fact
stringlengths 6
3.84k
| type
stringclasses 11
values | library
stringclasses 32
values | imports
listlengths 1
14
| filename
stringlengths 20
95
| symbolic_name
stringlengths 1
90
| docstring
stringlengths 7
20k
⌀ |
|---|---|---|---|---|---|---|
toAddGroupTopology (t : RingTopology R) : AddGroupTopology R where
toTopologicalSpace := t.toTopologicalSpace
toIsTopologicalAddGroup :=
@IsTopologicalRing.to_topologicalAddGroup _ _ t.toTopologicalSpace t.toIsTopologicalRing
|
def
|
Topology
|
[
"Mathlib.Algebra.Order.AbsoluteValue.Basic",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.Topology.Algebra.Group.GroupTopology"
] |
Mathlib/Topology/Algebra/Ring/Basic.lean
|
toAddGroupTopology
|
The forgetful functor from ring topologies on `a` to additive group topologies on `a`.
|
toAddGroupTopology.orderEmbedding : OrderEmbedding (RingTopology R) (AddGroupTopology R) :=
OrderEmbedding.ofMapLEIff toAddGroupTopology fun _ _ => Iff.rfl
|
def
|
Topology
|
[
"Mathlib.Algebra.Order.AbsoluteValue.Basic",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.Topology.Algebra.Group.GroupTopology"
] |
Mathlib/Topology/Algebra/Ring/Basic.lean
|
toAddGroupTopology.orderEmbedding
|
The order embedding from ring topologies on `a` to additive group topologies on `a`.
|
AbsoluteValue.comp {R S T : Type*} [Semiring T] [Semiring R] [Semiring S] [PartialOrder S]
(v : AbsoluteValue R S) {f : T →+* R} (hf : Function.Injective f) : AbsoluteValue T S where
toMulHom := v.1.comp f
nonneg' _ := v.nonneg _
eq_zero' _ := v.eq_zero.trans (map_eq_zero_iff f hf)
add_le' _ _ := (congr_arg v (map_add f _ _)).trans_le (v.add_le _ _)
|
def
|
Topology
|
[
"Mathlib.Algebra.Order.AbsoluteValue.Basic",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.Ring.Subring.Basic",
"Mathlib.Topology.Algebra.Group.GroupTopology"
] |
Mathlib/Topology/Algebra/Ring/Basic.lean
|
AbsoluteValue.comp
|
Construct an absolute value on a semiring `T` from an absolute value on a semiring `R`
and an injective ring homomorphism `f : T →+* R`
|
finite_of_compactSpace_of_t2Space [IsArtinianRing R] :
Finite R := by
obtain ⟨n, hn⟩ := IsArtinianRing.isNilpotent_jacobson_bot (R := R)
have H : (∏ p : PrimeSpectrum R, p.asIdeal) ^ n = ⊥ := by
rw [← le_bot_iff, ← Ideal.zero_eq_bot, ← hn]
gcongr
rw [Ideal.jacobson_bot, Ring.jacobson_eq_sInf_isMaximal, le_sInf_iff]
exact fun I hI ↦ Ideal.prod_le_inf.trans
(Finset.inf_le (b := PrimeSpectrum.mk I hI.isPrime) (by simp))
have := Ideal.finite_quotient_prod (R := R) PrimeSpectrum.asIdeal Finset.univ
(fun _ _ ↦ IsNoetherian.noetherian _) (fun _ _ ↦ inferInstance)
have := Ideal.finite_quotient_pow (IsNoetherian.noetherian (∏ p : PrimeSpectrum R, p.asIdeal)) n
rw [H] at this
exact .of_equiv _ (RingEquiv.quotientBot R).toEquiv
|
theorem
|
Topology
|
[
"Mathlib.RingTheory.DedekindDomain.Factorization",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.HopkinsLevitzki",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.LocalRing.Quotient",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Ideal"
] |
Mathlib/Topology/Algebra/Ring/Compact.lean
|
finite_of_compactSpace_of_t2Space
|
Compact Hausdorff Artinian (commutative) rings are finite. This is not an instance, as it would
apply to every `Finite` goal, causing slowly failing typeclass search in some cases.
