'`, `Action.resEquiv_inverse`, `FractionalIdeal.map_canonicalEquiv_mk0`, `WeierstrassCurve.inv_map_Δ'`, `OrderMonoidHom.coe_monoidHom`, `RingHom.coe_monoidHom_mk`, `Action.resEquiv_functor`, `CommRingCat.commMon_forget₂_map`, `SkewMonoidAlgebra.ringHom_ext'_iff`, `Subgroup.comap_normalClosure`, `RingEquiv.coe_monoidHom_trans`, `AlgHom.toRingHom_toMonoidHom`, `Submonoid.topEquiv_toMonoidHom`, `ClassGroup.mk_canonicalEquiv`, `MulEquiv.subgroupMap_symm_apply`, `IsDedekindDomain.HeightOneSpectrum.valuation_of_unit_mod_eq`, `MulEquiv.coe_monoidHom_comp_coe_monoidHom_symm`, `LinearOrderedCommGroupWithZero.inl_apply`, `LinearOrderedCommGroupWithZero.inr_apply`, `MulEquiv.coe_monoidHom_refl`, `RingEquiv.coe_monoidHom_refl`, `GroupExtension.Splitting.coe_monoidHom_mk`, `WithZero.map'_id`, `ContAction.res_map`, `ValuationSubring.unitsModPrincipalUnitsEquivResidueFieldUnits_comp_quotientGroup_mk`, `Units.mapContinuousMulEquiv_apply`, `ClassGroup.mk_def`, `Subgroup.map_equiv_eq_comap_symm`, `WeierstrassCurve.VariableChange.map_u`, `Submonoid.LocalizationMap.lift₀_def`, `IsGaloisGroup.map_mulEquivAlgEquiv_fixingSubgroup`, `FDRep.endRingEquiv_comp_ρ`, `Units.map_neg_one`, `ContinuousMonoidHom.toMonoidHom_toContinuousMonoidHom`, `IsPrimitiveRoot.map_rootsOfUnity`, `AlgHom.coe_monoidHom_injective`, `ZMod.unitsMap_def`, `Subgroup.isCoatom_comap`, `MonoidAlgebra.lift_unique'`, `Submonoid.map_coe_toMonoidHom`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`toMulHomClass` 📖	mathematical	—	`MulHomClass` `MulOne.toMul`	—	—
`toOneHomClass` 📖	mathematical	—	`OneHomClass` `MulOne.toOne`	—	—

MulHom

Definitions

Name	Category	Theorems
`comp` 📖	CompOp	40 math math: `comp_coprod`, `Semigrp.hom_comp`, `prodMap_def`, `Semigrp.ofHom_comp`, `Magma.AssocQuotient.lift_comp_of'`, `comp_mul`, `comp_apply`, `SkewMonoidAlgebra.nonUnitalAlgHom_ext'_iff`, `prod_unique`, `WithOne.mapMulHom_comp`, `WithOne.mapMulHom_mapMulHom`, `comp_id`, `map_srange`, `comp_assoc`, `MonoidAlgebra.nonUnitalAlgHom_ext'_iff`, `AddMonoidAlgebra.nonUnitalAlgHom_ext'_iff`, `Subsemigroup.map_map`, `MonoidAlgebra.liftMagma_symm_apply`, `AddMonoidAlgebra.liftMagma_symm_apply`, `snd_comp_prod`, `WithOne.map_map`, `coe_comp`, `cancel_right`, `fst_comp_prod`, `Magma.AssocQuotient.lift_comp_of`, `MonoidAlgebra.mapDomainNonUnitalRingHom_comp`, `Magma.AssocQuotient.hom_ext_iff`, `cancel_left`, `WithOne.map_comp`, `mul_comp`, `prod_comp_prodMap`, `MagmaCat.hom_comp`, `Subsemigroup.comap_comap`, `comp_noncommCoprod`, `Magma.AssocQuotient.lift_symm_apply`, `FreeMagma.toFreeSemigroup_comp_map`, `NonUnitalRingHom.coe_comp_mulHom`, `id_comp`, `WithOne.lift_unique`, `MagmaCat.ofHom_comp`
`copy` 📖	CompOp	2 math math: `coe_copy_eq`, `coe_copy`
`funLike` 📖	CompOp	122 math math: `coe_snd`, `WithOne.coeMulHom_apply`, `id_apply`, `MonoidAlgebra.liftMagma_apply_apply`, `coe_srange`, `MulMemClass.subtype_apply`, `FreeMagma.lift_of`, `coe_fst`, `FreeMagma.toFreeSemigroup_of`, `smulMulHom_apply`, `subsemigroupComap_apply_coe`, `AbsoluteValue.coe_mk`, `op_symm_apply_apply`, `WithOne.lift_coe`, `Pi.constMulHom_apply`, `coe_srangeRestrict`, `coe_prodMap`, `FreeMagma.hom_ext_iff`, `coe_prod`, `Con.mkMulHom_apply`, `Semigrp.hom_inv_apply`, `mul_apply`, `Semigrp.inv_hom_apply`, `Subsemigroup.mem_map`, `mclosure_preimage_le`, `map_multiset_ne_zero_prod`, `comp_apply`, `WithOne.mapMulHom_coe`, `Subsemigroup.mem_map_of_mem`, `restrict_apply`, `Finset.coe_singletonMulHom`, `srange_eq_top_iff_surjective`, `toFun_eq_coe`, `Semigrp.id_apply`, `mem_eqLocus`, `Con.ker_coeMulHom`, `MulMemClass.coe_subtype`, `MonoidAlgebra.ofMagma_apply`, `coe_ofDense`, `coe_coe`, `coe_mk`, `MagmaCat.coe_comp`, `subsemigroupMap_surjective`, `mulHomOfTendsto_apply`, `FreeSemigroup.lift_symm_apply`, `Semigrp.forget₂_full`, `toMulEquiv_symm_apply`, `Semigrp.ext_iff`, `Semigrp.comp_apply`, `prod_apply`, `Semigrp.forgetReflectsIsos`, `compLeft_apply`, `MagmaCat.id_apply`, `Subsemigroup.coe_comap`, `isClosed_range_coe`, `mem_srange`, `toMulEquiv_apply`, `Subsemigroup.mem_comap