Documentation Verification Report

Subsemiring

πŸ“ Source: Mathlib/RingTheory/GradedAlgebra/Homogeneous/Subsemiring.lean

Statistics

MetricCount
DefinitionsHomogeneousSubsemiring, instPartialOrder, setLike, toSubsemiring, Subsemiring, homogeneousCore
6
Theoremsmem_iff, ext, ext', ext_iff, isHomogeneous, is_homogeneous', mem_iff, subsemiringClass, toSubsemiring_injective, subsemiringClosure, subsemiringClosure_of_isHomogeneousElem, homogeneousCore_mono, isHomogeneous_iff_forall_subset, isHomogeneous_iff_subset_iInter, toSubsemiring_homogeneousCore_le
15
Total21

DirectSum.SetLike.IsHomogeneous

Theorems

NameKindAssumesProvesValidatesDepends On
mem_iff πŸ“–mathematicalDirectSum.SetLike.IsHomogeneous
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
GradedRing.toDecomposition
Subsemiring
Subsemiring.instSetLike
SetLike.instMembership
DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
Equiv
DirectSum
AddSubmonoidClass.toAddCommMonoid
EquivLike.toFunLike
Equiv.instEquivLike
DirectSum.decompose
β€”DirectSum.AddSubmonoidClass.IsHomogeneous.mem_iff
SubsemiringClass.toAddSubmonoidClass
Subsemiring.instSubsemiringClass

HomogeneousSubsemiring

Definitions

NameCategoryTheorems
instPartialOrder πŸ“–CompOp
1 mathmath: Subsemiring.homogeneousCore_mono
setLike πŸ“–CompOp
3 mathmath: mem_iff, isHomogeneous, subsemiringClass
toSubsemiring πŸ“–CompOp
5 mathmath: ext_iff, mem_iff, Subsemiring.toSubsemiring_homogeneousCore_le, toSubsemiring_injective, is_homogeneous'

Theorems

NameKindAssumesProvesValidatesDepends On
ext πŸ“–β€”toSubsemiringβ€”β€”toSubsemiring_injective
ext' πŸ“–β€”HomogeneousSubsemiring
SetLike.instMembership
setLike
β€”β€”DirectSum.AddSubmonoidClass.IsHomogeneous.ext
SubsemiringClass.toAddSubmonoidClass
subsemiringClass
is_homogeneous'
ext_iff πŸ“–mathematicalβ€”toSubsemiringβ€”ext
isHomogeneous πŸ“–mathematicalβ€”DirectSum.SetLike.IsHomogeneous
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
GradedRing.toDecomposition
HomogeneousSubsemiring
setLike
β€”is_homogeneous'
is_homogeneous' πŸ“–mathematicalβ€”DirectSum.SetLike.IsHomogeneous
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
GradedRing.toDecomposition
Subsemiring
Subsemiring.instSetLike
toSubsemiring
β€”β€”
mem_iff πŸ“–mathematicalβ€”Subsemiring
Semiring.toNonAssocSemiring
SetLike.instMembership
Subsemiring.instSetLike
toSubsemiring
HomogeneousSubsemiring
setLike
β€”β€”
subsemiringClass πŸ“–mathematicalβ€”SubsemiringClass
HomogeneousSubsemiring
Semiring.toNonAssocSemiring
setLike
β€”Subsemiring.mul_mem
Subsemiring.one_mem
Subsemiring.add_mem
Subsemiring.zero_mem
toSubsemiring_injective πŸ“–mathematicalβ€”HomogeneousSubsemiring
Subsemiring
Semiring.toNonAssocSemiring
toSubsemiring
β€”β€”