|
Ideal.isOpen_of_isMaximal (I : Ideal R) [I.IsMaximal] : IsOpen (X := R) I :=
have : I.toAddSubgroup.FiniteIndex :=
@AddSubgroup.finiteIndex_of_finite_quotient _ _ _
(inferInstanceAs (Finite (R ⧸ I)))
I.toAddSubgroup.isOpen_of_isClosed_of_finiteIndex (inferInstanceAs (IsClosed (X := R) I))
|
lemma
|
Topology
|
[
"Mathlib.RingTheory.DedekindDomain.Factorization",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.HopkinsLevitzki",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.LocalRing.Quotient",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Ideal"
] |
Mathlib/Topology/Algebra/Ring/Compact.lean
|
Ideal.isOpen_of_isMaximal
| null |
Ideal.isOpen_pow_of_isMaximal (I : Ideal R) [I.IsMaximal] (n : ℕ) :
IsOpen (X := R) ↑(I ^ n) :=
have : (I ^ n).toAddSubgroup.FiniteIndex :=
@AddSubgroup.finiteIndex_of_finite_quotient _ _ _
(Ideal.finite_quotient_pow (IsNoetherian.noetherian _) _)
(I ^ n).toAddSubgroup.isOpen_of_isClosed_of_finiteIndex
(Ideal.isCompact_of_fg (IsNoetherian.noetherian _)).isClosed
|
lemma
|
Topology
|
[
"Mathlib.RingTheory.DedekindDomain.Factorization",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.HopkinsLevitzki",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.LocalRing.Quotient",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Ideal"
] |
Mathlib/Topology/Algebra/Ring/Compact.lean
|
Ideal.isOpen_pow_of_isMaximal
| null |
isOpen_maximalIdeal_pow (n : ℕ) :
IsOpen (X := R) ↑(maximalIdeal R ^ n) :=
Ideal.isOpen_pow_of_isMaximal _ _
variable (R) in
|
lemma
|
Topology
|
[
"Mathlib.RingTheory.DedekindDomain.Factorization",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.HopkinsLevitzki",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.LocalRing.Quotient",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Ideal"
] |
Mathlib/Topology/Algebra/Ring/Compact.lean
|
isOpen_maximalIdeal_pow
| null |
isOpen_maximalIdeal : IsOpen (X := R) ↑(maximalIdeal R) :=
Ideal.isOpen_of_isMaximal _
|
lemma
|
Topology
|
[
"Mathlib.RingTheory.DedekindDomain.Factorization",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.HopkinsLevitzki",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.LocalRing.Quotient",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Ideal"
] |
Mathlib/Topology/Algebra/Ring/Compact.lean
|
isOpen_maximalIdeal
| null |
finite_residueField_of_compactSpace : Finite (ResidueField R) :=
inferInstanceAs (Finite (R ⧸ _))
|
instance
|
Topology
|
[
"Mathlib.RingTheory.DedekindDomain.Factorization",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.HopkinsLevitzki",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.LocalRing.Quotient",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Ideal"
] |
Mathlib/Topology/Algebra/Ring/Compact.lean
|
finite_residueField_of_compactSpace
| null |
isOpen_iff_finite_quotient {I : Ideal R} :
IsOpen (X := R) I ↔ Finite (R ⧸ I) := by
refine ⟨AddSubgroup.quotient_finite_of_isOpen I.toAddSubgroup, fun H ↦ ?_⟩
obtain ⟨n, hn⟩ := exists_maximalIdeal_pow_le_of_isArtinianRing_quotient I
exact AddSubgroup.isOpen_mono (H₁ := (maximalIdeal R ^ n).toAddSubgroup)
(H₂ := I.toAddSubgroup) hn (isOpen_maximalIdeal_pow R n)
|
lemma
|
Topology
|
[
"Mathlib.RingTheory.DedekindDomain.Factorization",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.HopkinsLevitzki",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.LocalRing.Quotient",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Ideal"
