Documentation Verification Report

Pointwise

πŸ“ Source: Mathlib/Algebra/Group/Submonoid/Pointwise.lean

Statistics

MetricCount
DefinitionsinvolutiveNeg, neg, negOrderIso, inv, invOrderIso, involutiveInv, pointwiseMulAction
7
Theoremsadd_subset, add_subset_closure, closure_add_le, closure_neg, closure_nsmul, closure_nsmul_le, coe_add_self_eq, coe_neg, coe_negOrderIso_apply, coe_negOrderIso_symm_apply, coe_sup, mem_closure_neg, mem_neg, neg_bot, neg_iInf, neg_iSup, neg_inf, neg_le, neg_le_neg, neg_sup, neg_top, nsmul_vadd_mem_closure_vadd, sup_eq_closure_add, addSubmonoid_closure, submonoid_closure, closure_inv, closure_mul_le, closure_pow, closure_pow_le, coe_inv, coe_invOrderIso_apply, coe_invOrderIso_symm_apply, coe_mul_self_eq, coe_pointwise_smul, coe_sup, instCovariantClassHSMulLe, inv_bot, inv_iInf, inv_iSup, inv_inf, inv_le, inv_le_inv, inv_sup, inv_top, mem_closure_inv, mem_inv, mem_inv_pointwise_smul_iff, mem_pointwise_smul_iff_inv_smul_mem, mem_smul_pointwise_iff_exists, mul_subset, mul_subset_closure, pointwise_isCentralScalar, pointwise_smul_le_pointwise_smul_iff, pointwise_smul_subset_iff, pow_smul_mem_closure_smul, smul_bot, smul_closure, smul_mem_pointwise_smul, smul_mem_pointwise_smul_iff, smul_sup, subset_pointwise_smul_iff, sup_eq_closure_mul, coe_add_coe, coe_mul_coe, coe_set_nsmul, coe_set_pow
66
Total73

AddSubmonoid

Definitions

NameCategoryTheorems
involutiveNeg πŸ“–CompOpβ€”
neg πŸ“–CompOp
15 mathmath: neg_iInf, neg_le, neg_inf, neg_bot, mem_neg, neg_sup, mk_add_mk_neg_eq_zero, coe_neg, mk_neg_add_mk_eq_zero, neg_top, closure_neg, leftNeg_eq_neg, neg_le_neg, neg_iSup, Submodule.neg_toAddSubmonoid
negOrderIso πŸ“–CompOp
2 mathmath: coe_negOrderIso_symm_apply, coe_negOrderIso_apply