IsHomogeneous

Theorems

NameKindAssumesProvesValidatesDepends On
subsemiringClosure πŸ“–mathematicalSet
Set.instMembership
SetLike.instMembership
DFunLike.coe
DFinsupp
AddZero.toZero
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddCommMonoid.toAddMonoid
DFinsupp.instDFunLike
Equiv
DirectSum
AddSubmonoidClass.toAddCommMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
EquivLike.toFunLike
Equiv.instEquivLike
DirectSum.decompose
GradedRing.toDecomposition
DirectSum.SetLike.IsHomogeneous
Subsemiring
Subsemiring.instSetLike
Subsemiring.closure
β€”Subsemiring.closure_induction
Subsemiring.subset_closure
DirectSum.decompose_zero
AddSubmonoidClass.toZeroMemClass
SubsemiringClass.toAddSubmonoidClass
Subsemiring.instSubsemiringClass
SetLike.GradedMonoid.toGradedOne
GradedRing.toGradedMonoid
DirectSum.decompose_one
DirectSum.one_def
eq_or_ne
DirectSum.of_eq_same
AddSubmonoidWithOneClass.toOneMemClass
SubsemiringClass.addSubmonoidWithOneClass
DirectSum.of_eq_of_ne
DirectSum.decompose_add
AddMemClass.add_mem
AddSubmonoidClass.toAddMemClass
SetLike.GradedMonoid.toGradedMul
DirectSum.decompose_mul
DirectSum.mul_eq_dfinsuppSum
DFinsupp.sum_apply
DFinsupp.sum.eq_1
AddSubmonoidClass.coe_finset_sum
sum_mem
SubmonoidClass.toMulMemClass
SubsemiringClass.toSubmonoidClass
subsemiringClosure_of_isHomogeneousElem πŸ“–mathematicalSetLike.IsHomogeneousElemDirectSum.SetLike.IsHomogeneous
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
GradedRing.toDecomposition
Subsemiring
Subsemiring.instSetLike
Subsemiring.closure
β€”Subsemiring.closure_insert_zero
subsemiringClosure
Set.mem_insert_iff
DirectSum.decompose_zero
eq_or_ne
DirectSum.decompose_of_mem
DirectSum.of_eq_same
DirectSum.of_eq_of_ne

Subsemiring

Definitions

NameCategoryTheorems
homogeneousCore πŸ“–CompOp
2 mathmath: homogeneousCore_mono, toSubsemiring_homogeneousCore_le

Theorems

NameKindAssumesProvesValidatesDepends On
homogeneousCore_mono πŸ“–mathematicalβ€”Monotone
Subsemiring
Semiring.toNonAssocSemiring
HomogeneousSubsemiring
PartialOrder.toPreorder
instPartialOrder
HomogeneousSubsemiring.instPartialOrder
homogeneousCore
β€”closure_mono
Set.image_mono
isHomogeneous_iff_forall_subset πŸ“–mathematicalβ€”DirectSum.SetLike.IsHomogeneous
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
GradedRing.toDecomposition
Subsemiring
instSetLike
Set
Set.instHasSubset
SetLike.coe
Set.preimage
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddMonoidWithOne.toAddMonoid
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
AddMonoidHom.instFunLike
GradedRing.proj
β€”β€”
isHomogeneous_iff_subset_iInter πŸ“–mathematicalβ€”DirectSum.SetLike.IsHomogeneous
NonUnitalNonAssocSemiring.toAddCommMonoid
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
GradedRing.toDecomposition
Subsemiring
instSetLike
Set
Set.instHasSubset
SetLike.coe
Set.iInter
Set.preimage
DFunLike.coe
AddMonoidHom
AddZeroClass.toAddZero
AddMonoid.toAddZeroClass
AddMonoidWithOne.toAddMonoid
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
AddMonoidHom.instFunLike
GradedRing.proj
β€”Set.subset_iInter_iff
toSubsemiring_homogeneousCore_le πŸ“–mathematicalβ€”Subsemiring
Semiring.toNonAssocSemiring
Preorder.toLE
PartialOrder.toPreorder
instPartialOrder
HomogeneousSubsemiring.toSubsemiring
homogeneousCore
β€”closure_le
Set.image_preimage_subset

(root)

