Documentation Verification Report

Commute

📁 Source: Mathlib/Data/Nat/Cast/Commute.lean

Statistics

MetricCount
DefinitionsCommute, Commute
2
TheoremsnatCast_mul_left, natCast_mul_natCast_mul, natCast_mul_right, natCast_mul_self, ofNat_left, ofNat_right, self_natCast_mul, self_natCast_mul_natCast_mul, cast_comm, cast_commute, commute_cast, natCast_mul_left, natCast_mul_natCast_mul, natCast_mul_right
14
Total16

Commute

Theorems

NameKindAssumesProvesValidatesDepends On
natCast_mul_left 📖mathematicalCommute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Commute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
SemiconjBy.natCast_mul_left
natCast_mul_natCast_mul 📖mathematicalCommute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Commute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
natCast_mul_right 📖mathematicalCommute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
Commute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
SemiconjBy.natCast_mul_right
natCast_mul_self 📖mathematicalCommute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
natCast_mul_left
refl
ofNat_left 📖mathematicalCommute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
instOfNatAtLeastTwo
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Nat.cast_commute
ofNat_right 📖mathematicalCommute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
instOfNatAtLeastTwo
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Nat.commute_cast
self_natCast_mul 📖mathematicalCommute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
natCast_mul_right
refl
self_natCast_mul_natCast_mul 📖mathematicalCommute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
natCast_mul_natCast_mul
refl

Function

Definitions

NameCategoryTheorems
Commute 📖MathDef
33 mathmath: Commute.iterate_iterate, CircleDeg1Lift.commute_nat_add, Commute.id_left, CircleDeg1Lift.commute_iff_commute, CliffordAlgebra.reverse_involute_commute, Commute.set_image, Commute.comp_left, Commute.finset_map, Commute.refl, Commute.finset_image, Commute.iterate_self, Commute.iterate_iterate_self, CircleDeg1Lift.commute_add_int, CircleDeg1Lift.commute_sub_int, CircleDeg1Lift.commute_add_nat, Commute.function_commute_mul_left, Commute.comp_right, CircleDeg1Lift.commute_int_add, Commute.iterate_right, Commute.self_iterate, iterate_commute, Commute.iterate_left, Commute.filter_map, Semiconj.commute, Commute.option_map, WittVector.verschiebung_frobenius_comm, Commute.function_commute_mul_right, Commute.filter_comap, AddCommute.function_commute_add_left, AddCommute.function_commute_add_right, CircleDeg1Lift.commute_sub_nat, Commute.symm, Commute.id_right

Nat

Theorems

NameKindAssumesProvesValidatesDepends On
cast_comm 📖mathematicalDistrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Commute.eq
cast_commute
cast_commute 📖mathematicalCommute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
cast_zero
Commute.zero_left
cast_succ
Commute.add_left
Commute.one_left
commute_cast 📖mathematicalCommute
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Commute.symm
cast_commute

SemiconjBy

Theorems

NameKindAssumesProvesValidatesDepends On
natCast_mul_left 📖mathematicalSemiconjBy
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
SemiconjBy
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
mul_left
Nat.cast_commute
natCast_mul_natCast_mul 📖mathematicalSemiconjBy
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
SemiconjBy
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
natCast_mul_right 📖mathematicalSemiconjBy
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
SemiconjBy
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
AddMonoidWithOne.toNatCast
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
mul_right
Nat.commute_cast

(root)