`, `MagmaCat.coe_id`, `map_mclosure`, `MagmaCat.ofHom_apply`, `coeFn_apply`, `MonoidAlgebra.mapDomainNonUnitalRingHom_apply`, `mulHomClass`, `AddMonoidAlgebra.liftMagma_apply_apply`, `AbsoluteValue.coe_toMulHom`, `Set.singletonMulHom_apply`, `AddMonoidAlgebra.ofMagma_apply`, `Filter.coe_pureMulHom`, `MagmaCat.hom_inv_apply`, `MagmaCat.ext_iff`, `coe_mul`, `Set.coe_singletonMulHom`, `ext_iff`, `coe_comp`, `FreeMagma.lift_symm_apply`, `Finset.imageMulHom_apply`, `MagmaCat.forget_map`, `FreeMagma.toFreeSemigroup_map`, `Magma.AssocQuotient.lift_of`, `FreeSemigroup.lift_comp_of'`, `FreeMagma.toFreeSemigroup_comp_of`, `FreeMagma.lift_comp_of`, `Filter.pureMulHom_apply`, `FreeMagma.length_toFreeSemigroup`, `MagmaCat.inv_hom_apply`, `FreeMagma.map_of`, `op_apply_apply`, `Subsemigroup.apply_coe_mem_map`, `mulMulHom_apply`, `srangeRestrict_surjective`, `map_mul`, `Pi.evalMulHom_apply`, `subsemigroupMap_apply_coe`, `map_dvd`, `Finset.singletonMulHom_apply`, `Magma.AssocQuotient.map_of`, `single_apply`, `Semigrp.coe_id`, `coe_id`, `NonUnitalAlgHom.coe_to_mulHom`, `MulEquiv.coe_toMulHom`, `FreeMagma.lift_comp_of'`, `Semigrp.ofHom_apply`, `Semigrp.forget_map`, `FreeSemigroup.length_map`, `MagmaCat.forgetReflectsIsos`, `NonUnitalRingHom.coe_toMulHom`, `FreeSemigroup.lift_of_mul`, `FreeSemigroup.hom_ext_iff`, `mulHomOfMemClosureRangeCoe_apply`, `FreeSemigroup.lift_comp_of`, `MagmaCat.comp_apply`, `Semigrp.coe_comp`, `Subsemigroup.coe_map`, `coprod_apply`, `MulMemClass.subtype_injective`, `FreeSemigroup.map_of`, `nhdsMulHom_apply`, `Pi.mulHom_apply`, `FreeSemigroup.lift_of`, `Con.ker_mkMulHom_eq`
`id` 📖	CompOp	14 math math: `id_apply`, `Subsemigroup.map_id`, `WithOne.map_id`, `MonoidAlgebra.mapDomainNonUnitalRingHom_id`, `Subsemigroup.comap_id`, `Semigrp.hom_id`, `WithOne.mapMulHom_id`, `comp_id`, `Semigrp.ofHom_id`, `NonUnitalRingHom.coe_mulHom_id`, `coe_id`, `MagmaCat.ofHom_id`, `MagmaCat.hom_id`, `id_comp`
`inverse` 📖	CompOp	1 math math: `inverse_apply`
`toFun` 📖	CompOp	16 math math: `MonoidHom.toMulHom_coe`, `AddMonoidAlgebra.mapDomainNonUnitalAlgHom_apply`, `Finset.imageMonoidHom_apply`, `map_mul'`, `toFun_eq_coe`, `Submonoid.LocalizationMap.isLocalizationMap`, `NonUnitalRingHom.map_zero'`, `mulMonoidHom_apply`, `AbsoluteValue.nonneg'`, `smulMonoidHom_apply`, `MonoidAlgebra.mapDomainNonUnitalAlgHom_apply`, `Pi.evalNonUnitalStarAlgHom_apply`, `AbsoluteValue.eq_zero'`, `NonUnitalRingHom.map_add'`, `AbsoluteValue.add_le'`, `NonUnitalStarRingHom.map_star'`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`cancel_left` 📖	mathematical	`DFunLike.coe` `MulHom` `funLike`	`comp`	—	`ext` `comp_apply`
`cancel_right` 📖	mathematical	`DFunLike.coe` `MulHom` `funLike`	`comp`	—	`ext` `Function.Surjective.forall` `DFunLike.ext_iff`
`coe_coe` 📖	mathematical	—	`DFunLike.coe` `MulHom` `funLike` `MulHomClass.toMulHom`	—	—
`coe_comp` 📖	mathematical	—	`DFunLike.coe` `MulHom` `funLike` `comp`	—	—
`coe_copy` 📖	mathematical	`DFunLike.coe` `MulHom` `funLike`	`copy`	—	—
`coe_copy_eq` 📖	mathematical	`DFunLike.coe` `MulHom` `funLike`	`copy`	—	`DFunLike.ext'`
`coe_id` 📖	mathematical	—	`DFunLike.coe` `MulHom` `funLike` `id`	—	—
`coe_mk` 📖	mathematical	—	`DFunLike.coe` `MulHom` `funLike`	—	—
`comp_apply` 📖	mathematical	—	`DFunLike.coe` `MulHom` `funLike` `comp`	—	—
`comp_assoc` 📖	mathematical	—	`comp`	—	—
`comp_id` 📖	mathematical	—	`comp` `id`	—	`ext`
`ext` 📖	—	`DFunLike.coe` `MulHom` `funLike`	—	—	`DFunLike.ext`
`ext_iff` 📖	mathematical	—	`DFunLike.coe` `MulHom` `funLike`	—	`ext`
`id_apply` 📖	mathematical	—	`DFunLike.coe` `MulHom` `funLike` `id`	—	—
`id_comp` 📖	mathematical	—	`comp` `id`	—	`ext`
`inverse_apply` 📖	mathematical	`DFunLike.coe` `MulHom` `funLike`	`inverse`	—	—
`map_mul` 📖	mathematical	—	`DFunLike.coe` `MulHom` `funLike`	—	`map_mul'`
`map_mul'` 📖	mathematical	—	`toFun`	—	—
`mk_coe` 📖	—	`DFunLike.coe` `MulHom` `funLike`	—	—	`ext`
`mulHomClass` 📖	mathematical	—	`MulHomClass` `MulHom` `funLike`	—	`map_mul'`
`toFun_eq_coe` 📖	mathematical	—	`toFun` `DFunLike.coe` `MulHom` `funLike`	—	—