Definitions

NameCategoryTheorems
HomogeneousSubsemiring πŸ“–CompData
5 mathmath: Subsemiring.homogeneousCore_mono, HomogeneousSubsemiring.mem_iff, HomogeneousSubsemiring.isHomogeneous, HomogeneousSubsemiring.toSubsemiring_injective, HomogeneousSubsemiring.subsemiringClass
Subsemiring πŸ“–CompData
315 mathmath: RingCone.coe_set_mk, Subsemiring.mem_sumSq, Subring.range_snd, Subsemiring.mem_toNonUnitalSubsemiring, Subsemiring.mem_top, Subsemiring.coe_subtype, Subsemiring.comap_iInf, Polynomial.mem_lifts_iff_mem_alg, Subsemiring.closure_le, Subsemiring.centralizer_le, Subsemiring.eq_top_iff', Subsemiring.isClosed_topologicalClosure, RingPreordering.mem_mk, Subsemiring.coe_pow, Subsemiring.mem_carrier, Subsemiring.closure_mono, Subsemiring.op_injective, Subring.toSubsemiring_injective, Subsemiring.mem_iInf, Polynomial.monomial_mem_lifts, CentroidHom.star_centerToCentroidCenter, Subsemiring.mem_closure, Polynomial.X_mem_lifts, RingHom.rangeS_eq_top, Subsemiring.toIsOrderedRing, Subsemiring.map_sup, Subsemiring.mem_smul_pointwise_iff_exists, Subsemiring.op_eq_bot, StarSubsemiring.mem_toSubsemiring, RingPreordering.toSubsemiring_strictMono, Subsemiring.gc_map_comap, Subsemiring.smul_mem_pointwise_smul_iff, Subsemiring.op_eq_top, Algebra.toSubsemiring_eq_top, Subring.coe_toSubsemiring, Subsemiring.center_eq_top, RingPreordering.toSubsemiring_lt_toSubsemiring, Subsemiring.op_sup, Subsemiring.mem_bot, Subsemiring.nontrivial, Subsemiring.op_le_op_iff, Subsemiring.homogeneousCore_mono, Subsemiring.le_op_iff, Subsemiring.map_iInf, Submonoid.subsemiringClosure_coe, Subsemiring.map_bot, RingHom.rangeSRestrict_surjective, Subsemiring.centerCongr_symm_apply_coe, IsScalarTower.subsemiring, mem_subalgebraOfSubsemiring, Polynomial.Splits.mem_lift_of_roots_mem_range, Subsemiring.coe_prod, RingEquiv.ofLeftInverseS_symm_apply, Subsemiring.op_sInf, Subsemiring.addEquivOp_apply_coe, Subring.toSubsemiring_mono, RingHom.rangeS_toSubmonoid, Subsemiring.isScalarTower, Subring.toSubsemiring_lt_toSubsemiring, Subring.mopRingEquivOp_apply_coe, Subsemiring.pointwise_smul_toAddSubmonoid, Subsemiring.prod_top, RingCon.comapQuotientEquivRangeS_symm_mk, RingHom.coe_rangeSRestrict, Subsemiring.smul_def, Subsemiring.smul_bot, Subsemiring.coe_equivMapOfInjective_apply, Subalgebra.mopAlgEquivOp_symm_apply, Subsemiring.unop_le_unop_iff, Algebra.sInf_toSubsemiring, Subsemiring.coe_sInf, Subsemiring.toIsStrictOrderedRing, Subsemiring.coe_add, Subsemiring.coe_one, isArtinianRing_rangeS, Subring.mem_toSubsemiring, StarSubsemiring.toSubsemiring_le_iff, Polynomial.C_mem_lifts, Subsemiring.mem_inf, Subsemiring.unop_eq_top, RingCone.mem_mk, Subsemiring.prod_mono_right, Subsemiring.op_iSup, Subsemiring.mem_mk', Subsemiring.opEquiv_apply, RingPreordering.toSubsemiring_mono, Subsemiring.unop_sup, Subsemiring.coe_toSubmonoid, Subsemiring.subset_closure, Subring.toSubsemiring_le_toSubsemiring, StarSubsemiring.toSubsemiring_injective, Subsemiring.centerCongr_apply_coe, Algebra.sup_toSubsemiring, Subsemiring.toSubmonoid_mono, Subsemiring.one_mem, StarSubsemiring.coe_toSubsemiring, Subsemiring.closure_le_centralizer_centralizer, Subring.toSubsemiring_top, Polynomial.mem_lifts, Subsemiring.op_le_iff, Subsemiring.unop_sInf, Subsemiring.noZeroDivisors, Subsemiring.pointwise_central_scalar, Subalgebra.linearEquivOp_apply_coe, Subsemiring