Documentation Verification Report

Basic

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

Statistics

MetricCount
DefinitionssubInvSMul, resolvent, resolventSet, spectrum
4
Theoremsspectrum_eq, apply_mem_spectrum, mem_resolventSet_apply, spectrum_apply_subset, val_inv_subInvSMul, val_subInvSMul, zero_notMem_spectrum, add_mem_add_iff, add_mem_iff, add_singleton_eq, algebraMap_mem, algebraMap_mem_iff, inv_mem_iff, inv_mem_resolventSet, invβ‚€_mem, invβ‚€_mem_iff, invβ‚€_mem_inv, invβ‚€_mem_inv_iff, isUnit_of_zero_notMem, isUnit_resolvent, map_inv, map_star, mem_iff, mem_resolventSet_iff, mem_resolventSet_of_left_right_inverse, ne_zero_of_mem_of_unit, neg_eq, nonzero_mul_comm, notMem_iff, not_isUnit_of_zero_mem, of_algebraMap_mem, of_invβ‚€_mem, of_invβ‚€_mem_inv, of_subsingleton, one_eq, preimage_algebraMap, preimage_units_mul_comm, resolventSet_of_subsingleton, resolvent_eq, scalar_eq, setOf_isUnit_inter_mul_comm, singleton_add_eq, singleton_sub_eq, smul_eq_smul, smul_mem_smul_iff, star_mem_resolventSet_iff, sub_singleton_eq, subset_singleton_zero_compl, subset_subalgebra, unit_mem_mul_comm, unit_smul_eq_smul, units_conjugate, units_conjugate', units_smul_resolvent, units_smul_resolvent_self, vadd_eq, zero_eq, zero_mem, zero_mem_iff, zero_mem_resolventSet_of_unit, zero_notMem, zero_notMem_iff
62
Total66

AlgEquiv

Theorems

NameKindAssumesProvesValidatesDepends On
spectrum_eq πŸ“–mathematicalβ€”spectrum
DFunLike.coe
EquivLike.toFunLike
β€”Set.Subset.antisymm
AlgHom.spectrum_apply_subset
AlgEquivClass.toAlgHomClass
coe_coe_symm_apply_coe_apply
instAlgEquivClass

AlgHom

Theorems

NameKindAssumesProvesValidatesDepends On
apply_mem_spectrum πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
CommRing.toCommSemiring
DFunLike.coe
β€”RingHom.instRingHomClass
AlgHomClass.toRingHomClass
map_sub
RingHomClass.toAddMonoidHomClass
AlgHomClass.commutes
sub_self
coe_subset_nonunits
RingHom.ker_ne_top
mem_resolventSet_apply πŸ“–mathematicalSet
Set.instMembership
resolventSet
DFunLike.coeβ€”map_sub
DistribMulActionSemiHomClass.toAddMonoidHomClass
NonUnitalAlgSemiHomClass.toDistribMulActionSemiHomClass
instNonUnitalAlgHomClassOfAlgHomClass
AlgHomClass.commutes
IsUnit.map
MonoidWithZeroHomClass.toMonoidHomClass
RingHomClass.toMonoidWithZeroHomClass
AlgHomClass.toRingHomClass
spectrum_apply_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
spectrum
DFunLike.coe
β€”mem_resolventSet_apply

IsUnit

Definitions

NameCategoryTheorems
subInvSMul πŸ“–CompOp
2 mathmath: val_subInvSMul, val_inv_subInvSMul

Theorems

NameKindAssumesProvesValidatesDepends On
val_inv_subInvSMul πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
Units
CommSemiring.toSemiring
Units.instSMul
Algebra.toSMul
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
algebraMap
Units.val
Units.instInv
subInvSMul
unit
β€”β€”
val_subInvSMul πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
Units
CommSemiring.toSemiring
Units.instSMul
Algebra.toSMul
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
RingHom.instFunLike
algebraMap
Units.val
subInvSMul
Units.instInv
β€”β€”

Units

Theorems

NameKindAssumesProvesValidatesDepends On
zero_notMem_spectrum πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
β€”spectrum.zero_notMem
isUnit

(root)