Theorems

NameKindAssumesProvesValidatesDepends On
add_subset πŸ“–mathematicalSet
Set.instHasSubset
SetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
instSetLike
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
β€”Set.add_subset_iff
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
instAddSubmonoidClass
add_subset_closure πŸ“–mathematicalSet
Set.instHasSubset
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SetLike.coe
AddSubmonoid
instSetLike
closure
β€”add_subset
Set.Subset.trans
subset_closure
closure_add_le πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
closure
Set
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”sInf_le
add_mem
SetLike.le_def
instIsConcreteLE
le_sup_left
subset_closure
le_sup_right
closure_neg πŸ“–mathematicalβ€”closure
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Set
Set.neg
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
AddSubmonoid
neg
β€”le_antisymm
closure_le
coe_neg
Set.neg_subset
neg_neg
subset_closure
neg_le
closure_nsmul πŸ“–mathematicalSet
Set.instMembership
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
closure
Set.NSMul
AddZero.toAdd
β€”LE.le.antisymm
closure_nsmul_le
closure_mono
Set.subset_nsmul
closure_nsmul_le πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
closure
Set
AddMonoid.toNatSMul
Set.addMonoid
β€”one_nsmul
le_refl
succ_nsmul
closure_add_le
sup_le_sup
sup_idem
coe_add_self_eq πŸ“–mathematicalβ€”Set
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SetLike.coe
AddSubmonoid
instSetLike
β€”coe_add_coe
instAddSubmonoidClass
coe_neg πŸ“–mathematicalβ€”SetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
instSetLike
neg
Set
Set.neg
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”β€”
coe_negOrderIso_apply πŸ“–mathematicalβ€”SetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
instSetLike
DFunLike.coe
RelIso
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
RelIso.instFunLike
negOrderIso
Set
Set.neg
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”β€”
coe_negOrderIso_symm_apply πŸ“–mathematicalβ€”SetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
instSetLike
DFunLike.coe
RelIso
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
RelIso.instFunLike
RelIso.symm
negOrderIso
Set
Set.neg
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”β€”
coe_sup πŸ“–mathematicalβ€”SetLike.coe
AddSubmonoid
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
Set
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
β€”Set.ext
SetLike.mem_coe
mem_sup
Set.mem_add
mem_closure_neg πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
SetLike.instMembership
instSetLike
closure
Set
Set.neg
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”closure_neg
mem_neg
mem_neg πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
SetLike.instMembership
instSetLike
neg
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
AddGroup.toSubtractionMonoid
β€”β€”
neg_bot πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
neg
Bot.bot
instBot
β€”SetLike.coe_injective
Set.neg_singleton
neg_zero
neg_iInf πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
neg
iInf
instInfSet
β€”OrderIso.map_iInf
neg_iSup πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
neg
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”OrderIso.map_iSup
neg_inf πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
neg
instMin
β€”SetLike.coe_injective
Set.inter_neg
neg_le πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
neg
β€”SetLike.coe_subset_coe
instIsConcreteLE
Set.neg_subset
neg_le_neg πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
neg
β€”SetLike.coe_subset_coe
instIsConcreteLE
Set.neg_subset_neg
neg_sup πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
neg
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”OrderIso.map_sup
neg_top πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SubNegMonoid.toAddMonoid
AddGroup.toSubNegMonoid
neg
Top.top
instTop
β€”SetLike.coe_injective
Set.neg_univ
nsmul_vadd_mem_closure_vadd πŸ“–mathematicalAddSubmonoid
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
SetLike.instMembership
instSetLike
closure
HVAdd.hVAdd
Set
instHVAdd
Set.vaddSet
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
AddAction.toAddSemigroupAction
AddMonoid.toNatSMul
β€”closure_induction
subset_closure
one_nsmul
zero_nsmul
zero_vadd
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
instAddSubmonoidClass
add_nsmul
AddSemigroupAction.add_vadd
vadd_add_assoc
add_comm
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
sup_eq_closure_add πŸ“–mathematicalβ€”AddSubmonoid
AddMonoid.toAddZeroClass
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
closure
Set
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
SetLike.coe
instSetLike
β€”le_antisymm
sup_le
subset_closure
zero_mem
add_zero
zero_add
LE.le.trans
closure_add_le
closure_eq
le_refl

Set.IsPWO

Theorems

NameKindAssumesProvesValidatesDepends On
addSubmonoid_closure πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
Set.IsPWO
SetLike.coe
AddSubmonoid
AddSubmonoid.instSetLike
AddSubmonoid.closure
β€”AddSubmonoid.closure_eq_image_sum
Set.PartiallyWellOrderedOn.image_of_monotone_on
Set.PartiallyWellOrderedOn.partiallyWellOrderedOn_sublistForallβ‚‚
instIsPreorderLe
List.SublistForallβ‚‚.sum_le_sum
covariant_swap_add_of_covariant_add
IsOrderedAddMonoid.toAddLeftMono
IsOrderedCancelAddMonoid.toIsOrderedAddMonoid
submonoid_closure πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
CommMonoid.toMonoid
Set.IsPWO
SetLike.coe
Submonoid
Submonoid.instSetLike
Submonoid.closure
β€”Submonoid.closure_eq_image_prod
Set.PartiallyWellOrderedOn.image_of_monotone_on
Set.PartiallyWellOrderedOn.partiallyWellOrderedOn_sublistForallβ‚‚
instIsPreorderLe
List.SublistForallβ‚‚.prod_le_prod'
covariant_swap_mul_of_covariant_mul
IsOrderedMonoid.toMulLeftMono
IsOrderedCancelMonoid.toIsOrderedMonoid

Submonoid

Definitions

NameCategoryTheorems
inv πŸ“–CompOp
14 mathmath: inv_top, mem_inv, inv_bot, mk_mul_mk_inv_eq_one, mk_inv_mul_mk_eq_one, inv_inf, inv_sup, coe_inv, inv_iSup, leftInv_eq_inv, inv_le_inv, inv_le, inv_iInf, closure_inv
invOrderIso πŸ“–CompOp
2 mathmath: coe_invOrderIso_apply, coe_invOrderIso_symm_apply
involutiveInv πŸ“–CompOpβ€”
pointwiseMulAction πŸ“–CompOp
21 mathmath: smul_mem_pointwise_smul_iffβ‚€, smul_mem_pointwise_smul, pointwise_smul_le_pointwise_smul_iffβ‚€, mem_smul_pointwise_iff_exists, pointwise_smul_le_pointwise_smul_iff, mem_inv_pointwise_smul_iff, coe_pointwise_smul, smul_mem_pointwise_smul_iff, smul_sup, le_pointwise_smul_iffβ‚€, pointwise_smul_le_iffβ‚€, mem_inv_pointwise_smul_iffβ‚€, mem_pointwise_smul_iff_inv_smul_memβ‚€, pointwise_smul_subset_iff, mem_pointwise_smul_iff_inv_smul_mem, pointwise_isCentralScalar, instCovariantClassHSMulLe, subset_pointwise_smul_iff, smul_closure, Subgroup.pointwise_smul_toSubmonoid, smul_bot

