Documentation Verification Report

Basic

πŸ“ Source: Mathlib/Algebra/DirectSum/Basic.lean

Statistics

MetricCount
DefinitionsIsInternal, addEquivProd, addEquivProdDirectSum, coeAddMonoidHom, coeFnAddMonoidHom, equivCongrLeft, fromAddMonoid, id, instAddCommGroup, map, mk, of, setToSet, sigmaCurry, sigmaCurryEquiv, sigmaUncurry, toAddMonoid, unique, uniqueOfIsEmpty, «term⨁_,_», instAddCommMonoidDirectSum, instCoeFunDirectSumForall, instDFunLikeDirectSum, instDecidableEqDirectSum, instInhabitedDirectSum
25
TheoremsaddSubmonoid_iSup_eq_top, addEquivProdDirectSum_apply, addEquivProdDirectSum_symm_apply_support', addEquivProdDirectSum_symm_apply_toFun, addHom_ext, addHom_ext', addHom_ext'_iff, add_apply, coeAddMonoidHom_eq_dfinsuppSum, coeAddMonoidHom_of, coeFnAddMonoidHom_apply, coe_of_apply, equivCongrLeft_apply, ext, ext_iff, finite_support, fromAddMonoid_of, fromAddMonoid_of_apply, induction_on, map_apply, map_comp, map_eq_iff, map_id, map_injective, map_of, map_surjective, mk_apply_of_mem, mk_apply_of_notMem, mk_injective, of_apply, of_eq_of_ne, of_eq_same, of_injective, sigmaCurry_apply, sigmaUncurry_apply, sub_apply, sum_support_of, sum_univ_of, support_of, support_of_subset, support_subset, support_zero, unique, toAddMonoid_inj, toAddMonoid_injective, toAddMonoid_of, zero_apply
47
Total72