Definitions

NameCategoryTheorems
resolvent πŸ“–CompOp
8 mathmath: spectrum.units_smul_resolvent_self, spectrum.resolvent_eq, spectrum.eventually_isUnit_resolvent, spectrum.hasDerivAt_resolvent, spectrum.resolvent_tendsto_cobounded, spectrum.units_smul_resolvent, spectrum.resolvent_isBigO_inv, spectrum.isUnit_resolvent
resolventSet πŸ“–CompOp
10 mathmath: spectrum.isOpen_resolventSet, spectrum.mem_resolventSet_of_left_right_inverse, spectrum.zero_mem_resolventSet_of_unit, spectrum.mem_resolventSet_iff, spectrum.star_mem_resolventSet_iff, spectrum.mem_resolventSet_of_norm_lt, spectrum.mem_resolventSet_of_spectralRadius_lt, spectrum.resolventSet_of_subsingleton, spectrum.isUnit_resolvent, spectrum.mem_resolventSet_of_norm_lt_mul
spectrum πŸ“–CompOp
188 mathmath: unitary.spectrum.unitary_conjugate, spectrum.inv_mem_iff, quasispectrum_eq_spectrum_union, spectrum_subset_quasispectrum, Matrix.IsHermitian.isClosedEmbedding_cfcAux, IsometricContinuousFunctionalCalculus.isGreatest_nnnorm_spectrum, IsSelfAdjoint.commute_cfcHom, Subalgebra.spectrum_sUnion_connectedComponentIn, Module.End.HasEigenvalue.mem_spectrum, spectrum.subset_closedBall_norm_mul, spectrumRestricts_iff, spectrum.instCompactSpace, Matrix.spectrum_toEuclideanLin, spectrum.invβ‚€_mem_iff, ContinuousFunctionalCalculus.compactSpace_spectrum, Module.End.spectrum_intrinsicStar, CStarAlgebra.norm_or_neg_norm_mem_spectrum, Matrix.spectrum_toLpLin, StarAlgebra.elemental.characterSpaceToSpectrum_coe, WeakDual.CharacterSpace.apply_mem_spectrum, Matrix.mem_spectrum_iff_isRoot_charpoly, cfcHom_continuous, cfcL_apply, LinearMap.spectrum_toMatrix', spectrum.nonempty_of_isAlgClosed_of_finiteDimensional, cfcHom_nonneg_iff, cfc_apply_mkD, spectrum_diagonal, unitary.spectrum.unitary_conjugate', AlgHom.apply_mem_spectrum, NNReal.spectrum_nonempty, cfcHom_isClosedEmbedding, SpectrumRestricts.compactSpace, Pi.spectrum_eq, spectrum.zero_notMem_iff, ContinuousFunctionalCalculus.isCompact_spectrum, IsIdempotentElem.spectrum_subset, cfcHom_le_iff, cfcHom_id, Unitary.spectrum_star_left_conjugate, IsSelfAdjoint.coe_mem_spectrum_complex, spectrum.zero_mem_iff, spectrum.gelfandTransform_eq, spectrum.zero_mem, spectrum.zero_notMem, ContinuousMap.spectrum_eq_preimage_range, Matrix.mem_spectrum_of_isRoot_charpoly, Unitary.norm_sub_one_lt_two_iff, spectrum.isCompact, spectrum.algebraMap_mem_iff, CFC.spectrum_algebraMap_eq, SpectrumRestricts.algebraMap_image, continuousOn_cfc_setProd, spectrum.smul_mem_smul_iff, Matrix.finite_spectrum, IsSelfAdjoint.val_re_map_spectrum, Unitization.zero_mem_spectrum_inr, Module.End.hasUnifEigenvalue_iff_mem_spectrum, coe_mem_spectrum_real_of_nonneg, Prod.spectrum_eq, IsSelfAdjoint.isConnected_spectrum_compl, spectrum.one_eq, Matrix.instFiniteSpectrum, StarAlgebra.elemental.bijective_characterSpaceToSpectrum, Unitary.spectrum_star_right_conjugate, continuousOn_cfc_nnreal, spectrum.map_pow, ContinuousFunctionalCalculus.exists_cfc_of_predicate, spectrum.pow_image_subset, spectrum.units_conjugate, Units.continuousOn_zpowβ‚€_spectrum, spectrum.subset_subalgebra, cfc_def, IsSelfAdjoint.spectrum_nonempty, lipschitzOnWith_cfc_fun, StarSubalgebra.spectrum_eq, spectrum.invβ‚€_mem_inv_iff, SpectrumRestricts.subset_preimage, Matrix.IsHermitian.cfcAux_apply, spectrum.preimage_algebraMap, Matrix.instFiniteElemRealSpectrum, unitary.spectrum_subset_circle, spectrum.isClosed, mem_quasispectrum_iff, LinearMap.spectrum_toMatrix, spectrum.isBounded, Unitization.mem_spectrum_inr_of_not_isUnit, Subalgebra.frontier_subset_frontier, spectrum.exists_mem_of_not_isUnit_aeval_prod, spectrum.subset_singleton_zero_compl, AlgEquiv.spectrum_eq, spectrum.nonempty, spectrum_subset_unitary_of_mem_unitary, spectrum.add_singleton_eq, cfc_eq_cfcL_mkD, range_cfc_eq_range_cfcHom, NormedRing.algEquivComplexOfComplete_symm_apply, Matrix.IsHermitian.spectrum_eq_image_range, Unitary.spectrum_subset_slitPlane_iff_norm_lt_two, spectrum.singleton_add_eq, spectrum.singleton_sub_eq, spectrum.map_star, Module.End.finite_spectrum, upperHemicontinuous_spectrum_nnreal, spec_cfcβ‚™Aux, cfcHom_eq_cfc_extend, spectrum.subset_polynomial_aeval, Matrix.spectrum_toLin, SpectrumRestricts.starAlgHom_apply, Matrix.finite_real_spectrum, spectrum.map_inv, Commute.cfcHom, AlgHom.spectrum_apply_subset, spectrum.mem_iff, spectrum.add_mem_add_iff, spectrum.map_polynomial_aeval_of_degree_pos, ContinuousMap.spectrum_eq_range, StarSubalgebra.mem_spectrum_iff, Matrix.IsHermitian.spectrum_real_eq_range_eigenvalues, cfcHom_isStrictlyPositive_iff, WeakDual.CharacterSpace.mem_spectrum_iff_exists, spectrum.add_mem_iff, CFC.spectrum_abs, Units.continuousOn_invβ‚€_spectrum, Units.zero_notMem_spectrum, cfc_apply_pi, SpectrumRestricts.image, spectrum.units_conjugate', IsSelfAdjoint.map_spectrum_real, Matrix.IsHermitian.cfcAux_id, spectrum.scalar_eq, unitary_iff_isStarNormal_and_spectrum_subset_unitary, spectrum.of_subsingleton, Subalgebra.frontier_spectrum, spectrum.setOf_isUnit_inter_mul_comm, Unitization.quasispectrum_eq_spectrum_inr', spectrum.isCompact_nnreal, spectrum.unit_mem_mul_comm, selfAdjoint.val_re_map_spectrum, Matrix.IsHermitian.eigenvalues_mem_spectrum_real, cfcHom_predicate, spectrum.instCompactSpaceNNReal, spectrum.preimage_units_mul_comm, norm_cfcHom, spectrum.exists_nnnorm_eq_spectralRadius, unitary.norm_sub_one_lt_two_iff, spectrum.nonzero_mul_comm, quasispectrum_eq_spectrum_union_zero, continuousOn_cfc_nnreal_setProd, SpectrumRestricts.rightInvOn, range_cfcHom, Module.End.mem_spectrum_iff_isRoot_charpoly, nnnorm_cfcHom, continuousOn_cfc, StarAlgebra.elemental.continuous_characterSpaceToSpectrum, spectrum.neg_eq, CStarAlgebra.norm_mem_spectrum_of_nonneg, Module.End.hasEigenvalue_iff_mem_spectrum, CFC.spectrum_one_eq, IsometricContinuousFunctionalCalculus.isGreatest_norm_spectrum, spectrum.unit_smul_eq_smul, CFC.spectrum_nonempty, CFC.spectrum_algebraMap_subset, IsometricContinuousFunctionalCalculus.isometric, spectrum.map_polynomial_aeval, ContinuousFunctionalCalculus.spectrum_nonempty, Unitization.quasispectrum_eq_spectrum_inr, spectrum.map_pow_of_pos, cfcHom_eq_of_isStarNormal, upperHemicontinuous_spectrum, cfcHom_map_spectrum, spectrum.zero_eq, spectrum.subset_circle_of_unitary, spectrum.subset_closedBall_norm, spectrum.vadd_eq, IsometricContinuousFunctionalCalculus.isGreatest_spectrum, Module.End.HasUnifEigenvalue.mem_spectrum, continuousFunctionalCalculus_map_id, spectrum.sub_singleton_eq, CStarAlgebra.nnnorm_mem_spectrum_of_nonneg, Unitary.spectrum_subset_circle, cfcHom_apply_mem_elemental, Matrix.posSemidef_iff_isHermitian_and_spectrum_nonneg, spectrum.notMem_iff, isometry_cfcHom, CFC.spectrum_zero_eq, unitary.spectrum_subset_slitPlane_iff_norm_lt_two, Matrix.spectrum_toLin'