] |
Mathlib/Topology/Algebra/Ring/Compact.lean
|
isOpen_iff_finite_quotient
| null |
IsDedekindDomain.isOpen_of_ne_bot
[IsDedekindDomain R] {I : Ideal R} (hI : I ≠ ⊥) :
IsOpen (X := R) I := by
rw [← Ideal.finprod_heightOneSpectrum_factorization hI,
finprod_eq_finset_prod_of_mulSupport_subset _
(s := (Ideal.finite_mulSupport hI).toFinset) (by simp)]
refine @AddSubgroup.isOpen_of_isClosed_of_finiteIndex _ _ _ _ (Submodule.toAddSubgroup _)
?_ (IsNoetherianRing.isClosed_ideal _)
refine @AddSubgroup.finiteIndex_of_finite_quotient _ _ _ ?_
refine Ideal.finite_quotient_prod _ _ (fun _ _ ↦ IsNoetherian.noetherian _) fun _ _ ↦ ?_
exact Ideal.finite_quotient_pow (IsNoetherian.noetherian _) _
|
lemma
|
Topology
|
[
"Mathlib.RingTheory.DedekindDomain.Factorization",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.HopkinsLevitzki",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.LocalRing.Quotient",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Ideal"
] |
Mathlib/Topology/Algebra/Ring/Compact.lean
|
IsDedekindDomain.isOpen_of_ne_bot
| null |
IsDedekindDomain.isOpen_iff
[IsDedekindDomain R] (hR : ¬ IsField R) {I : Ideal R} :
IsOpen (X := R) I ↔ I ≠ ⊥ := by
refine ⟨?_, IsDedekindDomain.isOpen_of_ne_bot⟩
rintro H rfl
have := discreteTopology_iff_isOpen_singleton_zero.mpr H
exact hR (Finite.isField_of_domain R)
|
lemma
|
Topology
|
[
"Mathlib.RingTheory.DedekindDomain.Factorization",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.HopkinsLevitzki",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.LocalRing.Quotient",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Ideal"
] |
Mathlib/Topology/Algebra/Ring/Compact.lean
|
IsDedekindDomain.isOpen_iff
| null |
IsDiscreteValuationRing.isOpen_iff
[IsDomain R] [IsDiscreteValuationRing R] {I : Ideal R} :
IsOpen (X := R) I ↔ I ≠ ⊥ :=
IsDedekindDomain.isOpen_iff (not_isField R)
|
lemma
|
Topology
|
[
"Mathlib.RingTheory.DedekindDomain.Factorization",
"Mathlib.RingTheory.DiscreteValuationRing.Basic",
"Mathlib.RingTheory.HopkinsLevitzki",
"Mathlib.RingTheory.IntegralDomain",
"Mathlib.RingTheory.LocalRing.Quotient",
"Mathlib.Topology.Algebra.Group.ClosedSubgroup",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Topology.Algebra.Module.Compact",
"Mathlib.Topology.Algebra.OpenSubgroup",
"Mathlib.Topology.Algebra.Ring.Ideal"
] |
Mathlib/Topology/Algebra/Ring/Compact.lean
|
IsDiscreteValuationRing.isOpen_iff
| null |
protected Ideal.closure (I : Ideal R) : Ideal R :=
{
AddSubmonoid.topologicalClosure
I.toAddSubmonoid with
carrier := closure I
smul_mem' := fun c _ hx => map_mem_closure (mulLeft_continuous _) hx fun _ => I.mul_mem_left c }
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] |
Mathlib/Topology/Algebra/Ring/Ideal.lean
|
Ideal.closure
|
The closure of an ideal in a topological ring as an ideal.
|
Ideal.coe_closure (I : Ideal R) : (I.closure : Set R) = closure I :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] |
Mathlib/Topology/Algebra/Ring/Ideal.lean
|
Ideal.coe_closure
| null |
Ideal.closure_eq_of_isClosed (I : Ideal R) (hI : IsClosed (I : Set R)) : I.closure = I :=
SetLike.ext' hI.closure_eq
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] |
Mathlib/Topology/Algebra/Ring/Ideal.lean
|
Ideal.closure_eq_of_isClosed
|
This is not `@[simp]` since otherwise it causes timeouts downstream as `simp` tries and fails to
generate an `IsClosed` instance.