DirectSum

Definitions

NameCategoryTheorems
IsInternal πŸ“–MathDef
21 mathmath: Submodule.isInternal_prime_power_torsion_of_pid, isInternal_ne_bot_iff, isInternal_submodule_iff_iSupIndep_and_iSup_eq_top, Submodule.isInternal_prime_power_torsion_of_is_torsion_by_ideal, OrthogonalFamily.isInternal_iff_of_isComplete, AddMonoidAlgebra.gradeBy.isInternal, OrthogonalFamily.isInternal_iff, isInternal_submodule_of_iSupIndep_of_iSup_eq_top, Decomposition.isInternal, Submodule.exists_isInternal_prime_power_torsion_of_pid, Submodule.torsionBySet_isInternal, LinearMap.IsSymmetric.directSum_isInternal_of_commute, LinearMap.IsSymmetric.direct_sum_isInternal, HahnEmbedding.ArchimedeanStrata.isInternal_stratum', Submodule.exists_isInternal_prime_power_torsion, AddMonoidAlgebra.grade.isInternal, Submodule.isInternal_prime_power_torsion, LinearMap.IsSymmetric.LinearMap.IsSymmetric.directSum_isInternal_of_pairwise_commute, Submodule.torsionBy_isInternal, isInternal_biSup_submodule_of_iSupIndep, isInternal_submodule_iff_isCompl
addEquivProd πŸ“–CompOpβ€”
addEquivProdDirectSum πŸ“–CompOp
5 mathmath: addEquivProdDirectSum_symm_apply_toFun, addEquivProdDirectSum_apply, addEquivProdDirectSum_symm_apply_support', lequivProdDirectSum_symm_apply, lequivProdDirectSum_apply
coeAddMonoidHom πŸ“–CompOp
6 mathmath: coeAddMonoidHom_eq_dfinsuppSum, MvPolynomial.DirectSum.coeAddMonoidHom_eq_support_sum, coeAddMonoidHom_of, Decomposition.left_inv, Decomposition.right_inv, OrthogonalFamily.projection_directSum_coeAddHom
coeFnAddMonoidHom πŸ“–CompOp
5 mathmath: mker_map, mrange_map, ker_map, range_map, coeFnAddMonoidHom_apply
equivCongrLeft πŸ“–CompOp
1 mathmath: equivCongrLeft_apply
fromAddMonoid πŸ“–CompOp
2 mathmath: fromAddMonoid_of_apply, fromAddMonoid_of
id πŸ“–CompOpβ€”
instAddCommGroup πŸ“–CompOp
34 mathmath: Module.Presentation.directSum_var, ModuleCat.ΞΉ_coprodIsoDirectSum_hom_apply, lie_module_bracket_apply, sub_apply, AddChar.directSum_apply, toAddMonoidAlgebra_sub, lie_of_of_ne, lie_of, instLieModule, AdicCompletion.sumInv_apply, Module.Presentation.directSum_relation, AdicCompletion.sum_lof, ModuleCat.lof_coprodIsoDirectSum_inv, Module.Relations.Solution.directSum_var, Module.Presentation.directSum_G, AdicCompletion.sumInv_comp_sum, ModuleCat.lof_coprodIsoDirectSum_inv_apply, Module.Relations.Solution.IsPresentation.directSum, ker_map, range_map, AdicCompletion.sum_of, ModuleCat.ΞΉ_coprodIsoDirectSum_hom, AddChar.directSum_injective, AddMonoidAlgebra.toDirectSum_sub, AdicCompletion.sumEquivOfFintype_apply, AddMonoidAlgebra.toDirectSum_neg, TensorProduct.gradedCommAux_lof_tmul, AdicCompletion.component_sumInv, Module.Presentation.directSum_R, toAddMonoidAlgebra_neg, Module.FaithfullyFlat.directSum, TensorProduct.gradedComm_of_tmul_of, AdicCompletion.sumEquivOfFintype_symm_apply, AdicCompletion.sum_comp_sumInv
map πŸ“–CompOp
15 mathmath: mker_map, mrange_map, coe_congr_addEquiv, map_injective, lmap_eq_map, map_of, ker_map, range_map, map_apply, map_comp, coe_congrAddEquiv, map_eq_iff, toAddMonoidHom_lmap, map_surjective, map_id
mk πŸ“–CompOpβ€”
of πŸ“–CompOp
85 mathmath: TensorAlgebra.ofDirectSum_of_tprod, mulHom_of_of, ExteriorAlgebra.GradedAlgebra.liftΞΉ_eq, Gmodule.smulAddMonoidHom_apply_of_of, qExpansionRingHom_apply, of_smul, fromAddMonoid_of_apply, TensorAlgebra.toDirectSum_tensorPower_tprod, toAddMonoid.unique, fromAddMonoid_of, decompose_of_mem, coe_of_mul_apply_aux, Gmodule.of_smul_of, sum_univ_of, lie_of_of_ne, Module.DirectLimit.quotMk_of, lie_of, support_of, algebraMap_toAddMonoid_hom, AddMonoidAlgebra.toDirectSum_single, qExpansion_of_mul, coeLinearMap_of, coe_of_mul_apply_add, AddMonoidAlgebra.decomposeAux_coe, ofPow, of_zero_smul, one_def, of_mul_of, coe_mul_of_apply_of_not_le, GradedAlgebra.proj_recompose, algebraMap_apply, lof_eq_of, of_zero_mul, coeAlgHom_of, CliffordAlgebra.GradedAlgebra.ΞΉ_apply, lieAlgebraOf_apply, map_of, coe_of_mul_apply_of_not_le, AddMonoidAlgebra.GradesBy.decompose_single, coeAddMonoidHom_of, of_injective, AdicCompletion.sum_of, coe_mul_of_apply_of_mem_zero, of_eq_same, coeRingHom_of, coe_mul_of_apply_aux, lie_of_same, TensorAlgebra.GradedAlgebra.ΞΉ_apply, toSemiring_of, toAddMonoidAlgebra_of, AddMonoidAlgebra.grade.decompose_single, addHom_ext'_iff, of_zero_pow, of_natCast, coe_mul_of_apply_add, ringHom_ext'_iff, coe_of_mul_apply_of_le, support_of_subset, coe_mul_of_apply_of_le, coe_of_mul_apply_of_mem_zero, of_apply, of_intCast, mul_eq_sum_support_ghas_mul, liftRingHom_symm_apply_coe, coe_mul_of_apply, mul_eq_dfinsuppSum, of_eq_of_gradedMonoid_eq, ofList_dProd, decompose_coe, AddMonoidAlgebra.decomposeAux_single, list_prod_ofFn_of_eq_dProd, coe_of_apply, CliffordAlgebra.GradedAlgebra.lift_ΞΉ_eq, TensorAlgebra.toDirectSum_ΞΉ, lmap_of, of_zero_one, decompose_symm_of, ExteriorAlgebra.GradedAlgebra.ΞΉ_apply, sum_support_of, qExpansion_of_pow, toAddMonoid_of, of_zero_ofNat, coe_of_mul_apply, GradedRing.proj_recompose, of_eq_of_ne
setToSet πŸ“–CompOpβ€”
sigmaCurry πŸ“–CompOp
1 mathmath: sigmaCurry_apply
sigmaCurryEquiv πŸ“–CompOpβ€”
sigmaUncurry πŸ“–CompOp
1 mathmath: sigmaUncurry_apply
toAddMonoid πŸ“–CompOp
8 mathmath: coe_toModule_eq_coe_toAddMonoid, toAddMonoid.unique, AddChar.directSum_apply, toSemiring_apply, toSemiring_coe_addMonoidHom, toAddMonoid_injective, toAddMonoid_of, toAddMonoid_inj
unique πŸ“–CompOpβ€”
uniqueOfIsEmpty πŸ“–CompOpβ€”
Β«term⨁_,_Β» πŸ“–CompOpβ€”