MulHomClass

Definitions

Name	Category	Theorems
`toMulHom` 📖	CompOp	19 math math: `Semigrp.mulEquiv_coe_eq`, `Con.ker_coeMulHom`, `NonUnitalRingHom.coe_mulHom_injective`, `Subsemigroup.topEquiv_toMulHom`, `MulHom.coe_coe`, `MagmaCat.mulEquiv_coe_eq`, `NonUnitalAlgHom.coe_mulHom_mk`, `Subsemigroup.mem_map_equiv`, `Subsemigroup.map_equiv_eq_comap_symm`, `NonUnitalAlgHom.toMulHom_eq_coe`, `MulEquiv.toMulHom_eq_coe`, `Subsemigroup.map_equiv_top`, `NonUnitalRingHom.coe_mulHom_id`, `NonUnitalAlgHom.coe_to_mulHom`, `MulEquiv.subsemigroupMap_apply_coe`, `MulEquiv.subsemigroupMap_symm_apply_coe`, `NonUnitalRingHom.coe_comp_mulHom`, `NonUnitalRingHom.coe_mulHom_mk`, `Subsemigroup.comap_equiv_eq_map_symm`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`map_mul` 📖	mathematical	—	`DFunLike.coe`	—	—

OneHom

Definitions

Name	Category	Theorems
`comp` 📖	CompOp	12 math math: `one_comp`, `coe_comp`, `mul_comp`, `id_comp`, `comp_one`, `cancel_right`, `cancel_left`, `div_comp`, `comp_id`, `inv_comp`, `comp_assoc`, `comp_apply`
`copy` 📖	CompOp	2 math math: `coe_copy_eq`, `coe_copy`
`funLike` 📖	CompOp	31 math math: `withTopMap_apply`, `inv_apply`, `oneHomClass`, `id_apply`, `map_one`, `ext_iff`, `Set.coe_singletonOneHom`, `coe_mk`, `MonoidHom.toOneHom_coe`, `Finset.coe_singletonOneHom`, `coe_comp`, `coe_mul`, `one_apply`, `mul_apply`, `coe_div`, `coe_mulSingle`, `Filter.pureOneHom_apply`, `MonoidHom.coe_mk`, `mulSingle_apply`, `pow_apply`, `coe_inv`, `coe_coe`, `Filter.coe_pureOneHom`, `Finset.imageOneHom_apply`, `withBotMap_apply`, `Finset.singletonOneHom_apply`, `zpow_apply`, `comp_apply`, `toFun_eq_coe`, `div_apply`, `coe_id`
`id` 📖	CompOp	4 math math: `id_apply`, `id_comp`, `comp_id`, `coe_id`
`inverse` 📖	CompOp	1 math math: `inverse_apply`
`toFun` 📖	CompOp	38 math math: `RingHom.map_add'`, `LinearEquiv.algEquivOfRing_apply`, `MulChar.toMonoidWithZeroHom_apply`, `unitsNonZeroDivisorsEquiv_apply`, `smulMonoidWithZeroHom_apply`, `GroupExtension.Splitting.rightInverse_rightHom`, `ContMDiffMonoidMorphism.contMDiff_toFun`, `AlgHom.commutes'`, `Algebra.directSumGAlgebra_toFun_apply`, `Submonoid.leftInvEquiv_apply`, `MonoidHom.toFun_eq_coe`, `mulMonoidWithZeroHom_apply`, `AddMonoidAlgebra.mapRangeAlgEquiv_apply`, `OrderMonoidHom.toFun_eq_coe`, `MonoidAlgebra.mapRangeAlgEquiv_apply`, `OrderRingHom.toFun_eq_coe`, `RingHom.toFun_eq_coe`, `ContinuousMonoidHom.continuous_toFun`, `MonoidWithZeroHom.toMonoidHom_coe`, `Ideal.quotientEquiv_apply`, `OrderRingHom.monotone'`, `OrderMonoidHom.monotone'`, `RingHom.toSemilinearMap_apply`, `Complex.uniformContinuous_ringHom_eq_id_or_conj`, `map_one'`, `RingHom.map_zero'`, `MonoidHom.map_mul'`, `divMonoidWithZeroHom_apply`, `LaurentSeries.comparePkg_eq_extension`, `ContinuousAlgHom.cont`, `DividedPowers.DPMorphism.ext_iff`, `MulChar.map_nonunit'`, `StarMonoidHom.map_star'`, `StarAlgHom.map_star'`, `AlgHom.toFun_eq_coe`, `AlgEquiv.algHomUnitsEquiv_apply_apply`, `toFun_eq_coe`, `IsCyclotomicExtension.autEquivPow_apply`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`cancel_left` 📖	mathematical	`DFunLike.coe` `OneHom` `funLike`	`comp`	—	`ext` `comp_apply`
`cancel_right` 📖	mathematical	`DFunLike.coe` `OneHom` `funLike`	`comp`	—	`ext` `Function.Surjective.forall` `DFunLike.ext_iff`
`coe_coe` 📖	mathematical	—	`DFunLike.coe` `OneHom` `funLike` `OneHomClass.toOneHom`	—	—
`coe_comp` 📖	mathematical	—	`DFunLike.coe` `OneHom` `funLike` `comp`	—	—
`coe_copy` 📖	mathematical	`DFunLike.coe` `OneHom` `funLike`	`copy`	—	—
`coe_copy_eq` 📖	mathematical	`DFunLike.coe` `OneHom` `funLike`	`copy`	—	`DFunLike.ext'`
`coe_id` 📖	mathematical	—	`DFunLike.coe` `OneHom` `funLike` `id`	—	—
`coe_mk` 📖	mathematical	—	`DFunLike.coe` `OneHom` `funLike`	—	—
`comp_apply` 📖	mathematical	—	`DFunLike.coe` `OneHom` `funLike` `comp`	—	—
`comp_assoc` 📖	mathematical	—	`comp`	—	—
`comp_id` 📖	mathematical	—	`comp` `id`	—	`ext`
`comp_one` 📖	mathematical	—	`comp` `OneHom` `instOneOneHom`	—	`ext` `map_one` `oneHomClass`
`ext` 📖	—	`DFunLike.coe` `OneHom` `funLike`	—	—	`DFunLike.ext`
`ext_iff` 📖	mathematical	—	`DFunLike.coe` `OneHom` `funLike`	—	`ext`
`id_apply` 📖	mathematical	—	`DFunLike.coe` `OneHom` `funLike` `id`	—	—
`id_comp` 📖	mathematical	—	`comp` `id`	—	`ext`
`inverse_apply` 📖	mathematical	`DFunLike.coe` `OneHom` `funLike`	`inverse`	—	—
`map_one` 📖	mathematical	—	`DFunLike.coe` `OneHom` `funLike`	—	`map_one'`
`map_one'` 📖	mathematical	—	`toFun`	—	—
`mk_coe` 📖	—	`DFunLike.coe` `OneHom` `funLike`	—	—	`ext`
`oneHomClass` 📖	mathematical	—	`OneHomClass` `OneHom` `funLike`	—	`map_one'`
`one_apply` 📖	mathematical	—	`DFunLike.coe` `OneHom` `funLike` `instOneOneHom`	—	—
`one_comp` 📖	mathematical	—	`comp` `OneHom` `instOneOneHom`	—	—
`toFun_eq_coe` 📖	mathematical	—	`toFun` `DFunLike.coe` `OneHom` `funLike`	—	—