spectrum

Theorems

NameKindAssumesProvesValidatesDepends On
add_mem_add_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
NonAssocSemiring.toNonUnitalNonAssocSemiring
β€”add_mem_iff
neg_add_cancel_left
add_mem_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Ring.toAddCommGroup
DFunLike.coe
RingHom
Ring.toSemiring
RingHom.instFunLike
algebraMap
β€”map_add
AddMonoidHomClass.toAddHomClass
RingHomClass.toAddMonoidHomClass
RingHom.instRingHomClass
sub_neg_eq_add
add_singleton_eq πŸ“–mathematicalβ€”Set
Set.add
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
spectrum
CommRing.toCommSemiring
Set.instSingletonSet
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
β€”singleton_add_eq
add_comm
algebraMap_mem πŸ“–mathematicalSet
Set.instMembership
spectrum
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
algebraMap
β€”algebraMap_mem_iff
algebraMap_mem_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
algebraMap
β€”Algebra.algebraMap_eq_smul_one
smul_assoc
one_smul
inv_mem_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
CommSemiring.toSemiring
Units
Units.instInv
β€”not_iff_not
inv_mem_resolventSet
inv_mem_resolventSet πŸ“–mathematicalSet
Set.instMembership
resolventSet
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
CommSemiring.toSemiring
Units
Units.instInv
β€”mem_resolventSet_iff
Algebra.algebraMap_eq_smul_one
Units.smul_def
IsUnit.smul_sub_iff_sub_inv_smul
Units.instIsScalarTower
IsScalarTower.right
Units.smulCommClass_left
Algebra.to_smulCommClass
inv_inv
IsUnit.sub_iff
mul_sub
mul_smul_comm
Units.mul_inv
mul_one
sub_mul
smul_mul_assoc
Units.inv_mul
one_mul
Commute.isUnit_mul_iff
invβ‚€_mem πŸ“–mathematicalSet
Set.instMembership
spectrum
Semifield.toCommSemiring
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Units
Units.instInv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
β€”invβ‚€_mem_iff
invβ‚€_mem_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
Semifield.toCommSemiring
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
Units
Units.instInv
β€”eq_or_ne
inv_zero
CanLift.prf
instCanLiftUnitsValIsUnit
Ne.isUnit
inv_inv
Units.val_inv_eq_inv_val
invβ‚€_mem_inv πŸ“–mathematicalSet
Set.instMembership
spectrum
Semifield.toCommSemiring
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Units
Units.instInv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
β€”invβ‚€_mem_inv_iff
invβ‚€_mem_inv_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
Semifield.toCommSemiring
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Units
Units.instInv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
β€”inv_inv
isUnit_of_zero_notMem πŸ“–mathematicalSet
Set.instMembership
spectrum
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
IsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
β€”zero_notMem_iff
isUnit_resolvent πŸ“–mathematicalβ€”Set
Set.instMembership
resolventSet
IsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
resolvent
β€”isUnit_ringInverse
map_inv πŸ“–mathematicalβ€”Set
Set.inv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
Field.toSemifield
spectrum
Semifield.toCommSemiring
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Units
Units.instInv
β€”Set.ext
map_star πŸ“–mathematicalβ€”spectrum
Star.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
StarRing.toStarAddMonoid
Set
Set.star
β€”Set.ext
star_mem_resolventSet_iff
mem_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
IsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
algebraMap
β€”β€”
mem_resolventSet_iff πŸ“–mathematicalβ€”Set
Set.instMembership
resolventSet
IsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