Definitions

NameCategoryTheorems
Commute 📖MathDef
287 mathmath: Commute.exp_right, Units.commute_iff_inv_mul_cancel, Commute.units_zpow_right, Commute.realPart_imaginaryPart, Nat.commute_cast, IsSelfAdjoint.commute_cfcHom, Units.commute_inv_coe, Polynomial.smeval_commute, Matrix.Commute.zpow_self, LaurentPolynomial.commute_T, Commute.cfcₙ_nnreal, Prod.commute_iff, Commute.inv_right_iff₀, LinearMap.IsIdempotentElem.commute_iff, Algebra.commute_of_mem_adjoin_singleton_of_commute, Commute.mul_right, cfc_commute_cfc, Commute.units_inv_right_iff, Commute.div_right, MvPowerSeries.commute_X_pow, commute_star_star, CircleDeg1Lift.commute_iff_commute, Commute.ofNat_left, NNRat.cast_commute, Commute.zpow_right, Commute.conj, Commute.pow_right, Matrix.mem_range_scalar_iff_commute_transvectionStruct, Commute.inv_left_iff, Commute.neg_right_iff, MulAction.not_commute_of_disjoint_movedBy_preimage, commute_invOf, CliffordAlgebra.commute_map_mul_map_of_isOrtho_of_mem_evenOdd_zero_right, Commute.intCast_right, Commute.exp_left, Equiv.Perm.Disjoint.commute, Multiset.noncommProd_commute, card_comm_eq_card_conjClasses_mul_card, Equiv.Perm.IsCycle.commute_iff', Pi.commute_iff, Multiset.map_noncommProd_aux, commute_lmul_rmul, IsStarNormal.star_comm_self, Commute.cfcAbs_cfcAbs, Commute.zero_left, Commute.sub_right, DihedralGroup.card_commute_odd, Algebra.commute_of_mem_adjoin_of_forall_mem_commute, Commute.prod, Module.Basis.mem_center_iff, LinearMap.commute_mulLeft_right, Commute.zpow_right₀, Equiv.Perm.cycleFactorsFinset_eq_finset, Algebra.commute_of_mem_adjoin_self, Multiset.noncommProd_add, Equiv.Perm.commute_iff_of_mem_cycleFactorsFinset, Commute.invOf_right, Finset.noncommProd_union_of_disjoint, AddMonoidAlgebra.of'_commute, LinearMap.restrict_commute, NonUnitalAlgebra.commute_of_mem_adjoin_of_forall_mem_commute, Commute.cfc_real, AddMonoidAlgebra.single_commute_single, Int.commute_cast, Commute.list_prod_right, MvPowerSeries.commute_X, Commute.units_of_val, isStarNormal_iff, MulOpposite.commute_unop, Commute.inv_left_iff₀, Commute.list_sum_left, PowerSeries.commute_X_pow, commute_lmul_sq_rmul, Commute.exp, Equiv.Perm.self_mem_cycle_factors_commute, commute_iff_eq, Commute.pow_pow, Commute.smul_left, DihedralGroup.oddCommuteEquiv_symm_apply, OrderMonoidHom.commute_inl_inr, Commute.div_left, MonoidHom.noncommPiCoprod_apply, IsSelfAdjoint.commute_iff, Commute.smul_left_iff, DihedralGroup.oddCommuteEquiv_apply, Finset.noncommProd_lemma, Commute.inv_inv_iff, Commute.of_map, Commute.neg_left_iff, Commute.natCast_mul_right, Equiv.Perm.support_noncommProd, commute_lmul_rmul_sq, Commute.zero_right, NonUnitalStarAlgebra.commute_of_mem_adjoin_singleton_of_commute, IsMulCentral.comm, Commute.neg_left, Commute.self_zpow, isMulCentral_iff, Matrix.mem_range_scalar_iff_commute_single, Commute.cfcₙHom, Matrix.exp_sum_of_commute, Matrix.Commute.zpow_zpow, Commute.tsum_right, Commute.units_val, LinearMap.IsIdempotentElem.commute_iff_of_isUnit, Commute.instRefl, NonUnitalNonAssocCommSemiring.mem_center_iff, Commute.mul_left, Units.commute_iff_inv_mul_cancel_assoc, Commute.units_inv_left_iff, Commute.zpow_zpow_self₀, Commute.pow_left, Commute.multiset_sum_right, Commute.symm_iff, Matrix.PosDef.commute_iff, Commute.ringInverse_ringInverse, Matrix.scalar_commute, Finset.sum_pow_of_commute, IsLprojection.commute, Matrix.Commute.zpow_zpow_self, NonUnitalAlgebra.commute_of_mem_adjoin_self, Commute.smul_right_iff, Units.commute_coe_inv, Commute.zpow_left, commute_lmul_lmul_sq, Equiv.Perm.commute_of_mem_cycleFactorsFinset_commute, Commute.pow_pow_self, Commute.zpow_self₀, Matrix.Commute.zpow_right, Commute.zpow_zpow_self, Commute.add_left, Commute.cfcAbs_right, Matrix.IsHermitian.commute_iff, Rat.commute_cast, Equiv.Perm.OnCycleFactors.mem_ker_toPermHom_iff, Commute.self_intCast_mul_intCast_mul, AddMonoidHom.mulRight_eq_mulLeft_iff_forall_commute, Matrix.mem_range_scalar_iff_commute_single', Commute.units_zpow_left, MonoidAlgebra.single_commute, Commute.tmul, instInfiniteProdSubtypeCommute, QuaternionAlgebra.coe_commute, Commute.list_prod_left, commute_iff_mul_nonneg, Commute.zpow_left₀, Commute.invOf_left, CliffordAlgebra.commute_map_mul_map_of_isOrtho_of_mem_evenOdd_zero_left, Commute.units_inv_left, Finset.sum_pow_eq_sum_piAntidiag_of_commute, Commute.zpow_zpow, Commute.natCast_mul_left, Commute.smul_right_iff₀, Commute.add_right, Polynomial.commute_X, Commute.smul_right, Commute.pow_self, Algebra.commute_algebraMap_left, commutatorElement_eq_one_iff_commute, Matrix.Commute.self_zpow, Module.End.commute_id_right, Matrix.Commute.zpow_left, Commute.inv_left, Commute.on_refl, NonUnitalStarAlgebra.commute_of_mem_adjoin_self, Commute.inv_right_iff, Commute.symm, MonoidHom.commute_inl_inr, Commute.multiset_sum_left, OrderMonoidHom.commute_inlₗ_inrₗ, Commute.refl, Commute.cfcHom, IsStrictlyPositive.commute_iff, LieModule.commute_toEnd_of_mem_center_right, NonUnitalAlgebra.commute_of_mem_adjoin_singleton_of_commute, Finset.noncommProd_cons', Commute.units_inv_right, Commute.of_orderOf_dvd_two, Multiset.noncommProd_cons', CFC.commute_abs_self, MulOpposite.commute_op, Commute.cfc, MonoidAlgebra.of_commute, Algebra.commute_algebraMap_right, PiTensorProduct.tprod_noncommProd, Commute.neg_one_left, Commute.neg_right, Finset.noncommProd_mul_distrib_aux, IsSelfAdjoint.commute_cfcₙHom, Commute.neg_one_right, DualNumber.commute_eps_left, ContinuousLinearMap.IsIdempotentElem.commute_iff_of_isUnit, Commute.units_val_iff, Finset.map_noncommProd, ContinuousLinearMap.IsIdempotentElem.commute_iff, Quaternion.coe_commute, IsStarProjection.commute_of_le, DualNumber.commute_eps_right, Commute.cfcₙ, Commute.ofNat_right, Commute.cfcAbs_left, Commute.conj_iff, Int.cast_commute, Equiv.Perm.cycleFactorsFinset_mem_commute, Commute.intCast_mul_left, Pi.mulSingle_apply_commute, Units.commute_iff_mul_inv_cancel_assoc, MonoidAlgebra.single_commute_single, Commute.map, Commute.natCast_mul_self, Commute.sum_right, Matrix.commute_diagonal, IsIdempotentElem.commute_of_anticommute, Commute.tprod, Commute.star_right, Commute.self_zpow₀, Finset.noncommProd_insert_of_notMem', Subgroup.commute_of_normal_of_disjoint, commute_map_iff, Commute.cfc_nnreal, Equiv.Perm.IsCycle.forall_commute_iff, commute_star_comm, Multiset.noncommProd_cons, NNRat.commute_cast, Commute.natCast_mul_natCast_mul, Equiv.Perm.IsCycle.commute_iff, Commute.zpow_zpow₀, IsSelfAdjoint.commute_cfcₙ, LieAlgebra.commute_ad_of_commute, Commute.smul_left_iff₀, Commute.self_natCast_mul_natCast_mul, Commute.tsum_left, Commute.inv_right₀, Commute.self_intCast_mul, NonUnitalStarAlgebra.commute_of_mem_adjoin_of_forall_mem_commute, Equiv.Perm.cycleFactorsFinset_mem_commute', Commute.inv_inv, Subgroup.commute_subtype_of_commute, Commute.unop, NormedSpace.exp_sum_of_commute, Commute.op, Commute.star_left, Commute.list_sum_right, Commute.pi, Polynomial.commute_X_pow, Rat.cast_commute, Commute.all, MonoidHom.FixedPointFree.commute_all_of_involutive, Commute.one_right, Matrix.scalar_commute_iff, Finset.noncommProd_commute, Commute.inv_left₀, Commute.intCast_mul_intCast_mul, AddMonoidAlgebra.single_commute, MonoidHom.commute_noncommPiCoprod, RootPairing.isOrthogonal_comm, LieModule.commute_toEnd_of_mem_center_left, MonoidWithZeroHom.commute_inl_inr, Units.commute_iff_mul_inv_cancel, cfcₙ_commute_cfcₙ, Polynomial.smeval_commute_left, commute_iff_lie_eq, Commute.intCast_mul_self, Commute.intCast_mul_right, Finset.noncommProd_insert_of_notMem, Commute.sum_left, commProb_def, Commute.cfcₙ_real, Commute.self_pow, AddMonoidHom.mulLeft_eq_mulRight_iff_forall_commute, IsSelfAdjoint.commute_cfc, Commute.sub_left, Nat.cast_commute, PowerSeries.commute_X, Commute.expUnitary, commute_rmul_rmul_sq, Finset.noncommProd_cons, Module.End.commute_id_left, Commute.self_natCast_mul, isStarNormal_iff_commute_realPart_imaginaryPart, Commute.intCast_left, Equiv.Perm.commute_ofSubtype_noncommPiCoprod, Commute.zpow_self, Commute.inv_right, MvPowerSeries.commute_monomial, Commute.star_star, Commute.one_left

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