Theorems

NameKindAssumesProvesValidatesDepends On
addEquivProdDirectSum_apply πŸ“–mathematicalβ€”DFunLike.coe
AddEquiv
DirectSum
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
instAddCommMonoidDirectSum
Prod.instAdd
EquivLike.toFunLike
AddEquiv.instEquivLike
addEquivProdDirectSum
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
DFinsupp.comapDomain
Option.some_injective
β€”β€”
addEquivProdDirectSum_symm_apply_support' πŸ“–mathematicalβ€”DFinsupp.support'
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFunLike.coe
AddEquiv
DirectSum
Prod.instAdd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
instAddCommMonoidDirectSum
EquivLike.toFunLike
AddEquiv.instEquivLike
AddEquiv.symm
addEquivProdDirectSum
Trunc.map
Multiset
Multiset.cons
Multiset.map
DFinsupp
β€”β€”
addEquivProdDirectSum_symm_apply_toFun πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
AddEquiv
DirectSum
Prod.instAdd
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
instAddCommMonoidDirectSum
EquivLike.toFunLike
AddEquiv.instEquivLike
AddEquiv.symm
addEquivProdDirectSum
β€”β€”
addHom_ext πŸ“–β€”DFunLike.coe
AddMonoidHom
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
of
β€”β€”DFinsupp.addHom_ext
addHom_ext' πŸ“–β€”AddMonoidHom.comp
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
of
β€”β€”addHom_ext
DFunLike.congr_fun
addHom_ext'_iff πŸ“–mathematicalβ€”AddMonoidHom.comp
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
of
β€”addHom_ext'
add_apply πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
DirectSum
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
instAddCommMonoidDirectSum
β€”β€”
coeAddMonoidHom_eq_dfinsuppSum πŸ“–mathematicalβ€”DFunLike.coe
AddMonoidHom
DirectSum
SetLike.instMembership
AddSubmonoidClass.toAddCommMonoid
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
coeAddMonoidHom
DFinsupp.sum
AddZero.toZero
β€”DFinsupp.sumAddHom_apply
coeAddMonoidHom_of πŸ“–mathematicalβ€”DFunLike.coe
AddMonoidHom
DirectSum
SetLike.instMembership
AddSubmonoidClass.toAddCommMonoid
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
coeAddMonoidHom
of
β€”toAddMonoid_of
coeFnAddMonoidHom_apply πŸ“–mathematicalβ€”DFunLike.coe
AddMonoidHom
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
Pi.addZeroClass
AddMonoidHom.instFunLike
coeFnAddMonoidHom
DFinsupp
AddZero.toZero
DFinsupp.instDFunLike
β€”β€”
coe_of_apply πŸ“–mathematicalβ€”SetLike.instMembership
DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
AddMonoidHom
DirectSum
AddSubmonoidClass.toAddCommMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
of
ZeroMemClass.zero
AddSubmonoidClass.toZeroMemClass
β€”AddSubmonoidClass.toZeroMemClass
Decidable.eq_or_ne
of_eq_same
of_eq_of_ne
ZeroMemClass.coe_zero
equivCongrLeft_apply πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
Equiv.symm
DFinsupp.instDFunLike
AddEquiv
DirectSum
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
instAddCommMonoidDirectSum
AddEquiv.instEquivLike
equivCongrLeft
β€”DFinsupp.comapDomain'_apply
Equiv.right_inv
ext πŸ“–β€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
β€”β€”DFunLike.ext
ext_iff πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
β€”ext
finite_support πŸ“–mathematicalβ€”Set.Finite
Function.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SetLike.instMembership
DFunLike.coe
DFinsupp
DFinsupp.instDFunLike
β€”Set.Finite.subset
Finset.finite_toSet
support_subset
fromAddMonoid_of πŸ“–mathematicalβ€”DFunLike.coe
AddMonoidHom
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
AddMonoidHom.instAddCommMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
fromAddMonoid
of
AddMonoidHom.comp
β€”fromAddMonoid.eq_1
toAddMonoid_of
fromAddMonoid_of_apply πŸ“–mathematicalβ€”DFunLike.coe
AddMonoidHom
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
AddMonoidHom.instAddCommMonoid
fromAddMonoid
of
β€”fromAddMonoid_of
AddMonoidHom.coe_comp
induction_on πŸ“–β€”DirectSum
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
DFunLike.coe
AddMonoidHom
AddMonoidHom.instFunLike
of
AddCommMagma.toAdd
AddCommSemigroup.toAddCommMagma
AddCommMonoid.toAddCommSemigroup
β€”β€”DFinsupp.induction
map_apply πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
AddMonoidHom
DirectSum
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
map
β€”DFinsupp.mapRange_apply
map_zero
AddMonoidHomClass.toZeroHomClass
AddMonoidHom.instAddMonoidHomClass
map_comp πŸ“–mathematicalβ€”map
AddMonoidHom.comp
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DirectSum
instAddCommMonoidDirectSum
β€”DFinsupp.mapRange.addMonoidHom_comp
map_eq_iff πŸ“–mathematicalβ€”DFunLike.coe
AddMonoidHom
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
map
DFinsupp
AddZero.toZero
DFinsupp.instDFunLike
β€”map_apply
map_id πŸ“–mathematicalβ€”map
AddMonoidHom.id
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DirectSum