algebraMap
β€”β€”
mem_resolventSet_of_left_right_inverse πŸ“–mathematicalDistrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Set
Set.instMembership
resolventSet
β€”Units.isUnit
left_inv_eq_right_inv
ne_zero_of_mem_of_unit πŸ“–β€”Set
Set.instMembership
spectrum
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
β€”β€”zero_mem_resolventSet_of_unit
neg_eq πŸ“–mathematicalβ€”Set
Set.neg
NegZeroClass.toNeg
SubNegZeroMonoid.toNegZeroClass
SubtractionMonoid.toSubNegZeroMonoid
SubtractionCommMonoid.toSubtractionMonoid
AddCommGroup.toDivisionAddCommMonoid
Ring.toAddCommGroup
CommRing.toRing
spectrum
CommRing.toCommSemiring
β€”Set.ext
map_neg
RingHomClass.toAddMonoidHomClass
RingHom.instRingHomClass
sub_neg_eq_add
nonzero_mul_comm πŸ“–mathematicalβ€”Set
Set.instSDiff
spectrum
Semifield.toCommSemiring
Field.toSemifield
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Set.instSingletonSet
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
β€”unit_mem_mul_comm
Set.eq_of_subset_of_subset
notMem_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
IsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
algebraMap
β€”β€”
not_isUnit_of_zero_mem πŸ“–mathematicalSet
Set.instMembership
spectrum
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
IsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
β€”zero_mem_iff
of_algebraMap_mem πŸ“–β€”Set
Set.instMembership
spectrum
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
algebraMap
β€”β€”algebraMap_mem_iff
of_invβ‚€_mem πŸ“–mathematicalSet
Set.instMembership
spectrum
Semifield.toCommSemiring
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
Units
Units.instInv
β€”invβ‚€_mem_iff
of_invβ‚€_mem_inv πŸ“–β€”Set
Set.instMembership
spectrum
Semifield.toCommSemiring
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Units
Units.instInv
InvOneClass.toInv
DivInvOneMonoid.toInvOneClass
DivisionMonoid.toDivInvOneMonoid
DivisionCommMonoid.toDivisionMonoid
CommGroupWithZero.toDivisionCommMonoid
Semifield.toCommGroupWithZero
β€”β€”invβ‚€_mem_inv_iff
of_subsingleton πŸ“–mathematicalβ€”spectrum
Set
Set.instEmptyCollection
β€”eq_1
resolventSet_of_subsingleton
Set.compl_univ
one_eq πŸ“–mathematicalβ€”spectrum
Semifield.toCommSemiring
Field.toSemifield
AddMonoidWithOne.toOne
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
Set
Set.instSingletonSet
DivisionRing.toRing
Field.toDivisionRing
β€”Algebra.algebraMap_eq_smul_one
one_smul
scalar_eq
preimage_algebraMap πŸ“–mathematicalβ€”Set.preimage
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
algebraMap
spectrum
β€”Set.ext
algebraMap_mem_iff
preimage_units_mul_comm πŸ“–mathematicalβ€”Set.preimage
Units
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
Units.val
spectrum
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”Set.ext
unit_mem_mul_comm
resolventSet_of_subsingleton πŸ“–mathematicalβ€”resolventSet
Set.univ
β€”β€”
resolvent_eq πŸ“–mathematicalSet
Set.instMembership
resolventSet
resolvent
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Units
Units.instInv
IsUnit.unit
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
RingHom.instFunLike
algebraMap
β€”Ring.inverse_unit
scalar_eq πŸ“–mathematicalβ€”spectrum
Semifield.toCommSemiring
Field.toSemifield
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
Set
Set.instSingletonSet
β€”add_zero
singleton_add_eq
zero_eq
Set.singleton_add_singleton
setOf_isUnit_inter_mul_comm πŸ“–mathematicalβ€”Set
Set.instInter
setOf
IsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
spectrum
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
β€”Set.ext