.mem_map_equiv, Polynomial.lifts_iff_ringHom_rangeS, Subsemiring.closure_singleton_one, Subring.topEquiv_apply, RingEquiv.subsemiringMap_apply_coe, Subsemiring.mem_op, Subring.coe_set_mk, Subsemiring.top_prod_top, Algebra.iSup_toSubsemiring, HomogeneousSubsemiring.mem_iff, Subring.mopRingEquivOp_symm_apply, Subsemiring.sInf_toAddSubmonoid, Subsemiring.comap_top, Subsemiring.centerToMulOpposite_symm_apply_coe, RingPreordering.toSubsemiring_le_toSubsemiring, IsHomogeneous.subsemiringClosure, Subsemiring.addEquivOp_symm_apply_coe, Subsemiring.coe_carrier_toSubmonoid, Subsemiring.closure_singleton_natCast, Polynomial.C'_mem_lifts, Subsemiring.smul_mem_pointwise_smul_iffβ‚€, Subsemiring.map_comap_eq, Subsemiring.mem_centralizer_iff, Subsemiring.centerToMulOpposite_apply_coe, Subsemiring.isHomogeneous_iff_subset_iInter, Subsemiring.pointwise_smul_subset_iff, RingPreordering.coe_toSubsemiring, Algebra.sSup_toSubsemiring, RingEquiv.subsemiringMap_symm_apply_coe, RingCon.comapQuotientEquivRangeS_mk, Subsemiring.toSubmonoid_strictMono, Subring.topEquiv_symm_apply_coe, Subsemiring.closure_iUnion, Subsemiring.ext_iff, Subsemiring.center.smulCommClass_right, RingCon.coe_quotientKerEquivRangeS_mk, Subsemiring.rangeS_subtype, Subsemiring.comap_toSubmonoid, RingHom.rangeS_eq_map, Subsemiring.coe_sumSq, Subsemiring.range_fst, Subsemiring.mem_inv_pointwise_smul_iffβ‚€, RingHom.mem_rangeS, RingHom.rangeS_top_iff_surjective, Subsemiring.closure_sUnion, Subsemiring.opEquiv_symm_apply, Algebra.iInf_toSubsemiring, Subalgebra.mem_mk, Subsemiring.closure_eq, Subsemiring.mem_comap, Subsemiring.map_inf, Algebra.top_toSubsemiring, Subsemiring.unop_injective, StarSubsemiring.coe_mk, Subring.toSubsemiring_strictMono, Subsemiring.pointwise_smul_def, RingHom.sclosure_preimage_le, Subalgebra.coe_toSubsemiring, StarSubalgebra.rangeS_le, CharP.subsemiring, Subsemiring.mopRingEquivOp_apply_coe, Subsemiring.op_inf, IsLocalization.scaleRoots_commonDenom_mem_lifts, Subsemiring.zero_mem, Subsemiring.instSMulCommClassSubtypeMemCenter_1, RingHom.rangeS_top_of_surjective, Subsemiring.pointwise_smul_le_pointwise_smul_iff, Subsemiring.comap_inf, Subsemiring.coe_inf, Subsemiring.instSubsemiringClass, Subsemiring.coe_toNonUnitalSubsemiring, Subsemiring.isHomogeneous_iff_forall_subset, Subsemiring.coe_top, Subsemiring.unop_inf, Subalgebra.coe_valA, Subsemiring.unop_top, RingHom.mem_rangeS_self, Subsemiring.range_snd, Subalgebra.toSubsemiring_injective, Subsemiring.smulCommClass_left, Subsemiring.toSubsemiring_homogeneousCore_le, Subalgebra.algEquivOpMop_apply, Polynomial.lifts_iff_coeff_lifts, Subsemiring.mem_toSubmonoid, Subsemiring.instCovariantClassHSMulLe, Subsemiring.map_iSup, Subsemiring.coe_centralizer, Subsemiring.toNonUnitalSubsemiring_injective, Subring.mk_le_mk, Subsemiring.op_sSup, Subsemiring.coe_pointwise_smul, Subring.ringEquivOpMop_symm_apply_coe, Subalgebra.mem_toSubsemiring, Subsemiring.mem_closure_iff_exists_list, Subsemiring.mem_toAddSubmonoid, Subsemiring.map_toSubmonoid, isNoetherianRing_rangeS, RingPreordering.mem_toSubsemiring, Subsemiring.topologicalSemiring, Algebra.algebraMap_ofSubsemiring, Subsemiring.ringEquivOpMop_symm_apply_coe, Subsemiring.instSMulCommClassSubtypeMemCenter, Polynomial.X_pow_mem_lifts, HomogeneousSubsemiring.toSubsemiring_injective, Subsemiring.coe_closure_eq, Subsemiring