instAddCommMonoidDirectSum
β€”DFinsupp.mapRange.addMonoidHom_id
map_injective πŸ“–mathematicalβ€”DirectSum
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
map
β€”DFinsupp.mapRange_injective
map_zero
AddMonoidHomClass.toZeroHomClass
AddMonoidHom.instAddMonoidHomClass
map_of πŸ“–mathematicalβ€”DFunLike.coe
AddMonoidHom
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
map
of
β€”DFinsupp.mapRange_single
map_zero
AddMonoidHomClass.toZeroHomClass
AddMonoidHom.instAddMonoidHomClass
map_surjective πŸ“–mathematicalβ€”DirectSum
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
map
β€”DFinsupp.mapRange_surjective
map_zero
AddMonoidHomClass.toZeroHomClass
AddMonoidHom.instAddMonoidHomClass
mk_apply_of_mem πŸ“–mathematicalFinset
Finset.instMembership
DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
AddMonoidHom
DirectSum
Pi.addZeroClass
Set.Elem
SetLike.coe
Finset.instSetLike
Set
Set.instMembership
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
β€”β€”
mk_apply_of_notMem πŸ“–mathematicalFinset
Finset.instMembership
DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
AddMonoidHom
DirectSum
Pi.addZeroClass
Set.Elem
SetLike.coe
Finset.instSetLike
Set
Set.instMembership
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
β€”β€”
mk_injective πŸ“–mathematicalβ€”DirectSum
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
Pi.addZeroClass
Set.Elem
SetLike.coe
Finset
Finset.instSetLike
Set
Set.instMembership
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
β€”DFinsupp.mk_injective
of_apply πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
AddMonoidHom
DirectSum
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
of
β€”DFinsupp.single_apply
of_eq_of_ne πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
AddMonoidHom
DirectSum
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
of
β€”DFinsupp.single_eq_of_ne
of_eq_same πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
AddMonoidHom
DirectSum
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
of
β€”DFinsupp.single_eq_same
of_injective πŸ“–mathematicalβ€”DirectSum
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
of
β€”DFinsupp.single_injective
sigmaCurry_apply πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
DirectSum
AddMonoidHom
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
sigmaCurry
β€”DFinsupp.sigmaCurry_apply
sigmaUncurry_apply πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
AddMonoidHom
DirectSum
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
sigmaUncurry
β€”DFinsupp.sigmaUncurry_apply
sub_apply πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
DirectSum
AddCommGroup.toAddCommMonoid
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddCommGroup.toAddGroup
instAddCommGroup
β€”β€”
sum_support_of πŸ“–mathematicalβ€”Finset.sum
DirectSum
instAddCommMonoidDirectSum
DFinsupp.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFunLike.coe
AddMonoidHom
AddMonoidHom.instFunLike
of
DFinsupp
DFinsupp.instDFunLike
β€”DFinsupp.sum_single
sum_univ_of πŸ“–mathematicalβ€”Finset.sum
DirectSum
instAddCommMonoidDirectSum
Finset.univ
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
AddMonoidHom.instFunLike
of
DFinsupp
AddZero.toZero
DFinsupp.instDFunLike
β€”DFinsupp.ext
DFinsupp.finset_sum_apply
Finset.sum_congr
of_apply
Finset.sum_dite_eq'
support_of πŸ“–mathematicalβ€”DFinsupp.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFunLike.coe
AddMonoidHom
DirectSum
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
of
Finset
Finset.instSingleton
β€”DFinsupp.support_single_ne_zero
support_of_subset πŸ“–mathematicalβ€”Finset
Finset.instHasSubset
DFinsupp.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFunLike.coe
AddMonoidHom
DirectSum
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
of
Finset.instSingleton
β€”DFinsupp.support_single_subset
support_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
Function.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SetLike.instMembership
DFunLike.coe
DFinsupp
DFinsupp.instDFunLike
SetLike.coe
Finset
Finset.instSetLike
DFinsupp.support
β€”AddSubmonoidClass.toZeroMemClass
support_zero πŸ“–mathematicalβ€”DFinsupp.support
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DirectSum
instAddCommMonoidDirectSum
Finset
Finset.instEmptyCollection
β€”DFinsupp.support_zero
toAddMonoid_inj πŸ“–mathematicalβ€”toAddMonoidβ€”toAddMonoid_injective
toAddMonoid_injective πŸ“–mathematicalβ€”AddMonoidHom
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
toAddMonoid
β€”AddEquiv.injective
toAddMonoid_of πŸ“–mathematicalβ€”DFunLike.coe
AddMonoidHom
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
toAddMonoid
of
β€”DFinsupp.liftAddHom_apply_single
zero_apply πŸ“–mathematicalβ€”DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
DirectSum
instAddCommMonoidDirectSum
β€”β€”