unit_mem_mul_comm
singleton_add_eq πŸ“–mathematicalβ€”Set
Set.add
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Set.instSingletonSet
spectrum
CommRing.toCommSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
β€”Set.ext
Set.singleton_add
Set.image_add_left
Set.mem_preimage
add_comm
add_mem_iff
map_neg
RingHomClass.toAddMonoidHomClass
RingHom.instRingHomClass
neg_neg
singleton_sub_eq πŸ“–mathematicalβ€”Set
Set.sub
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
CommRing.toRing
Set.instSingletonSet
spectrum
CommRing.toCommSemiring
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
β€”sub_eq_add_neg
neg_eq
singleton_add_eq
smul_eq_smul πŸ“–mathematicalSet.Nonempty
spectrum
Semifield.toCommSemiring
Field.toSemifield
Algebra.toSMul
Ring.toSemiring
Set
Set.smulSet
CommSemiring.toSemiring
Algebra.id
β€”eq_or_ne
zero_smul
zero_eq
Set.zero_smul_set
unit_smul_eq_smul
smul_mem_smul_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
Units
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
Units.instSMul
Algebra.toSMul
Ring.toSemiring
Algebra.id
β€”Algebra.algebraMap_eq_smul_one
smul_assoc
Units.instIsScalarTower
IsScalarTower.left
Units.smulCommClass_left
Algebra.to_smulCommClass
IsScalarTower.right
star_mem_resolventSet_iff πŸ“–mathematicalβ€”Set
Set.instMembership
resolventSet
Star.star
InvolutiveStar.toStar
StarAddMonoid.toInvolutiveStar
AddCommMonoid.toAddMonoid
NonUnitalNonAssocSemiring.toAddCommMonoid
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
StarRing.toStarAddMonoid
β€”Algebra.algebraMap_eq_smul_one
star_sub
StarModule.star_smul
star_star
star_one
IsUnit.star
sub_singleton_eq πŸ“–mathematicalβ€”Set
Set.sub
SubNegMonoid.toSub
AddGroup.toSubNegMonoid
AddGroupWithOne.toAddGroup
Ring.toAddGroupWithOne
CommRing.toRing
spectrum
CommRing.toCommSemiring
Set.instSingletonSet
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
β€”neg_sub
neg_eq
singleton_sub_eq
subset_singleton_zero_compl πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Set
Set.instHasSubset
spectrum
Compl.compl
Set.instCompl
Set.instSingletonSet
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
β€”Set.subset_compl_singleton_iff
zero_notMem
subset_subalgebra πŸ“–mathematicalβ€”Set
Set.instHasSubset
spectrum
SetLike.instMembership
SubringClass.toRing
SubalgebraClass.toAlgebra
Ring.toSemiring
SubringClass.toSubsemiringClass
β€”SubringClass.toSubsemiringClass
Set.compl_subset_compl
IsUnit.map
MonoidWithZeroHomClass.toMonoidHomClass
RingHomClass.toMonoidWithZeroHomClass
AlgHomClass.toRingHomClass
AlgHom.algHomClass
unit_mem_mul_comm πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
β€”sub_eq_add_neg
IsUnit.unit.congr_simp
mul_assoc
mul_add
Distrib.leftDistribClass
mul_one
add_mul
Distrib.rightDistribClass
one_mul
neg_mul
mul_neg
IsUnit.mul_val_inv
sub_add_cancel
Mathlib.Tactic.Abel.subst_into_addg
Mathlib.Tactic.Abel.term_atomg
Mathlib.Tactic.Abel.term_add_constg
zero_add
Mathlib.Tactic.Abel.subst_into_negg
Mathlib.Tactic.Abel.term_neg
neg_zero
Mathlib.Tactic.Abel.const_add_termg
IsUnit.val_inv_mul
Algebra.algebraMap_eq_smul_one
Units.instIsScalarTower
IsScalarTower.right
Units.smulCommClass_left
Algebra.to_smulCommClass
smul_mul_assoc
mul_smul_comm
unit_smul_eq_smul πŸ“–mathematicalβ€”spectrum
Units
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
Units.instSMul
Algebra.toSMul
Ring.toSemiring
Set
Set.smulSet
Algebra.id
β€”Set.ext
smul_inv_smul
smul_mem_smul_iff
inv_smul_smul
units_conjugate πŸ“–mathematicalβ€”spectrum
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Units
Units.instInv
β€”mem_iff
Mathlib.Tactic.Contrapose.contraposeβ‚„