OneHomClass

Definitions

Name	Category	Theorems
`toOneHom` 📖	CompOp	1 math math: `OneHom.coe_coe`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`map_one` 📖	mathematical	—	`DFunLike.coe`	—	—

Subsingleton

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`of_oneHomClass` 📖	—	—	—	—	`DFunLike.ext` `map_one`
`of_zeroHomClass` 📖	—	—	—	—	`DFunLike.ext` `map_zero`

ZeroHom

Definitions

Name	Category	Theorems
`comp` 📖	CompOp	15 math math: `comp_zero`, `coe_comp`, `sub_comp`, `comp_id`, `smul_comp`, `comp_apply`, `cancel_right`, `neg_comp`, `zero_comp`, `id_comp`, `DFinsupp.sumZeroHom_piSingle`, `Finsupp.mapRange.zeroHom_comp`, `cancel_left`, `comp_assoc`, `add_comp`
`copy` 📖	CompOp	2 math math: `coe_copy`, `coe_copy_eq`
`funLike` 📖	CompOp	116 math math: `zsmul_apply`, `HahnSeries.orderTop_single_le`, `HahnSeries.map_single`, `AddMonoidHom.coe_mk`, `HahnSeries.cardSupp_single_of_ne`, `neg_apply`, `HahnModule.coeff_single_smul_vadd`, `Set.coe_singletonZeroHom`, `coe_comp`, `HahnSeries.coeff_single_mul_add`, `HahnSeries.orderTop_sub_pos`, `nsmul_apply`, `HahnSeries.single_zero_natCast`, `HahnSeries.single_zero_ofScientific`, `HahnSeries.cardSupp_single_le`, `HahnSeries.ofPowerSeries_X_pow`, `LaurentSeries.valuation_single_zpow`, `HahnSeries.single_pow`, `AddMonoidHom.toZeroHom_coe`, `HahnSeries.coeff_single_same`, `HahnSeries.single_zero_nnratCast`, `withTopMap_apply`, `coe_sub`, `LaurentSeries.ofPowerSeries_powerSeriesPart`, `comp_apply`, `LaurentSeries.single_order_mul_powerSeriesPart`, `HahnSeries.single_zero_intCast`, `RatFunc.coe_X`, `HahnSeries.single_mul_single`, `HahnSeries.single_zero_mul_eq_smul`, `MonoidWithZeroHom.coe_mk`, `HahnSeries.cardSupp_truncLT_le`, `HahnSeries.single_sub`, `HahnSeries.cardSupp_map_le`, `HahnSeries.single_eq_zero_iff`, `DFinsupp.sumZeroHom_apply`, `HahnSeries.order_single_mul_of_isRegular`, `HahnSeries.coeff_single_zero_mul`, `HahnSeries.coe_truncLTLinearMap`, `coe_smul`, `ext_iff`, `MonoidWithZeroHom.toZeroHom_coe`, `sub_apply`, `HahnSeries.truncLT_add`, `HahnSeries.support_truncLT`, `HahnSeries.embDomain_single`, `HahnSeries.lt_orderTop_single`, `HahnSeries.coeff_truncLT`, `Filter.coe_pureZeroHom`, `coe_id`, `HahnSeries.unit_aux`, `HahnSeries.ofPowerSeries_X`, `zeroHomClass`, `HahnSeries.coeff_single_of_ne`, `HahnSeries.cardSupp_mul_single_le`, `HahnSeries.coeff_truncLT_of_le`, `HahnSeries.cardSupp_single_mul_le`, `HahnSeries.coeff_ofFinsupp`, `coe_mk`, `HahnSeries.map_zero`, `HahnSeries.single_neg`, `HahnSeries.single_injective`, `smul_apply`, `HahnModule.coeff_single_zero_smul`, `HahnSeries.single_add`, `LaurentSeries.hasseDeriv_single_add`, `HahnSeries.coeff_single`, `HahnSeries.support_single_of_ne`, `HahnSeries.single_zero_one`, `NonarchAddGroupSeminorm.toZeroHom_eq_coe`, `single_apply`, `zero_apply`, `PowerSeries.coe_X`, `HahnSeries.single_div_single`, `HahnSeries.inv_def`, `HahnSeries.support_single_subset`, `Finsupp.mapRange.zeroHom_apply`, `HahnSeries.support_map_subset`, `RatFunc.single_one_eq_pow`, `HahnSeries.coeff_mul_single`, `HahnSeries.coeff_truncLT_of_lt`, `coe_neg`, `RatFunc.single_zpow`, `coe_single`, `HahnSeries.coeff_toOrderTopSubOnePos_pow`, `withBotMap_apply`, `map_zero`, `HahnSeries.single_eq_zero`, `DFinsupp.sumZeroHom_single`, `HahnSeries.coeff_mul_single_add`, `HahnSeries.coeff_mul_single_zero`, `HahnSeries.truncLT_smul`, `HahnSeries.inv_single`, `HahnSeries.C_apply`, `HahnSeries.support_truncLT_subset`, `id_apply`, `Finset.singletonZeroHom_apply`, `add_apply`, `HahnSeries.leadingCoeff_of_single`, `RatFunc.single_inv`, `HahnSeries.single_zero_ratCast`, `HahnEmbedding.Partial.apply_of_mem_stratum`, `HahnSeries.single_zero_ofNat`, `HahnSeries.SummableFamily.powerSeriesFamily_smul`, `coe_coe`, `SMulZeroClass.toZeroHom_apply`, `Finset.coe_singletonZeroHom`, `LaurentSeries.hasseDeriv_single`, `HahnSeries.order_single`, `HahnSeries.coeff_single_mul`, `Filter.pureZeroHom_apply`, `coe_add`, `HahnSeries.orderTop_single`, `HahnModule.single_zero_smul_eq_smul`, `Finset.imageZeroHom_apply`, `toFun_eq_coe`
`id` 📖	CompOp	5 math math: `comp_id`, `coe_id`, `id_comp`, `Finsupp.mapRange.zeroHom_id`, `id_apply`
`inverse` 📖	CompOp	1 math math: `inverse_apply`
`toFun` 📖	CompOp	39 math math: `CentroidHom.map_mul_left'`, `OrderMonoidWithZeroHom.toFun_eq_coe`, `Valuation.toFun_eq_coe`, `AddMonoidHom.toFun_eq_coe`, `Matrix.diagLinearMap_apply`, `Matrix.diagonalLinearMap_apply`, `MeasureTheory.Lp.compMeasurePreservingₗ_apply`, `HahnSeries.single.linearMap_apply`, `linearMapOfMemClosureRangeCoe_apply`, `AddMonoidAlgebra.singleZeroRingHom_apply`, `OrderMonoidWithZeroHom.monotone'`, `map_zero'`, `RingHom.ENatMap_apply`, `ContinuousMap.coeFnLinearMap_apply`, `NonUnitalRingHom.op_symm_apply_apply`, `OrderAddMonoidHom.monotone'`, `Valuation.map_add_le_max'`, `ContinuousLinearMap.compLeftContinuous_apply`, `AddSubmonoid.leftNegEquiv_apply`, `MonoidWithZeroHom.map_one'`, `RingHom.withTopMap_apply`, `CentroidHom.toFun_eq_coe`, `ContMDiffAddMonoidMorphism.contMDiff_toFun`, `NonUnitalRingHom.op_apply_apply`, `MonoidAlgebra.singleOneRingHom_apply`, `SeparationQuotient.normedMk_apply`, `HahnSeries.coeff.linearMap_apply`, `CentroidHom.map_mul_right'`, `AddMonoidHom.map_add'`, `MonoidWithZeroHom.map_mul'`, `NonarchAddGroupSeminorm.add_le_max'`, `OrderAddMonoidHom.toFun_eq_coe`, `SkewMonoidAlgebra.DistribMulActionHom.single_toFun`, `ContinuousAddMonoidHom.continuous_toFun`, `NonarchAddGroupSeminorm.neg'`, `ArithmeticFunction.toFun_eq`, `CategoryTheory.Functor.mapLinearMap_apply`, `AddGroupExtension.Splitting.rightInverse_rightHom`, `toFun_eq_coe`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`cancel_left` 📖	mathematical	`DFunLike.coe` `ZeroHom` `funLike`	`comp`	—	`ext` `comp_apply`
`cancel_right` 📖	mathematical	`DFunLike.coe` `ZeroHom` `funLike`	`comp`	—	`ext` `Function.Surjective.forall` `DFunLike.ext_iff`
`coe_coe` 📖	mathematical	—	`DFunLike.coe` `ZeroHom` `funLike` `ZeroHomClass.toZeroHom`	—	—
`coe_comp` 📖	mathematical	—	`DFunLike.coe` `ZeroHom` `funLike` `comp`	—	—
`coe_copy` 📖	mathematical	`DFunLike.coe` `ZeroHom` `funLike`	`copy`	—	—
`coe_copy_eq` 📖	mathematical	`DFunLike.coe` `ZeroHom` `funLike`	`copy`	—	`DFunLike.ext'`
`coe_id` 📖	mathematical	—	`DFunLike.coe` `ZeroHom` `funLike` `id`	—	—
`coe_mk` 📖	mathematical	—	`DFunLike.coe` `ZeroHom` `funLike`	—	—
`comp_apply` 📖	mathematical	—	`DFunLike.coe` `ZeroHom` `funLike` `comp`	—	—
`comp_assoc` 📖	mathematical	—	`comp`	—	—
`comp_id` 📖	mathematical	—	`comp` `id`	—	`ext`
`comp_zero` 📖	mathematical	—	`comp` `ZeroHom` `instZeroZeroHom`	—	`ext` `map_zero` `zeroHomClass`
`ext` 📖	—	`DFunLike.coe` `ZeroHom` `funLike`	—	—	`DFunLike.ext`
`ext_iff` 📖	mathematical	—	`DFunLike.coe` `ZeroHom` `funLike`	—	`ext`
`id_apply` 📖	mathematical	—	`DFunLike.coe` `ZeroHom` `funLike` `id`	—	—
`id_comp` 📖	mathematical	—	`comp` `id`	—	`ext`
`inverse_apply` 📖	mathematical	`DFunLike.coe` `ZeroHom` `funLike`	`inverse`	—	—
`map_zero` 📖	mathematical	—	`DFunLike.coe` `ZeroHom` `funLike`	—	`map_zero'`
`map_zero'` 📖	mathematical	—	`toFun`	—	—
`mk_coe` 📖	—	`DFunLike.coe` `ZeroHom` `funLike`	—	—	`ext`
`toFun_eq_coe` 📖	mathematical	—	`toFun` `DFunLike.coe` `ZeroHom` `funLike`	—	—
`zeroHomClass` 📖	mathematical	—	`ZeroHomClass` `ZeroHom` `funLike`	—	`map_zero'`
`zero_apply` 📖	mathematical	—	`DFunLike.coe` `ZeroHom` `funLike` `instZeroZeroHom`	—	—
`zero_comp` 📖	mathematical	—	`comp` `ZeroHom` `instZeroZeroHom`	—	—