DirectSum.IsInternal

Theorems

NameKindAssumesProvesValidatesDepends On
addSubmonoid_iSup_eq_top πŸ“–mathematicalDirectSum.IsInternal
AddSubmonoid
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
AddSubmonoid.instSetLike
AddSubmonoid.instAddSubmonoidClass
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
AddSubmonoid.instCompleteLattice
Top.top
AddSubmonoid.instTop
β€”AddSubmonoid.instAddSubmonoidClass
AddMonoidHom.instAddMonoidHomClass
AddSubmonoid.iSup_eq_mrange_dfinsuppSumAddHom
AddMonoidHom.mrange_eq_top
Function.Bijective.surjective

DirectSum.toAddMonoid

Theorems

NameKindAssumesProvesValidatesDepends On
unique πŸ“–mathematicalβ€”DFunLike.coe
AddMonoidHom
DirectSum
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instAddCommMonoidDirectSum
AddMonoidHom.instFunLike
DirectSum.toAddMonoid
AddMonoidHom.comp
DirectSum.of
β€”DFinsupp.addHom_ext'
DFinsupp.sumAddHom_comp_single

(root)

Definitions

NameCategoryTheorems
instAddCommMonoidDirectSum πŸ“–CompOp
273 mathmath: DirectSum.IsInternal.ofBijective_coeLinearMap_same, TensorAlgebra.ofDirectSum_of_tprod, DirectSum.mulHom_of_of, DirectSum.mker_map, MvPolynomial.DirectSum.coeLinearMap_eq_dfinsuppSum, ExteriorAlgebra.GradedAlgebra.liftΞΉ_eq, DirectSum.Gmodule.smulAddMonoidHom_apply_of_of, DirectSum.component.of, DirectSum.coe_toModule_eq_coe_toAddMonoid, DirectSum.lmap_id, qExpansionRingHom_apply, DirectSum.mrange_map, PiTensorProduct.ofDirectSumEquiv_tprod_lof, Module.Presentation.directSum_var, DirectSum.of_smul, DirectSum.fromAddMonoid_of_apply, DirectSum.instIsCentralScalar, TensorProduct.gradedComm_tmul_one, TensorAlgebra.toDirectSum_tensorPower_tprod, DirectSum.toAddMonoid.unique, LinearAlgebra.FreeProduct.mul_injections, AddCommGroup.equiv_directSum_zmod_of_finite', Representation.directSum_apply, TensorProduct.gradedMul_assoc, TensorProduct.directSum_lof_tmul_lof, DirectSum.lequivCongrLeft_apply, DirectSum.lmap_apply, DirectSum.lmap_comp, DirectSum.coeAddMonoidHom_eq_dfinsuppSum, DirectSum.fromAddMonoid_of, rank_directSum, TensorProduct.coe_directSumRight', DirectSum.mk_smul, DirectSum.mk_injective, Module.equiv_directSum_of_isTorsion, DirectSum.decompose_of_mem, DirectSum.coe_of_mul_apply_aux, DirectSum.Gmodule.of_smul_of, TensorProduct.gradedMul_def, DirectSum.sum_univ_of, MultilinearMap.fromDirectSumEquiv_symm_apply, DirectSum.lie_of_of_ne, GradedTensorProduct.auxEquiv_one, Module.DirectLimit.quotMk_of, IsBaseChange.directSumPow, DirectSum.lie_of, TensorProduct.gradedComm_algebraMap_tmul, DirectSum.instSMulCommClass, IsBaseChange.directSum, DirectSum.support_of, DirectSum.algebraMap_toAddMonoid_hom, AddMonoidAlgebra.toDirectSum_single, qExpansion_of_mul, DirectSum.ker_lmap, AdicCompletion.sumInv_apply, DirectSum.coeLinearMap_of, AdicCompletion.sum_lof, DirectSum.coeLinearMap_eq_dfinsuppSum, ExteriorAlgebra.GradedAlgebra.ΞΉ_sq_zero, DirectSum.coe_of_mul_apply_add, AddMonoidAlgebra.decomposeAux_coe, DirectSum.ofPow, DirectSum.of_zero_smul, DirectSum.coe_congr_addEquiv, LinearAlgebra.FreeProduct.