Theorems

NameKindAssumesProvesValidatesDepends On
closure_inv πŸ“–mathematicalβ€”closure
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Set
Set.inv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
Submonoid
inv
β€”le_antisymm
closure_le
coe_inv
Set.inv_subset
inv_inv
subset_closure
inv_le
closure_mul_le πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
closure
Set
Set.mul
MulOne.toMul
MulOneClass.toMulOne
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”sInf_le
mul_mem
SetLike.le_def
instIsConcreteLE
le_sup_left
subset_closure
le_sup_right
closure_pow πŸ“–mathematicalSet
Set.instMembership
MulOne.toOne
MulOneClass.toMulOne
Monoid.toMulOneClass
closure
Set.NPow
MulOne.toMul
β€”LE.le.antisymm
closure_pow_le
closure_mono
Set.subset_pow
closure_pow_le πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
closure
Set
Monoid.toNatPow
Set.monoid
β€”pow_one
pow_succ
closure_mul_le
sup_le_sup
le_refl
sup_idem
coe_inv πŸ“–mathematicalβ€”SetLike.coe
Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instSetLike
inv
Set
Set.inv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”β€”
coe_invOrderIso_apply πŸ“–mathematicalβ€”SetLike.coe
Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instSetLike
DFunLike.coe
RelIso
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
RelIso.instFunLike
invOrderIso
Set
Set.inv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”β€”
coe_invOrderIso_symm_apply πŸ“–mathematicalβ€”SetLike.coe
Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
instSetLike
DFunLike.coe
RelIso
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
RelIso.instFunLike
RelIso.symm
invOrderIso
Set
Set.inv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”β€”
coe_mul_self_eq πŸ“–mathematicalβ€”Set
Set.mul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
SetLike.coe
Submonoid
instSetLike
β€”coe_mul_coe
instSubmonoidClass
coe_pointwise_smul πŸ“–mathematicalβ€”SetLike.coe
Submonoid
Monoid.toMulOneClass
instSetLike
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
pointwiseMulAction
Set
Set.smulSet
MulDistribMulAction.toMulAction
β€”β€”
coe_sup πŸ“–mathematicalβ€”SetLike.coe
Submonoid
Monoid.toMulOneClass
CommMonoid.toMonoid
instSetLike
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
Set
Set.mul
MulOne.toMul
MulOneClass.toMulOne
β€”Set.ext
instCovariantClassHSMulLe πŸ“–mathematicalβ€”CovariantClass
Submonoid
Monoid.toMulOneClass
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
pointwiseMulAction
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
β€”Set.image_mono
inv_bot πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
inv
Bot.bot
instBot
β€”SetLike.coe_injective
Set.inv_singleton
inv_one
inv_iInf πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
inv
iInf
instInfSet
β€”OrderIso.map_iInf
inv_iSup πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
inv
iSup
ConditionallyCompletePartialOrderSup.toSupSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”OrderIso.map_iSup
inv_inf πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
inv
instMin
β€”SetLike.coe_injective
Set.inter_inv
inv_le πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
inv
β€”SetLike.coe_subset_coe
instIsConcreteLE
Set.inv_subset
inv_le_inv πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
inv
β€”SetLike.coe_subset_coe
instIsConcreteLE
Set.inv_subset_inv
inv_sup πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
inv
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”OrderIso.map_sup
inv_top πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
inv
Top.top
instTop
β€”SetLike.coe_injective
Set.inv_univ
mem_closure_inv πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
SetLike.instMembership
instSetLike
closure
Set
Set.inv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”closure_inv
mem_inv
mem_inv πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
DivInvMonoid.toMonoid
Group.toDivInvMonoid
SetLike.instMembership
instSetLike
inv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”β€”
mem_inv_pointwise_smul_iff πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
pointwiseMulAction
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
MulDistribMulAction.toMulAction
β€”Set.mem_inv_smul_set_iff
mem_pointwise_smul_iff_inv_smul_mem πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