ZeroHomClass

Definitions

Name	Category	Theorems
`toZeroHom` 📖	CompOp	1 math math: `ZeroHom.coe_coe`

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`map_zero` 📖	mathematical	—	`DFunLike.coe`	—	—

(root)

Definitions

Name	Category	Theorems
`AddHom` 📖	CompData	133 math math: `AddMagmaCat.coe_id`, `AddHom.coe_prodMap`, `Matrix.entryAddHom_apply`, `LinearMap.coe_mk`, `AddHom.coeFn_apply`, `AddHom.coe_srangeRestrict`, `AddHom.subsemigroupMap_apply_coe`, `AddMagma.FreeAddSemigroup.lift_symm_apply`, `FreeAddSemigroup.lift_comp_of`, `PNat.coeAddHom_apply`, `AddHom.withTopMap_apply`, `AddHom.toAddEquiv_apply`, `AddMagmaCat.forgetReflectsIsos`, `AddHom.toAddEquiv_symm_apply`, `FreeAddSemigroup.lift_symm_apply`, `AddHom.coe_prod`, `AddHom.srangeRestrict_surjective`, `AddHom.subsemigroupMap_surjective`, `AddHom.compLeft_apply`, `AddSemigrp.coe_id`, `AddMagma.FreeAddSemigroup.lift_comp_of`, `AddHom.ENatMap_apply`, `AddHom.comp_add`, `AddHom.op_apply_apply`, `AddSubsemigroup.mem_map_of_mem`, `FreeAddMagma.toFreeAddSemigroup_comp_of`, `AddCon.ker_coeAddHom`, `AddMagmaCat.forget_map`, `AddHom.srange_eq_top_iff_surjective`, `AddMagmaCat.ofHom_apply`, `FreeAddMagma.lift_comp_of`, `AddCon.ker_mkAddHom_eq`, `FreeAddSemigroup.lift_of`, `AddHom.addHomClass`, `AddMonoidAlgebra.mapDomainNonUnitalRingHom_apply`, `AddHom.toFun_eq_coe`, `TensorProduct.lift.tmul'`, `AddHom.mulOp_apply_apply`, `AddHom.mulLeft_apply`, `AddHom.prod_apply`, `AddMagma.FreeAddSemigroup.lift_comp_of'`, `AddHom.isClosed_range_coe`, `FreeAddMagma.map_of`, `AddMemClass.subtype_apply`, `AddMagmaCat.coe_comp`, `AddHom.op_symm_apply_apply`, `Pi.addHom_apply`, `WithZero.lift_coe`, `AddSemigrp.comp_apply`, `WithZero.mapAddHom_injective'`, `AddHom.id_apply`, `Set.coe_singletonAddHom`, `AddSubsemigroup.coe_map`, `AddHom.coe_id`, `AddHom.ext_iff`, `AddSemigrp.neg_hom_apply`, `AddHom.coe_comp`, `AddHom.coe_fst`, `AddSemigrp.hom_neg_apply`, `WithZero.mapAddHom_coe`, `FreeAddMagma.lift_comp_of'`, `AddHom.comp_apply`, `AddSemigrp.ofHom_apply`, `AddMagmaCat.ext_iff`, `AddHom.coe_add`, `FreeAddSemigroup.length_map`, `WithZero.lift_zero`, `AddHom.coe_coe`, `Finset.singletonAddHom_apply`, `AddMemClass.coe_subtype`, `AddSemigrp.forget₂_full`, `Finset.imageAddHom_apply`, `AddSubsemigroup.mem_comap`, `AddSubsemigroup.mem_map`, `AddCon.mkAddHom_apply`, `AddMonoidHom.toAddHom_injective`, `WithZero.map_injective'`, `AddMonoidAlgebra.le_infDegree_mul`, `AddHom.mulOp_symm_apply_apply`, `FreeAddSemigroup.map_of`, `FreeAddMagma.toFreeAddSemigroup_of`, `FreeAddMagma.toFreeAddSemigroup_map`, `Set.singletonAddHom_apply`, `nhdsAddHom_apply`, `addAddHom_apply`, `AddMagma.FreeAddSemigroup.lift_of`, `AddSemigrp.coe_comp`, `AddHom.subsemigroupComap_apply_coe`, `LinearMap.coe_toAddHom`, `AddSubsemigroup.coe_comap`, `addHomOfMemClosureRangeCoe_apply`, `Filter.pureAddHom_apply`, `FreeAddMagma.lift_symm_apply`, `WithZero.lift_unique`, `Pi.constAddHom_apply`, `FreeAddMagma.hom_ext_iff`, `AddHom.mulRight_apply`, `AddHom.coprod_apply`, `Finset.coe_singletonAddHom`, `AddMemClass.subtype_injective`, `AddHom.mem_eqLocus`, `AddMagmaCat.comp_apply`, `AddMagmaCat.id_apply`, `FreeAddSemigroup.hom_ext_iff`, `AddSemigrp.id_apply`, `AddHom.restrict_apply`, `AddHom.coe_snd`, `FreeAddMagma.lift_of`, `AddHom.mclosure_preimage_le`, `FreeAddMagma.length_toFreeAddSemigroup`, `AddHom.add_apply`, `AddHom.coe_mk`, `addHomOfTendsto_apply`, `FreeAddSemigroup.lift_of_add`, `AddHom.coe_ofDense`, `AddHom.mem_srange`, `AddEquiv.coe_toAddHom`, `AddMagma.FreeAddSemigroup.map_of`, `AddSubsemigroup.apply_coe_mem_map`, `AddMagmaCat.neg_hom_apply`, `Pi.evalAddHom_apply`, `WithZero.coeAddHom_apply`, `AddHom.add_comp`, `AddHom.withBotMap_apply`, `FreeAddSemigroup.lift_comp_of'`, `Filter.coe_pureAddHom`, `AddHom.coe_srange`, `AddHom.map_multiset_ne_zero_sum`, `AddHom.map_add`, `AddSemigrp.forgetReflectsIsos`, `AddHom.map_mclosure`, `AddMagmaCat.hom_neg_apply`, `AddSemigrp.ext_iff`
`AddHomClass` 📖	CompData	4 math math: `AddHom.addHomClass`, `AddEquivClass.instAddHomClass`, `AddMonoidHomClass.toAddHomClass`, `SemilinearMapClass.toAddHomClass`
`AddMonoidHomClass` 📖	CompData	16 math math: `AddEquivClass.instAddMonoidHomClass`, `RingHomClass.toAddMonoidHomClass`, `AddMonoidHom.instAddMonoidHomClass`, `ContinuousAddMonoidHom.instAddMonoidHomClass`, `SemilinearMapClass.instAddMonoidHomClass`, `AddMonoid.End.instAddMonoidHomClass`, `Derivation.instAddMonoidHomClass`, `AddGroupExtension.Splitting.instAddMonoidHomClass`, `SemiNormedGrp₁.instAddMonoidHomClassSubtypeNormedAddGroupHomCarrierNormNoninc`, `NormedAddGroupHom.toAddMonoidHomClass`, `CentroidHomClass.toAddMonoidHomClass`, `AddSubmonoid.LocalizationMap.instAddMonoidHomClass`, `instAddMonoidHomClassContMDiffAddMonoidMorphism`, `NonUnitalRingHomClass.toAddMonoidHomClass`, `OrderAddMonoidHom.instAddMonoidHomClass`, `DistribMulActionSemiHomClass.toAddMonoidHomClass`
`MonoidHomClass` 📖	CompData	15 math math: `MonoidHom.instMonoidHomClass`, `GroupExtension.Splitting.instMonoidHomClass`, `MulSemiringActionSemiHomClass.toMonoidHomClass`, `MulRingSeminormClass.toMonoidHomClass`, `OrderMonoidHom.instMonoidHomClass`, `MulCharClass.toMonoidHomClass`, `MonoidWithZeroHomClass.toMonoidHomClass`, `Submonoid.LocalizationMap.instMonoidHomClass`, `instMonoidHomClassContMDiffMonoidMorphism`, `StarMonoidHom.instMonoidHomClass`, `ContinuousMonoidHom.instMonoidHomClass`, `BialgHomClass.toMonoidHomClass`, `MulEquivClass.instMonoidHomClass`, `RingHomClass.toMonoidHomClass`, `Monoid.End.instMonoidHomClass`