ΞΉ_apply, PiTensorProduct.ofDirectSumEquiv_symm_lof_tprod, DirectSum.one_def, TensorProduct.directSumRight_comp_rTensor, DirectSum.addEquivProdDirectSum_symm_apply_toFun, DirectSum.of_mul_of, GradedTensorProduct.auxEquiv_symm_one, DirectSum.sigmaLcurry_apply, GradedTensorProduct.auxEquiv_comm, DirectSum.coe_mul_of_apply_of_not_le, AddCommGroup.equiv_free_prod_directSum_zmod, DirectSum.congr_linearEquiv_toAddEquiv, TensorProduct.gradedComm_algebraMap, GradedAlgebra.proj_recompose, DirectSum.algebraMap_apply, TensorProduct.gradedComm_tmul_of_zero, Module.Relations.Solution.directSum_var, DirectSum.lof_eq_of, DirectSum.map_injective, DirectSum.of_zero_mul, DirectSum.lmap_eq_map, DirectSum.coe_congr_linearEquiv, AddCommGroup.equiv_directSum_zmod_of_finite, TensorProduct.directSumLeft_tmul_lof, DirectSum.lieAlgebraComponent_apply, AdicCompletion.sumInv_comp_sum, DirectSum.mk_apply_of_notMem, TensorProduct.algebraMap_gradedMul, MvPolynomial.DirectSum.coeAddMonoidHom_eq_support_sum, AddMonoidAlgebra.toDirectSum_zero, DirectSum.congr_linearEquiv_toLinearMap, DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem_ne, TensorProduct.gradedComm_one, AddMonoidAlgebra.toDirectSum_add, DirectSum.linearEquivFunOnFintype_lof, DirectSum.coeAlgHom_of, TensorProduct.gradedComm_symm, CliffordAlgebra.GradedAlgebra.ΞΉ_apply, GradedTensorProduct.auxEquiv_mul, DirectSum.congrLinearEquiv_toLinearMap, DirectSum.lieAlgebraOf_apply, DirectSum.map_of, DirectSum.decompose_symm_add, DirectSum.lmap_surjective, DirectSum.lmap_lof, DirectSum.addEquivProdDirectSum_apply, LinearAlgebra.FreeProduct.identify_one, DirectSum.coe_of_mul_apply_of_not_le, DirectSum.linearMap_ext_iff, DirectSum.addEquivProdDirectSum_symm_apply_support', DirectSum.mk_apply_of_mem, TensorProduct.gradedMul_one, DirectSum.ext_component_iff, LinearAlgebra.FreeProduct.ΞΉ_def, DirectSum.ker_map, Module.Finite.instDirectSum, GradedTensorProduct.mulHom_apply, MultilinearMap.fromDirectSumEquiv_lof, TensorProduct.directSumRight_tmul_lof, AddMonoidAlgebra.GradesBy.decompose_single, DirectSum.IsInternal.ofBijective_coeLinearMap_of_mem, MvPolynomial.weightedHomogeneousComponent_directSum, DirectSum.coeAddMonoidHom_of, TensorProduct.directSumLeft_symm_lof_tmul, DirectSum.component.lof_self, DirectSum.of_injective, DirectSum.toSemiring_apply, DirectSum.toAddMonoidAlgebra_zero, AdicCompletion.sum_of, DirectSum.mulHom_apply, DirectSum.coe_mul_of_apply_of_mem_zero, DirectSum.of_eq_same, DirectSum.single_eq_lof, DirectSum.sigmaCurry_apply, DirectSum.coeRingHom_of, MultilinearMap.fromDirectSumEquiv_apply, finsuppLEquivDirectSum_single, GradedTensorProduct.auxEquiv_tmul, DirectSum.decomposeLinearEquiv_apply, TensorProduct.gradedComm_of_zero_tmul, DirectSum.decomposeLinearEquiv_symm_comp_lof, DirectSum.coe_mul_of_apply_aux, TensorProduct.directSumLeft_tmul, DirectSum.lie_of_same, DirectSum.decompose_zero, MvPolynomial.DirectSum.coeLinearMap_eq_finsum, TensorAlgebra.GradedAlgebra.ΞΉ_apply, DirectSum.toSemiring_of, DirectSum.decomposeLinearEquiv_symm_apply, DirectSum.toAddMonoidAlgebra_of, LinearAlgebra.FreeProduct.rel_id, Module.equiv_free_prod_directSum, AddMonoidAlgebra.grade.decompose_single, TensorProduct.directSum_symm_lof_tmul, DirectSum.addHom_ext'_iff, DirectSum.toModule_lof, MultilinearMap.directSum_ext_iff, TensorProduct.directSumRight_tmul, TensorProduct.gradedComm_tmul_algebraMap, DirectSum.of_zero_pow, DirectSum.coeFnAddMonoidHom_apply, MvPolynomial.weightedDecomposition.decompose'_eq, DirectSum.of_natCast, DirectSum.sigmaLuncurry_apply, DirectSum.coe_mul_of_apply_add, finsuppLEquivDirectSum_apply, DirectSum.lof_apply, DirectSum.map_apply, DirectSum.ringHom_ext'_iff, DirectSum.map_comp, DirectSum.apply_eq_component, DirectSum.