https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/!4.234852.20heartbeats.20of.20the.20linter
|
topologicalRingQuotientTopology : TopologicalSpace (R ⧸ N) :=
instTopologicalSpaceQuotient
variable [IsTopologicalRing R]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] |
Mathlib/Topology/Algebra/Ring/Ideal.lean
|
topologicalRingQuotientTopology
| null |
QuotientRing.isOpenMap_coe : IsOpenMap (mk N) :=
QuotientAddGroup.isOpenMap_coe
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] |
Mathlib/Topology/Algebra/Ring/Ideal.lean
|
QuotientRing.isOpenMap_coe
| null |
QuotientRing.isOpenQuotientMap_mk : IsOpenQuotientMap (mk N) :=
QuotientAddGroup.isOpenQuotientMap_mk
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] |
Mathlib/Topology/Algebra/Ring/Ideal.lean
|
QuotientRing.isOpenQuotientMap_mk
| null |
QuotientRing.isQuotientMap_coe_coe : IsQuotientMap fun p : R × R => (mk N p.1, mk N p.2) :=
((isOpenQuotientMap_mk N).prodMap (isOpenQuotientMap_mk N)).isQuotientMap
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] |
Mathlib/Topology/Algebra/Ring/Ideal.lean
|
QuotientRing.isQuotientMap_coe_coe
| null |
topologicalRing_quotient : IsTopologicalRing (R ⧸ N) where
__ := QuotientAddGroup.instIsTopologicalAddGroup _
continuous_mul := (QuotientRing.isQuotientMap_coe_coe N).continuous_iff.2 <|
continuous_quot_mk.comp continuous_mul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.Group.Quotient",
"Mathlib.RingTheory.Ideal.Quotient.Defs"
] |
Mathlib/Topology/Algebra/Ring/Ideal.lean
|
topologicalRing_quotient
| null |
Real.uniformContinuous_add : UniformContinuous fun p : ℝ × ℝ => p.1 + p.2 :=
Metric.uniformContinuous_iff.2 fun _ε ε0 =>
let ⟨δ, δ0, Hδ⟩ := rat_add_continuous_lemma abs ε0
⟨δ, δ0, fun _ _ h =>
let ⟨h₁, h₂⟩ := max_lt_iff.1 h
Hδ h₁ h₂⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Operations",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Bornology.Real",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Topology.Order.Real",
"Mathlib.Topology.UniformSpace.Real"
] |
Mathlib/Topology/Algebra/Ring/Real.lean
|
Real.uniformContinuous_add
| null |
Real.uniformContinuous_neg : UniformContinuous (@Neg.neg ℝ _) :=
Metric.uniformContinuous_iff.2 fun ε ε0 =>
⟨_, ε0, fun _ _ h => by simpa only [abs_sub_comm, Real.dist_eq, neg_sub_neg] using h⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Operations",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Bornology.Real",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Topology.Order.Real",
"Mathlib.Topology.UniformSpace.Real"
] |
Mathlib/Topology/Algebra/Ring/Real.lean
|
Real.uniformContinuous_neg
| null |
Real.uniformContinuous_const_mul {x : ℝ} : UniformContinuous (x * ·) :=
uniformContinuous_of_continuousAt_zero (DistribMulAction.toAddMonoidHom ℝ x)
(continuous_const_smul x).continuousAt
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Operations",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Bornology.Real",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Topology.Order.Real",
"Mathlib.Topology.UniformSpace.Real"
] |
Mathlib/Topology/Algebra/Ring/Real.lean
|
Real.uniformContinuous_const_mul
| null |
isEmbedding_coe : IsEmbedding ((↑) : ℝ≥0 → ℝ≥0∞) :=
coe_strictMono.isEmbedding_of_ordConnected <| by rw [range_coe']; exact ordConnected_Iio
@[simp, norm_cast]
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Operations",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Bornology.Real",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Topology.Order.Real",
"Mathlib.Topology.UniformSpace.Real"
] |
Mathlib/Topology/Algebra/Ring/Real.lean
|
isEmbedding_coe
| null |
tendsto_coe {f : Filter α} {m : α → ℝ≥0} {a : ℝ≥0} :
Tendsto (fun a => (m a : ℝ≥0∞)) f (𝓝 ↑a) ↔ Tendsto m f (𝓝 a) :=
isEmbedding_coe.tendsto_nhds_iff.symm
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Operations",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Bornology.Real",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Topology.Order.Real",
"Mathlib.Topology.UniformSpace.Real"
] |
Mathlib/Topology/Algebra/Ring/Real.lean
|
tendsto_coe
| null |
isOpenEmbedding_coe : IsOpenEmbedding ((↑) : ℝ≥0 → ℝ≥0∞) :=
⟨isEmbedding_coe, by rw [range_coe']; exact isOpen_Iio⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Operations",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Bornology.Real",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Topology.Order.Real",
"Mathlib.Topology.UniformSpace.Real"
] |
Mathlib/Topology/Algebra/Ring/Real.lean
|
isOpenEmbedding_coe
| null |
nhds_coe_coe {r p : ℝ≥0} :
𝓝 ((r : ℝ≥0∞), (p : ℝ≥0∞)) = (𝓝 (r, p)).map fun p : ℝ≥0 × ℝ≥0 => (↑p.1, ↑p.2) :=
((isOpenEmbedding_coe.prodMap isOpenEmbedding_coe).map_nhds_eq (r, p)).symm