`MulHom` 📖	CompData	135 math math: `MulHom.coe_snd`, `WithOne.coeMulHom_apply`, `MulHom.id_apply`, `MonoidAlgebra.liftMagma_apply_apply`, `MulHom.coe_srange`, `MulMemClass.subtype_apply`, `FreeMagma.lift_of`, `MulHom.coe_fst`, `FreeMagma.toFreeSemigroup_of`, `smulMulHom_apply`, `MulHom.subsemigroupComap_apply_coe`, `AbsoluteValue.coe_mk`, `MulHom.op_symm_apply_apply`, `WithOne.lift_coe`, `Pi.constMulHom_apply`, `MulHom.coe_srangeRestrict`, `MulHom.coe_prodMap`, `FreeMagma.hom_ext_iff`, `MulHom.coe_prod`, `Con.mkMulHom_apply`, `Semigrp.hom_inv_apply`, `MulHom.mul_apply`, `Magma.AssocQuotient.lift_comp_of'`, `Semigrp.inv_hom_apply`, `Subsemigroup.mem_map`, `WithOne.map_injective'`, `MulHom.mclosure_preimage_le`, `MulHom.comp_mul`, `MulHom.map_multiset_ne_zero_prod`, `MulHom.comp_apply`, `WithOne.mapMulHom_coe`, `Subsemigroup.mem_map_of_mem`, `MulHom.restrict_apply`, `Finset.coe_singletonMulHom`, `MulHom.srange_eq_top_iff_surjective`, `MulHom.toFun_eq_coe`, `Semigrp.id_apply`, `MulHom.mem_eqLocus`, `Con.ker_coeMulHom`, `MulMemClass.coe_subtype`, `NonUnitalRingHom.coe_mulHom_injective`, `MonoidAlgebra.ofMagma_apply`, `MulHom.coe_ofDense`, `MulHom.coe_coe`, `MulHom.coe_mk`, `MagmaCat.coe_comp`, `MulHom.subsemigroupMap_surjective`, `mulHomOfTendsto_apply`, `FreeSemigroup.lift_symm_apply`, `WithOne.lift_one`, `Semigrp.forget₂_full`, `MulHom.toMulEquiv_symm_apply`, `Semigrp.ext_iff`, `Semigrp.comp_apply`, `MulHom.prod_apply`, `Semigrp.forgetReflectsIsos`, `MulHom.compLeft_apply`, `MagmaCat.id_apply`, `Subsemigroup.coe_comap`, `MulHom.isClosed_range_coe`, `MulHom.mem_srange`, `MulHom.toMulEquiv_apply`, `Subsemigroup.mem_comap`, `MagmaCat.coe_id`, `MulHom.map_mclosure`, `MagmaCat.ofHom_apply`, `MulHom.coeFn_apply`, `MonoidAlgebra.mapDomainNonUnitalRingHom_apply`, `MulHom.mulHomClass`, `AddMonoidAlgebra.liftMagma_apply_apply`, `AbsoluteValue.coe_toMulHom`, `Set.singletonMulHom_apply`, `AddMonoidAlgebra.ofMagma_apply`, `Filter.coe_pureMulHom`, `MonoidAlgebra.liftMagma_symm_apply`, `AddMonoidAlgebra.liftMagma_symm_apply`, `MagmaCat.hom_inv_apply`, `MagmaCat.ext_iff`, `MulHom.coe_mul`, `Set.coe_singletonMulHom`, `MulHom.ext_iff`, `MulHom.coe_comp`, `FreeMagma.lift_symm_apply`, `Magma.AssocQuotient.lift_comp_of`, `Finset.imageMulHom_apply`, `MagmaCat.forget_map`, `WithOne.mapMulHom_injective'`, `FreeMagma.toFreeSemigroup_map`, `Magma.AssocQuotient.lift_of`, `FreeSemigroup.lift_comp_of'`, `MonoidHom.toMulHom_injective`, `FreeMagma.toFreeSemigroup_comp_of`, `FreeMagma.lift_comp_of`, `Filter.pureMulHom_apply`, `FreeMagma.length_toFreeSemigroup`, `MagmaCat.inv_hom_apply`, `FreeMagma.map_of`, `MulHom.op_apply_apply`, `Subsemigroup.apply_coe_mem_map`, `mulMulHom_apply`, `MulHom.srangeRestrict_surjective`, `MulHom.map_mul`, `Pi.evalMulHom_apply`, `MulHom.subsemigroupMap_apply_coe`, `MulHom.map_dvd`, `Finset.singletonMulHom_apply`, `Magma.AssocQuotient.map_of`, `MulHom.single_apply`, `MulHom.mul_comp`, `Semigrp.coe_id`, `MulHom.coe_id`, `NonUnitalAlgHom.coe_to_mulHom`, `MulEquiv.coe_toMulHom`, `FreeMagma.lift_comp_of'`, `Magma.AssocQuotient.lift_symm_apply`, `Semigrp.ofHom_apply`, `Semigrp.forget_map`, `FreeSemigroup.length_map`, `MagmaCat.forgetReflectsIsos`, `NonUnitalRingHom.coe_toMulHom`, `FreeSemigroup.lift_of_mul`, `FreeSemigroup.hom_ext_iff`, `mulHomOfMemClosureRangeCoe_apply`, `FreeSemigroup.lift_comp_of`, `MagmaCat.comp_apply`, `Semigrp.coe_comp`, `Subsemigroup.coe_map`, `MulHom.coprod_apply`, `WithOne.lift_unique`, `MulMemClass.subtype_injective`, `FreeSemigroup.map_of`, `nhdsMulHom_apply`, `Pi.mulHom_apply`, `FreeSemigroup.lift_of`, `Con.ker_mkMulHom_eq`
`MulHomClass` 📖	CompData	6 math math: `MulHom.mulHomClass`, `MonoidHomClass.toMulHomClass`, `AbsoluteValue.mulHomClass`, `MulEquivClass.instMulHomClass`, `NonUnitalAlgSemiHomClass.toMulHomClass`, `NonUnitalRingHomClass.toMulHomClass`
`OneHom` 📖	CompData	41 math math: `OneHom.withTopMap_apply`, `instSubsingletonOneHom`, `OneHom.inv_apply`, `OneHom.one_comp`, `OneHom.oneHomClass`, `instIsCancelMulOneHom`, `OneHom.id_apply`, `OneHom.map_one`, `OneHom.ext_iff`, `Set.coe_singletonOneHom`, `OneHom.coe_mk`, `MonoidHom.toOneHom_coe`, `Finset.coe_singletonOneHom`, `OneHom.coe_comp`, `OneHom.mul_comp`, `OneHom.coe_mul`, `OneHom.comp_one`, `OneHom.one_apply`, `OneHom.mul_apply`, `OneHom.div_comp`, `OneHom.coe_div`, `OneHom.coe_mulSingle`, `Filter.pureOneHom_apply`, `MonoidHom.coe_mk`, `OneHom.mulSingle_apply`, `OneHom.pow_apply`, `OneHom.inv_comp`, `OneHom.coe_inv`, `OneHom.coe_coe`, `Filter.coe_pureOneHom`, `Finset.imageOneHom_apply`, `OneHom.withBotMap_apply`, `Finset.singletonOneHom_apply`, `OneHom.zpow_apply`, `instIsRightCancelMulOneHom`, `OneHom.comp_apply`, `MonoidHom.toOneHom_injective`, `OneHom.toFun_eq_coe`, `OneHom.div_apply`, `OneHom.coe_id`, `instIsLeftCancelMulOneHom`
`OneHomClass` 📖	CompData	2 math math: `OneHom.oneHomClass`, `MonoidHomClass.toOneHomClass`
`ZeroHom` 📖	CompData	132 math math: `ZeroHom.zsmul_apply`, `ZeroHom.comp_zero`, `HahnSeries.orderTop_single_le`, `HahnSeries.map_single`, `AddMonoidHom.coe_mk`, `HahnSeries.cardSupp_single_of_ne`, `ZeroHom.neg_apply`, `ZeroHom.instSMulCommClass`, `HahnModule.coeff_single_smul_vadd`, `Set.coe_singletonZeroHom`, `ZeroHom.coe_comp`, `HahnSeries.coeff_single_mul_add`, `HahnSeries.orderTop_sub_pos`, `ZeroHom.nsmul_apply`, `HahnSeries.single_zero_natCast`, `HahnSeries.single_zero_ofScientific`, `HahnSeries.cardSupp_single_le`, `HahnSeries.ofPowerSeries_X_pow`, `LaurentSeries.valuation_single_zpow`, `HahnSeries.single_pow`, `AddMonoidHom.toZeroHom_injective`, `ZeroHom.sub_comp`, `AddMonoidHom.toZeroHom_coe`, `HahnSeries.coeff_single_same`, `HahnSeries.single_zero_nnratCast`, `ZeroHom.withTopMap_apply`, `ZeroHom.smul_comp`, `ZeroHom.coe_sub`, `LaurentSeries.ofPowerSeries_powerSeriesPart`, `ZeroHom.comp_apply`, `LaurentSeries.single_order_mul_powerSeriesPart`, `HahnSeries.single_zero_intCast`, `RatFunc.coe_X`, `HahnSeries.single_mul_single`, `HahnSeries.single_zero_mul_eq_smul`, `MonoidWithZeroHom.coe_mk`, `HahnSeries.cardSupp_truncLT_le`, `ZeroHom.instIsCentralScalar`, `HahnSeries.single_sub`, `HahnSeries.cardSupp_map_le`, `HahnSeries.single_eq_zero_iff`, `DFinsupp.sumZeroHom_apply`, `HahnSeries.order_single_mul_of_isRegular`, `HahnSeries.coeff_single_zero_mul`, `HahnSeries.coe_truncLTLinearMap`, `ZeroHom.coe_smul`, `ZeroHom.ext_iff`, `MonoidWithZeroHom.toZeroHom_coe`, `ZeroHom.sub_apply`, `HahnSeries.truncLT_add`, `HahnSeries.support_truncLT`, `HahnSeries.embDomain_single`, `HahnSeries.lt_orderTop_single`, `HahnSeries.coeff_truncLT`, `Filter.coe_pureZeroHom`, `instIsRightCancelAddZeroHom`, `ZeroHom.coe_id`, `HahnSeries.unit_aux`, `HahnSeries.ofPowerSeries_X`, `ZeroHom.zeroHomClass`, `HahnSeries.coeff_single_of_ne`, `HahnSeries.cardSupp_mul_single_le`, `HahnSeries.coeff_truncLT_of_le`, `ZeroHom.neg_comp`, `instIsLeftCancelAddZeroHom`, `HahnSeries.cardSupp_single_mul_le`, `ZeroHom.zero_comp`, `HahnSeries.coeff_ofFinsupp`, `ZeroHom.coe_mk`, `HahnSeries.map_zero`, `HahnSeries.single_neg`, `HahnSeries.single_injective`, `ZeroHom.smul_apply`, `HahnModule.coeff_single_zero_smul`, `HahnSeries.single_add`, `LaurentSeries.hasseDeriv_single_add`, `HahnSeries.coeff_single`, `HahnSeries.support_single_of_ne`, `HahnSeries.single_zero_one`, `DFinsupp.sumZeroHom_piSingle`, `NonarchAddGroupSeminorm.toZeroHom_eq_coe`, `ZeroHom.single_apply`, `ZeroHom.zero_apply`, `PowerSeries.coe_X`, `HahnSeries.single_div_single`, `HahnSeries.inv_def`, `HahnSeries.support_single_subset`, `Finsupp.mapRange.zeroHom_apply`, `HahnSeries.support_map_subset`, `RatFunc.single_one_eq_pow`, `HahnSeries.coeff_mul_single`, `instSubsingletonZeroHom`, `HahnSeries.coeff_truncLT_of_lt`, `ZeroHom.coe_neg`, `RatFunc.single_zpow`, `ZeroHom.instIsScalarTower`, `ZeroHom.coe_single`, `HahnSeries.coeff_toOrderTopSubOnePos_pow`, `ZeroHom.withBotMap_apply`, `ZeroHom.map_zero`, `HahnSeries.single_eq_zero`, `DFinsupp.sumZeroHom_single`, `HahnSeries.coeff_mul_single_add`, `HahnSeries.coeff_mul_single_zero`, `HahnSeries.truncLT_smul`, `HahnSeries.inv_single`, `HahnSeries.C_apply`, `HahnSeries.support_truncLT_subset`, `ZeroHom.id_apply`, `Finset.singletonZeroHom_apply`, `ZeroHom.add_apply`, `HahnSeries.leadingCoeff_of_single`, `RatFunc.single_inv`, `HahnSeries.single_zero_ratCast`, `instIsCancelAddZeroHom`, `HahnEmbedding.Partial.apply_of_mem_stratum`, `HahnSeries.single_zero_ofNat`, `HahnSeries.SummableFamily.powerSeriesFamily_smul`, `ZeroHom.coe_coe`, `SMulZeroClass.toZeroHom_apply`, `Finset.coe_singletonZeroHom`, `LaurentSeries.hasseDeriv_single`, `HahnSeries.order_single`, `MonoidWithZeroHom.toZeroHom_injective`, `HahnSeries.coeff_single_mul`, `Filter.pureZeroHom_apply`, `ZeroHom.coe_add`, `HahnSeries.orderTop_single`, `HahnModule.single_zero_smul_eq_smul`, `Finset.imageZeroHom_apply`, `ZeroHom.toFun_eq_coe`, `ZeroHom.add_comp`
`ZeroHomClass` 📖	CompData	8 math math: `QuadraticMap.zeroHomClass`, `AddGroupSeminormClass.toZeroHomClass`, `ZeroHom.zeroHomClass`, `AbsoluteValue.zeroHomClass`, `MulEquivClass.toZeroHomClass`, `AddMonoidHomClass.toZeroHomClass`, `MonoidWithZeroHomClass.toZeroHomClass`, `ContinuousMapZero.instZeroHomClass`
`instCoeTCAddHomOfAddHomClass` 📖	CompOp	—
`instCoeTCAddMonoidHomOfAddMonoidHomClass` 📖	CompOp	—
`instCoeTCMonoidHomOfMonoidHomClass` 📖	CompOp	—
`instCoeTCMulHomOfMulHomClass` 📖	CompOp	—
`instCoeTCOneHomOfOneHomClass` 📖	CompOp	—
`instCoeTCZeroHomOfZeroHomClass` 📖	CompOp	—
`instInhabitedAddHom` 📖	CompOp	—
`instInhabitedAddMonoidHom` 📖	CompOp	—
`instInhabitedMonoidHom` 📖	CompOp	—
`instInhabitedMulHom` 📖	CompOp	—
`instInhabitedOneHom` 📖	CompOp	—
`instInhabitedZeroHom` 📖	CompOp	—
`instOneMonoidHom` 📖	CompOp	40 math math: `AddChar.toMonoidHomEquiv_zero`, `groupHomology.map₁_one`, `sum_hom_units`, `Monoid.Coprod.fst_comp_inr`, `Subgroup.map_one_eq_bot`, `MonoidHom.ker_eq_top_iff`, `MonoidHom.one_comp`, `Polynomial.Gal.restrictDvd_def`, `LinearMap.det_def`, `MonoidHom.range_one`, `MonCat.hom_one`, `MonoidHom.range_eq_bot_iff`, `groupCohomology.map₁_one`, `MonoidHom.fst_comp_inr`, `groupCohomology.mapCocycles₁_one`, `QuotientGroup.mk'_comp_subtype`, `GroupSeminorm.comp_zero`, `Function.MulExact.monoidHom_comp_eq_zero`, `MonoidHom.restrict_eq_one_iff`, `AddChar.toMonoidHomEquiv_symm_one`, `SemidirectProduct.mulEquivProd_symm_apply_left`, `Rep.linearizationTrivialIso_inv_hom`, `Action.trivial_ρ`, `GroupExtension.rightHom_comp_inl`, `Monoid.Coprod.snd_comp_inl`, `Representation.smul_tprod_one_asModule`, `SemidirectProduct.rightHom_comp_inl`, `MonoidHom.snd_comp_inl`, `MonoidHom.one_apply`, `LinearMap.coe_det`, `Monoid.CoprodI.of_leftInverse`, `SemidirectProduct.mulEquivProd_apply`, `MonoidHom.mker_one`, `Representation.smul_one_tprod_asModule`, `SemidirectProduct.mulEquivProd_symm_apply_right`, `QuotientGroup.ker_le_range_iff`, `MonoidHom.range_le_ker_iff`, `MonoidHom.ker_one`, `Rep.linearizationTrivialIso_hom_hom`, `MonoidHom.comp_one`
`instOneMulHom` 📖	CompOp	—
`instOneOneHom` 📖	CompOp	3 math math: `OneHom.one_comp`, `OneHom.comp_one`, `OneHom.one_apply`
`instZeroAddHom` 📖	CompOp	—
`instZeroAddMonoidHom` 📖	CompOp	38 math math: `AddMonCat.hom_zero`, `AddCommGrpCat.hom_zero`, `AddMonoidHom.mapMatrix_zero`, `AddMonoidHom.fst_comp_inr`, `AddMonoidHom.restrict_eq_zero_iff`, `AddSubgroup.map_zero_eq_bot`, `AddMonoidHom.comp_zero`, `AddGroupSeminorm.comp_zero`, `AddMonoidHom.instSMulCommClass`, `AddMonoidHom.zero_apply`, `AddMonoidHom.smul_comp`, `QuotientAddGroup.ker_le_range_iff`, `QuotientAddGroup.mk'_comp_subtype`, `AddMonoidHom.zero_comp`, `AddMonoidHom.range_eq_bot_iff`, `AddMonoid.Coprod.fst_comp_inr`, `AddChar.toAddMonoidHomEquiv_symm_zero`, `AddMonoidHom.mker_zero`, `DFinsupp.sumAddHom_piSingle`, `AddMonoidHom.snd_comp_inl`, `AddMonoidHom.coe_smul`, `AddMonoidHom.instIsCentralScalar`, `AddMonoidHom.smul_apply`, `DomMulAct.instSMulCommClassAddMonoidHom_1`, `AddMonoidHom.associator_eq_zero_iff_associative`, `AddMonoidHom.range_le_ker_iff`, `DFinsupp.sumAddHom_zero`, `AddMonoidHom.associator_eq_zero`, `AddChar.toAddMonoidHomEquiv_zero`, `AddMonoidHom.ker_eq_top_iff`, `AddMonoidHom.range_zero`, `AddMonoid.Coprod.snd_comp_inl`, `AddGroupExtension.rightHom_comp_inl`, `Function.Exact.addMonoidHom_comp_eq_zero`, `AddMonoidHom.instIsScalarTower`, `AddMonoidHom.completion_zero`, `AddMonoidHom.ker_zero`, `AddMonoidHom.mapMatrix_smul`
`instZeroZeroHom` 📖	CompOp	10 math math: `ZeroHom.comp_zero`, `ZeroHom.instSMulCommClass`, `ZeroHom.smul_comp`, `ZeroHom.instIsCentralScalar`, `ZeroHom.coe_smul`, `ZeroHom.zero_comp`, `ZeroHom.smul_apply`, `DFinsupp.sumZeroHom_piSingle`, `ZeroHom.zero_apply`, `ZeroHom.instIsScalarTower`
`«term_→*_»` 📖	CompOp	—
`«term_→+_»` 📖	CompOp	—
`«term_→ₙ*_»` 📖	CompOp	—
`«term_→ₙ+_»` 📖	CompOp	—