mul_sub
sub_mul
Algebra.right_comm
Units.mul_inv
one_mul
IsUnit.mul
Units.isUnit
le_antisymm
mul_assoc
Units.inv_mul_cancel_left
inv_inv
Units.inv_mul
mul_one
units_conjugate' πŸ“–mathematicalβ€”spectrum
Distrib.toMul
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Units
Units.instInv
β€”inv_inv
units_conjugate
units_smul_resolvent πŸ“–mathematicalβ€”Units
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
Units.instSMul
Algebra.toSMul
Ring.toSemiring
resolvent
Units.instInv
Algebra.id
β€”Algebra.algebraMap_eq_smul_one
smul_assoc
Units.instIsScalarTower
IsScalarTower.left
smul_sub
mem_iff
smul_inv_smul
IsUnit.smul
Units.smulCommClass_left
Algebra.to_smulCommClass
IsScalarTower.right
Ring.inverse_non_unit
smul_zero
notMem_iff
IsUnit.val_subInvSMul
IsUnit.unit_spec
Ring.inverse_unit
IsUnit.val_inv_subInvSMul
IsUnit.unit.congr_simp
units_smul_resolvent_self πŸ“–mathematicalβ€”Units
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
CommSemiring.toSemiring
Units.instSMul
Algebra.toSMul
Ring.toSemiring
resolvent
Units.val
Units.instInv
AddMonoidWithOne.toOne
AddCommMonoidWithOne.toAddMonoidWithOne
NonAssocSemiring.toAddCommMonoidWithOne
Semiring.toNonAssocSemiring
β€”Units.inv_mul
units_smul_resolvent
vadd_eq πŸ“–mathematicalβ€”HVAdd.hVAdd
Set
instHVAdd
Set.vaddSet
AddSemigroupAction.toVAdd
AddMonoid.toAddSemigroup
AddMonoidWithOne.toAddMonoid
AddGroupWithOne.toAddMonoidWithOne
Ring.toAddGroupWithOne
CommRing.toRing
AddAction.toAddSemigroupAction
AddMonoid.toAddAction
spectrum
CommRing.toCommSemiring
Distrib.toAdd
NonUnitalNonAssocSemiring.toDistrib
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
DFunLike.coe
RingHom
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
Ring.toSemiring
RingHom.instFunLike
algebraMap
β€”Set.singleton_add
singleton_add_eq
zero_eq πŸ“–mathematicalβ€”spectrum
Semifield.toCommSemiring
Field.toSemifield
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring
NonAssocRing.toNonUnitalNonAssocRing
Ring.toNonAssocRing
Set
Set.instSingletonSet
NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing
NonUnitalCommRing.toNonUnitalNonAssocCommRing
CommRing.toNonUnitalCommRing
Field.toCommRing
β€”Set.Subset.antisymm
eq_1
Set.compl_subset_comm
Set.mem_compl_singleton_iff
IsUnit.smul
Units.smulCommClass_left
Algebra.to_smulCommClass
Units.instIsScalarTower
IsScalarTower.right
isUnit_one
Algebra.algebraMap_eq_smul_one
sub_zero
zero_smul
sub_self
NeZero.one
zero_mem πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Set
Set.instMembership
spectrum
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
β€”zero_mem_iff
zero_mem_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
IsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
β€”mem_iff
map_zero
MonoidWithZeroHomClass.toZeroHomClass
RingHomClass.toMonoidWithZeroHomClass
RingHom.instRingHomClass
zero_sub
IsUnit.neg_iff
zero_mem_resolventSet_of_unit πŸ“–mathematicalβ€”Set
Set.instMembership
resolventSet
Units.val
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
β€”Units.isUnit
zero_notMem πŸ“–mathematicalIsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
Set
Set.instMembership
spectrum
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
β€”zero_notMem_iff
zero_notMem_iff πŸ“–mathematicalβ€”Set
Set.instMembership
spectrum
MulZeroClass.toZero
NonUnitalNonAssocSemiring.toMulZeroClass
NonAssocSemiring.toNonUnitalNonAssocSemiring
Semiring.toNonAssocSemiring
CommSemiring.toSemiring
IsUnit
MonoidWithZero.toMonoid
Semiring.toMonoidWithZero
Ring.toSemiring
β€”zero_mem_iff

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