pointwiseMulAction
MulDistribMulAction.toMulAction
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”Set.mem_smul_set_iff_inv_smul_mem
mem_smul_pointwise_iff_exists πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
pointwiseMulAction
MulDistribMulAction.toMulAction
β€”Set.mem_smul_set
mul_subset πŸ“–mathematicalSet
Set.instHasSubset
SetLike.coe
Submonoid
Monoid.toMulOneClass
instSetLike
Set.mul
MulOne.toMul
MulOneClass.toMulOne
β€”Set.mul_subset_iff
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
instSubmonoidClass
mul_subset_closure πŸ“–mathematicalSet
Set.instHasSubset
Set.mul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
SetLike.coe
Submonoid
instSetLike
closure
β€”mul_subset
Set.Subset.trans
subset_closure
pointwise_isCentralScalar πŸ“–mathematicalβ€”IsCentralScalar
Submonoid
Monoid.toMulOneClass
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
pointwiseMulAction
MulOpposite
MulOpposite.instMonoid
β€”Monoid.End.instMonoidHomClass
MonoidHom.ext
IsCentralScalar.op_smul_eq_smul
pointwise_smul_le_pointwise_smul_iff πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
pointwiseMulAction
β€”Set.smul_set_subset_smul_set_iff
pointwise_smul_subset_iff πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
pointwiseMulAction
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”Set.smul_set_subset_iff_subset_inv_smul_set
pow_smul_mem_closure_smul πŸ“–mathematicalSubmonoid
Monoid.toMulOneClass
CommMonoid.toMonoid
SetLike.instMembership
instSetLike
closure
Set
Set.smulSet
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
Monoid.toNatPow
β€”closure_induction
subset_closure
pow_one
pow_zero
one_smul
SubmonoidClass.toOneMemClass
instSubmonoidClass
pow_add
SemigroupAction.mul_smul
smul_mul_assoc
mul_comm
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
smul_bot πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
pointwiseMulAction
Bot.bot
instBot
β€”map_bot
smul_closure πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
pointwiseMulAction
closure
Set
Set.smulSet
MulDistribMulAction.toMulAction
β€”MonoidHom.map_mclosure
smul_mem_pointwise_smul πŸ“–mathematicalSubmonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
pointwiseMulAction
MulDistribMulAction.toMulAction
β€”Set.smul_mem_smul_set
smul_mem_pointwise_smul_iff πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
SetLike.instMembership
instSetLike
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
pointwiseMulAction
MulDistribMulAction.toMulAction
β€”Set.smul_mem_smul_set_iff
smul_sup πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
SemigroupAction.toSMul
Monoid.toSemigroup
MulAction.toSemigroupAction
pointwiseMulAction
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
β€”map_sup
subset_pointwise_smul_iff πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
SemigroupAction.toSMul
Monoid.toSemigroup
DivInvMonoid.toMonoid
Group.toDivInvMonoid
MulAction.toSemigroupAction
pointwiseMulAction
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
Group.toDivisionMonoid
β€”Set.subset_smul_set_iff
sup_eq_closure_mul πŸ“–mathematicalβ€”Submonoid
Monoid.toMulOneClass
SemilatticeSup.toMax
Lattice.toSemilatticeSup
ConditionallyCompleteLattice.toLattice
CompleteLattice.toConditionallyCompleteLattice
instCompleteLattice
closure
Set
Set.mul
MulOne.toMul
MulOneClass.toMulOne
SetLike.coe
instSetLike
β€”le_antisymm
sup_le
subset_closure
one_mem
mul_one
one_mul
LE.le.trans
closure_mul_le
closure_eq
le_refl

(root)

Theorems

NameKindAssumesProvesValidatesDepends On
coe_add_coe πŸ“–mathematicalβ€”Set
Set.add
AddZero.toAdd
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
SetLike.coe
β€”Set.ext
Set.mem_add
SetLike.mem_coe
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
ZeroMemClass.zero_mem
AddSubmonoidClass.toZeroMemClass
zero_add
coe_mul_coe πŸ“–mathematicalβ€”Set
Set.mul
MulOne.toMul
MulOneClass.toMulOne
Monoid.toMulOneClass
SetLike.coe
β€”Set.ext
MulMemClass.mul_mem
SubmonoidClass.toMulMemClass
OneMemClass.one_mem
SubmonoidClass.toOneMemClass
one_mul
coe_set_nsmul πŸ“–mathematicalβ€”Set
AddMonoid.toNatSMul
Set.addMonoid
SetLike.coe
β€”one_nsmul
succ_nsmul
coe_add_coe
coe_set_pow πŸ“–mathematicalβ€”Set
Monoid.toNatPow
Set.monoid
SetLike.coe
β€”pow_one
pow_succ
coe_mul_coe

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