Theorems

Name	Kind	Assumes	Proves	Validates	Depends On
`instSubsingletonAddMonoidHom` 📖	mathematical	—	`AddMonoidHom`	—	`Subsingleton.of_zeroHomClass` `AddMonoidHomClass.toZeroHomClass` `AddMonoidHom.instAddMonoidHomClass`
`instSubsingletonMonoidHom` 📖	mathematical	—	`MonoidHom`	—	`Subsingleton.of_oneHomClass` `MonoidHomClass.toOneHomClass` `MonoidHom.instMonoidHomClass`
`instSubsingletonOneHom` 📖	mathematical	—	`OneHom`	—	`Subsingleton.of_oneHomClass` `OneHom.oneHomClass`
`instSubsingletonZeroHom` 📖	mathematical	—	`ZeroHom`	—	`Subsingleton.of_zeroHomClass` `ZeroHom.zeroHomClass`
`iterate_map_add` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe`	—	`Function.Semiconj₂.iterate` `map_add`
`iterate_map_div` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe` `DivInvMonoid.toDiv` `Group.toDivInvMonoid`	—	`Function.Semiconj₂.iterate` `map_div`
`iterate_map_inv` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe` `DivInvMonoid.toInv` `Group.toDivInvMonoid`	—	`Function.Commute.iterate_left` `map_inv`
`iterate_map_mul` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe`	—	`Function.Semiconj₂.iterate` `map_mul`
`iterate_map_neg` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe` `SubNegMonoid.toNeg` `AddGroup.toSubNegMonoid`	—	`Function.Commute.iterate_left` `map_neg`
`iterate_map_nsmul` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe` `AddMonoid.toNatSMul`	—	`Function.Commute.iterate_left` `map_nsmul`
`iterate_map_one` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe`	—	`Function.iterate_fixed` `map_one`
`iterate_map_pow` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe` `Monoid.toNatPow`	—	`Function.Commute.iterate_left` `map_pow`
`iterate_map_sub` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid`	—	`Function.Semiconj₂.iterate` `map_sub`
`iterate_map_zero` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe`	—	`Function.iterate_fixed` `map_zero`
`iterate_map_zpow` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe` `DivInvMonoid.toZPow` `Group.toDivInvMonoid`	—	`Function.Commute.iterate_left` `map_zpow`
`iterate_map_zsmul` 📖	mathematical	—	`Nat.iterate` `DFunLike.coe` `SubNegMonoid.toZSMul` `AddGroup.toSubNegMonoid`	—	`Function.Commute.iterate_left` `map_zsmul`
`map_add` 📖	mathematical	—	`DFunLike.coe`	—	`AddHomClass.map_add`
`map_add_eq_zero` 📖	mathematical	`AddZero.toAdd` `AddZero.toZero`	`DFunLike.coe`	—	`map_add` `AddMonoidHomClass.toAddHomClass` `map_zero` `AddMonoidHomClass.toZeroHomClass`
`map_add_neg` 📖	mathematical	—	`DFunLike.coe` `AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `SubNegMonoid.toAddMonoid` `AddGroup.toSubNegMonoid` `SubNegMonoid.toNeg` `SubtractionMonoid.toSubNegMonoid`	—	`map_add` `AddMonoidHomClass.toAddHomClass` `map_neg`
`map_comp_add` 📖	mathematical	—	`DFunLike.coe` `Pi.instAdd`	—	`map_add`
`map_comp_add_neg` 📖	mathematical	—	`DFunLike.coe` `Pi.instAdd` `AddZero.toAdd` `AddZeroClass.toAddZero` `AddMonoid.toAddZeroClass` `SubNegMonoid.toAddMonoid` `AddGroup.toSubNegMonoid` `Pi.instNeg` `SubNegMonoid.toNeg` `SubtractionMonoid.toSubNegMonoid`	—	`map_comp_add` `AddMonoidHomClass.toAddHomClass` `map_comp_neg`
`map_comp_div` 📖	mathematical	—	`DFunLike.coe` `Pi.instDiv` `DivInvMonoid.toDiv` `Group.toDivInvMonoid` `DivisionMonoid.toDivInvMonoid`	—	`map_div`
`map_comp_div'` 📖	mathematical	`DFunLike.coe` `DivInvMonoid.toInv`	`Pi.instDiv` `DivInvMonoid.toDiv`	—	`map_div'`
`map_comp_inv` 📖	mathematical	—	`DFunLike.coe` `Pi.instInv` `DivInvMonoid.toInv` `Group.toDivInvMonoid` `DivisionMonoid.toDivInvMonoid`	—	`map_inv`
`map_comp_mul` 📖	mathematical	—	`DFunLike.coe` `Pi.instMul`	—	`map_mul`
`map_comp_mul_inv` 📖	mathematical	—	`DFunLike.coe` `Pi.instMul` `MulOne.toMul` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `DivInvMonoid.toMonoid` `Group.toDivInvMonoid` `Pi.instInv` `DivInvMonoid.toInv` `DivisionMonoid.toDivInvMonoid`	—	`map_comp_mul` `MonoidHomClass.toMulHomClass` `map_comp_inv`
`map_comp_neg` 📖	mathematical	—	`DFunLike.coe` `Pi.instNeg` `SubNegMonoid.toNeg` `AddGroup.toSubNegMonoid` `SubtractionMonoid.toSubNegMonoid`	—	`map_neg`
`map_comp_nsmul` 📖	mathematical	—	`DFunLike.coe` `Pi.instSMul` `AddMonoid.toNatSMul`	—	`map_nsmul`
`map_comp_one` 📖	mathematical	—	`DFunLike.coe` `Pi.instOne`	—	`map_one`
`map_comp_pow` 📖	mathematical	—	`DFunLike.coe` `Pi.instPow` `Monoid.toNatPow`	—	`map_pow`
`map_comp_sub` 📖	mathematical	—	`DFunLike.coe` `Pi.instSub` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `SubtractionMonoid.toSubNegMonoid`	—	`map_sub`
`map_comp_sub'` 📖	mathematical	`DFunLike.coe` `SubNegMonoid.toNeg`	`Pi.instSub` `SubNegMonoid.toSub`	—	`map_sub'`
`map_comp_zero` 📖	mathematical	—	`DFunLike.coe` `Pi.instZero`	—	`map_zero`
`map_comp_zpow` 📖	mathematical	—	`DFunLike.coe` `Pi.instPow` `DivInvMonoid.toZPow` `Group.toDivInvMonoid` `DivisionMonoid.toDivInvMonoid`	—	`map_comp_zpow'` `map_inv`
`map_comp_zpow'` 📖	mathematical	`DFunLike.coe` `DivInvMonoid.toInv`	`Pi.instPow` `DivInvMonoid.toZPow`	—	`map_zpow'`
`map_comp_zsmul` 📖	mathematical	—	`DFunLike.coe` `Pi.instSMul` `SubNegMonoid.toZSMul` `AddGroup.toSubNegMonoid` `SubtractionMonoid.toSubNegMonoid`	—	`map_comp_zsmul'` `map_neg`
`map_comp_zsmul'` 📖	mathematical	`DFunLike.coe` `SubNegMonoid.toNeg`	`Pi.instSMul` `SubNegMonoid.toZSMul`	—	`map_zsmul'`
`map_div` 📖	mathematical	—	`DFunLike.coe` `DivInvMonoid.toDiv` `Group.toDivInvMonoid` `DivisionMonoid.toDivInvMonoid`	—	`map_div'` `MonoidHomClass.toMulHomClass` `map_inv`
`map_div'` 📖	mathematical	`DFunLike.coe` `DivInvMonoid.toInv`	`DivInvMonoid.toDiv`	—	—
`map_eq_one_iff` 📖	—	`DFunLike.coe`	—	—	`map_one`
`map_eq_zero_iff` 📖	—	`DFunLike.coe`	—	—	`map_zero`
`map_inv` 📖	mathematical	—	`DFunLike.coe` `DivInvMonoid.toInv` `Group.toDivInvMonoid` `DivisionMonoid.toDivInvMonoid`	—	`eq_inv_of_mul_eq_one_left` `map_mul_eq_one` `inv_mul_cancel`
`map_mul` 📖	mathematical	—	`DFunLike.coe`	—	`MulHomClass.map_mul`
`map_mul_eq_one` 📖	mathematical	`MulOne.toMul` `MulOne.toOne`	`DFunLike.coe`	—	`map_mul` `MonoidHomClass.toMulHomClass` `map_one` `MonoidHomClass.toOneHomClass`
`map_mul_inv` 📖	mathematical	—	`DFunLike.coe` `MulOne.toMul` `MulOneClass.toMulOne` `Monoid.toMulOneClass` `DivInvMonoid.toMonoid` `Group.toDivInvMonoid` `DivInvMonoid.toInv` `DivisionMonoid.toDivInvMonoid`	—	`map_mul` `MonoidHomClass.toMulHomClass` `map_inv`
`map_ne_one_iff` 📖	—	`DFunLike.coe`	—	—	`Iff.not` `map_eq_one_iff`
`map_ne_zero_iff` 📖	—	`DFunLike.coe`	—	—	`Iff.not` `map_eq_zero_iff`
`map_neg` 📖	mathematical	—	`DFunLike.coe` `SubNegMonoid.toNeg` `AddGroup.toSubNegMonoid` `SubtractionMonoid.toSubNegMonoid`	—	`eq_neg_of_add_eq_zero_left` `map_add_eq_zero` `neg_add_cancel`
`map_nsmul` 📖	mathematical	—	`DFunLike.coe` `AddMonoid.toNatSMul`	—	`zero_nsmul` `map_zero` `AddMonoidHomClass.toZeroHomClass` `succ_nsmul` `map_add` `AddMonoidHomClass.toAddHomClass`
`map_one` 📖	mathematical	—	`DFunLike.coe`	—	`OneHomClass.map_one`
`map_pow` 📖	mathematical	—	`DFunLike.coe` `Monoid.toNatPow`	—	`pow_zero` `map_one` `MonoidHomClass.toOneHomClass` `pow_succ` `map_mul` `MonoidHomClass.toMulHomClass`
`map_sub` 📖	mathematical	—	`DFunLike.coe` `SubNegMonoid.toSub` `AddGroup.toSubNegMonoid` `SubtractionMonoid.toSubNegMonoid`	—	`map_sub'` `AddMonoidHomClass.toAddHomClass` `map_neg`
`map_sub'` 📖	mathematical	`DFunLike.coe` `SubNegMonoid.toNeg`	`SubNegMonoid.toSub`	—	—
`map_zero` 📖	mathematical	—	`DFunLike.coe`	—	`ZeroHomClass.map_zero`
`map_zpow` 📖	mathematical	—	`DFunLike.coe` `DivInvMonoid.toZPow` `Group.toDivInvMonoid` `DivisionMonoid.toDivInvMonoid`	—	`map_zpow'` `map_inv`
`map_zpow'` 📖	mathematical	`DFunLike.coe` `DivInvMonoid.toInv`	`DivInvMonoid.toZPow`	—	`zpow_natCast` `map_pow` `zpow_negSucc`
`map_zsmul` 📖	mathematical	—	`DFunLike.coe` `SubNegMonoid.toZSMul` `AddGroup.toSubNegMonoid` `SubtractionMonoid.toSubNegMonoid`	—	`map_zsmul'` `map_neg`
`map_zsmul'` 📖	mathematical	`DFunLike.coe` `SubNegMonoid.toNeg`	`SubNegMonoid.toZSMul`	—	`natCast_zsmul` `map_nsmul` `negSucc_zsmul`
`ne_one_of_map` 📖	—	—	—	—	`map_one`
`ne_zero_of_map` 📖	—	—	—	—	`map_zero`

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