.topologicalClosure_coe, Subsemiring.mem_pointwise_smul_iff_inv_smul_mem, Subsemiring.closure_empty, Subsemiring.mem_map, NonUnitalNonAssocSemiring.mem_center_iff, Subsemiring.toAddSubmonoid_mono, HomogeneousSubsemiring.is_homogeneous', Subsemiring.coe_ofClass, Subsemiring.closure_univ, Subsemiring.closure_union, Subsemiring.coe_matrix, Subsemiring.prod_mono_left, Subsemiring.subset_pointwise_smul_iff, Subsemiring.toAddSubmonoid_injective, Polynomial.lifts_iff_liftsRing, Subsemiring.mem_mk, Subsemiring.prod_bot_sup_bot_prod, Subsemiring.topEquiv_symm_apply_coe, Subsemiring.coe_op, Subsemiring.coe_zero, Subsemiring.unop_bot, Subsemiring.coe_set_mk, Subsemiring.centralizer_eq_top_iff_subset, CentroidHom.centerToCentroidCenter_apply, Subsemiring.coe_mk', Subalgebra.linearEquivOp_symm_apply_coe, RingCon.rangeS_mk', Subsemiring.le_topologicalClosure, Subsemiring.smul_sup, Subring.mem_mk, Subring.range_fst, Subsemiring.map_le_iff_le_comap, Subsemiring.ringEquivOpMop_apply, Subsemiring.subtype_apply, Subsemiring.toAddSubmonoid_strictMono, Subsemiring.pointwise_smul_le_pointwise_smul_iffβ‚€, RingEquiv.ofLeftInverseS_apply, Subsemiring.top_prod, Subalgebra.pointwise_smul_toSubsemiring, Submonoid.subsemiringClosure_mem, Subsemiring.op_top, Subsemiring.unop_iInf, Subsemiring.continuousSMul, Subsemiring.subtype_injective, Subsemiring.unop_sSup, Subsemiring.unop_eq_bot, Subsemiring.sInf_toSubmonoid, Subsemiring.coe_toAddSubmonoid, Polynomial.lifts_iff_set_range, Subalgebra.mem_perfectClosure_iff, Subsemiring.smulCommClass_right, Subsemiring.smul_closure, Subsemiring.le_centralizer_centralizer, Subsemiring.pointwise_smul_le_iffβ‚€, Subsemiring.closure_singleton_zero, Subsemiring.mopRingEquivOp_symm_apply, Subsemiring.coe_iInf, Subsemiring.coe_center, Algebra.IsInvariant.charpoly_mem_lifts, Subsemiring.topEquiv_apply, Subsemiring.coe_map, Subsemiring.toSubmonoid_injective, Module.End.mem_subsemiringCenter_iff, Subsemiring.mem_closure_of_mem, Subsemiring.instCanLiftSetCoeAndMemOfNatForallForallForallForallHAddForallForallForallForallHMul, RingHom.mem_eqLocusS, Subsemiring.coe_mul, Subsemiring.mem_center_iff, Subalgebra.mopAlgEquivOp_apply_coe, Subalgebra.coe_mk, Subsemiring.faithfulSMul, RingHom.eqOn_sclosure, Subalgebra.algEquivOpMop_symm_apply_coe, Subalgebra.rangeS_le, Subring.ringEquivOpMop_apply, Subring.toSubsemiring_eq_top, Subsemiring.mem_pointwise_smul_iff_inv_smul_memβ‚€, Subring.pointwise_smul_toSubsemiring, Subsemiring.unop_iSup, Algebra.inf_toSubsemiring, Subsemiring.mem_prod, Subsemiring.mem_nonneg, Subsemiring.mem_unop, IsHomogeneous.subsemiringClosure_of_isHomogeneousElem, Polynomial.mem_lift_of_splits_of_roots_mem_range, integralClosure.mem_lifts_of_monic_of_dvd_map, Polynomial.lifts_iff_coeffs_subset_range, Subsemiring.mem_closure_iff, Polynomial.mem_map_rangeS, Subalgebra.coe_valA', Subsemiring.mem_inv_pointwise_smul_iff, Subsemiring.coe_nonneg, RingHom.coe_rangeS, Subsemiring.coe_unop, Subsemiring.op_iInf, Subsemiring.mem_sInf, RingHom.eqLocusS_same, RingPreordering.toSubsemiring_injective, Subsemiring.coe_comap, Subsemiring.le_pointwise_smul_iffβ‚€, RingHom.ker_rangeSRestrict, Subsemiring.coe_bot, Subsemiring.op_bot, RingPreordering.coe_set_mk, Subsemiring.center_le_centralizer, Subsemiring.center.smulCommClass_left

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