coe_of_mul_apply_of_le, DirectSum.support_of_subset, DirectSum.coe_mul_of_apply_of_le, DirectSum.coe_of_mul_apply_of_mem_zero, DirectSum.equivCongrLeft_apply, TensorProduct.directSumRight'_restrict, DirectSum.of_apply, DirectSum.toAddMonoidAlgebra_add, DirectSum.of_intCast, DirectSum.mul_eq_sum_support_ghas_mul, Module.finrank_directSum, DirectSum.coeFnLinearMap_apply, TensorProduct.gradedMul_algebraMap, AdicCompletion.sumEquivOfFintype_apply, Module.Flat.directSum, TensorProduct.gradedCommAux_comp_gradedCommAux, HahnEmbedding.Seed.hahnCoeff_apply, DirectSum.range_lmap, DirectSum.coe_mul_of_apply, DirectSum.mul_eq_dfinsuppSum, TensorProduct.tmul_of_gradedMul_of_tmul, TensorProduct.gradedComm_one_tmul, DirectSum.decomposeAddEquiv_symm_apply, DirectSum.of_eq_of_gradedMonoid_eq, DirectSum.support_smul, DirectSum.decompose_symm_zero, DirectSum.ofList_dProd, DirectSum.decomposeAddEquiv_apply, DirectSum.congrLinearEquiv_toAddEquiv, DirectSum.smul_apply, DirectSum.Decomposition.left_inv, DirectSum.support_zero, DirectSum.IsInternal.ofBijective_coeLinearMap_of_ne, DirectSum.Decomposition.right_inv, TensorProduct.gradedCommAux_lof_tmul, DirectSum.decompose_coe, DirectSum.sigmaUncurry_apply, AddMonoidAlgebra.decomposeAux_single, AdicCompletion.component_sumInv, DirectSum.linearEquivFunOnFintype_symm_single, LinearAlgebra.FreeProduct.lift_apply, DirectSum.list_prod_ofFn_of_eq_dProd, addMonoidAlgebraAddEquivDirectSum_symm_apply, DirectSum.coe_of_apply, DirectSum.coe_congrAddEquiv, CliffordAlgebra.GradedAlgebra.lift_ΞΉ_eq, TensorAlgebra.toDirectSum_ΞΉ, DirectSum.lmap_of, addMonoidAlgebraAddEquivDirectSum_apply, DirectSum.lmap_injective, PiTensorProduct.ofDirectSumEquiv_tprod_apply, DirectSum.toModule.unique, TensorProduct.one_gradedMul, lmap_finsuppLEquivDirectSum_eq, DirectSum.of_zero_one, DirectSum.decompose_symm_of, ExteriorAlgebra.GradedAlgebra.ΞΉ_apply, LinearAlgebra.FreeProduct.lof_map_one, DirectSum.sum_support_of, CliffordAlgebra.GradedAlgebra.ΞΉ_sq_scalar, TensorProduct.gradedComm_of_tmul_of, DirectSum.toAddMonoid_injective, qExpansion_of_pow, DirectSum.toAddMonoid_of, DirectSum.of_zero_ofNat, AdicCompletion.sumEquivOfFintype_symm_apply, Module.torsion_by_prime_power_decomposition, DirectSum.decompose_symm_sum, DirectSum.map_eq_iff, DirectSum.decompose_add, DirectSum.toLieAlgebra_apply, DirectSum.add_apply, DirectSum.decompose_sum, TensorProduct.gradedComm_gradedMul, DirectSum.toAddMonoidHom_lmap, Module.Flat.directSum_iff, DirectSum.instIsScalarTower, TensorProduct.directSumRight_symm_lof_tmul, DirectSum.coe_of_mul_apply, DirectSum.decompose_smul, DirectSum.lmap_eq_iff, DirectSum.range_coeLinearMap, DirectSum.lequivProdDirectSum_symm_apply, DirectSum.Gmodule.smul_def, DirectSum.linearEquivFunOnFintype_symm_coe, DirectSum.linearEquivFunOnFintype_apply, OrthogonalFamily.projection_directSum_coeAddHom, GradedRing.proj_recompose, DirectSum.zero_apply, finsuppLEquivDirectSum_symm_lof, AdicCompletion.sum_comp_sumInv, DirectSum.map_surjective, MvPolynomial.decomposition.decompose'_eq, DirectSum.coe_congrLinearEquiv, DirectSum.lequivProdDirectSum_apply, DirectSum.map_id, Module.Free.directSum, DirectSum.of_eq_of_ne
instCoeFunDirectSumForall πŸ“–CompOpβ€”
instDFunLikeDirectSum πŸ“–CompOpβ€”
instDecidableEqDirectSum πŸ“–CompOpβ€”
instInhabitedDirectSum πŸ“–CompOpβ€”

---

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