|
theorem
|
Topology
|
[
"Mathlib.Data.EReal.Operations",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Algebra.IsUniformGroup.Defs",
"Mathlib.Topology.Bornology.Real",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Topology.Order.Real",
"Mathlib.Topology.UniformSpace.Real"
] |
Mathlib/Topology/Algebra/Ring/Real.lean
|
nhds_coe_coe
| null |
@[to_additive]
instSMul : SMul M (SeparationQuotient X) where
smul c := Quotient.map' (c • ·) fun _ _ h ↦ h.const_smul c
@[to_additive (attr := simp)]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instSMul
| null |
mk_smul (c : M) (x : X) : mk (c • x) = c • mk x := rfl
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mk_smul
| null |
instContinuousConstSMul : ContinuousConstSMul M (SeparationQuotient X) where
continuous_const_smul c := isQuotientMap_mk.continuous_iff.2 <|
continuous_mk.comp <| continuous_const_smul c
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instContinuousConstSMul
| null |
instIsPretransitiveSMul [MulAction.IsPretransitive M X] :
MulAction.IsPretransitive M (SeparationQuotient X) where
exists_smul_eq := surjective_mk.forall₂.2 fun x y ↦
(MulAction.exists_smul_eq M x y).imp fun _ ↦ congr_arg mk
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instIsPretransitiveSMul
| null |
instIsCentralScalar [SMul Mᵐᵒᵖ X] [IsCentralScalar M X] :
IsCentralScalar M (SeparationQuotient X) where
op_smul_eq_smul a := surjective_mk.forall.2 (congr_arg mk <| op_smul_eq_smul a ·)
variable {N : Type*} [SMul N X]
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instIsCentralScalar
| null |
instSMulCommClass [ContinuousConstSMul N X] [SMulCommClass M N X] :
SMulCommClass M N (SeparationQuotient X) :=
surjective_mk.smulCommClass mk_smul mk_smul
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instSMulCommClass
| null |
instIsScalarTower [SMul M N] [ContinuousConstSMul N X] [IsScalarTower M N X] :
IsScalarTower M N (SeparationQuotient X) where
smul_assoc a b := surjective_mk.forall.2 fun x ↦ congr_arg mk <| smul_assoc a b x
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instIsScalarTower
| null |
instContinuousSMul {M X : Type*} [SMul M X] [TopologicalSpace M] [TopologicalSpace X]
[ContinuousSMul M X] : ContinuousSMul M (SeparationQuotient X) where
continuous_smul := by
rw [(IsOpenQuotientMap.id.prodMap isOpenQuotientMap_mk).isQuotientMap.continuous_iff]
exact continuous_mk.comp continuous_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instContinuousSMul
| null |
instSMulZeroClass {M X : Type*} [Zero X] [SMulZeroClass M X] [TopologicalSpace X]
[ContinuousConstSMul M X] : SMulZeroClass M (SeparationQuotient X) :=
ZeroHom.smulZeroClass ⟨mk, mk_zero⟩ mk_smul
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instSMulZeroClass
| null |
instMulAction {M X : Type*} [Monoid M] [MulAction M X] [TopologicalSpace X]
[ContinuousConstSMul M X] : MulAction M (SeparationQuotient X) :=
surjective_mk.mulAction mk mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instMulAction
| null |
@[to_additive]
instMul [Mul M] [ContinuousMul M] : Mul (SeparationQuotient M) where
mul := Quotient.map₂ (· * ·) fun _ _ h₁ _ _ h₂ ↦ Inseparable.mul h₁ h₂
@[to_additive (attr := simp)]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instMul
| null |
mk_mul [Mul M] [ContinuousMul M] (a b : M) : mk (a * b) = mk a * mk b := rfl
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mk_mul
| null |
instContinuousMul [Mul M] [ContinuousMul M] : ContinuousMul (SeparationQuotient M) where
continuous_mul := isQuotientMap_prodMap_mk.continuous_iff.2 <| continuous_mk.comp continuous_mul
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instContinuousMul
| null |
instCommMagma [CommMagma M] [ContinuousMul M] : CommMagma (SeparationQuotient M) :=
surjective_mk.commMagma mk mk_mul
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instCommMagma
| null |
instSemigroup [Semigroup M] [ContinuousMul M] : Semigroup (SeparationQuotient M) :=
surjective_mk.semigroup mk mk_mul
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instSemigroup
| null |
instCommSemigroup [CommSemigroup M] [ContinuousMul M] :
CommSemigroup (SeparationQuotient M) :=
surjective_mk.commSemigroup mk mk_mul
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instCommSemigroup
| null |
instMulOneClass [MulOneClass M] [ContinuousMul M] :
MulOneClass (SeparationQuotient M) :=
surjective_mk.mulOneClass mk mk_one mk_mul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instMulOneClass
| null |
@[to_additive (attr := simps) /-- `SeparationQuotient.mk` as an `AddMonoidHom`. -/]
mkMonoidHom [MulOneClass M] [ContinuousMul M] : M →* SeparationQuotient M where
toFun := mk
map_mul' := mk_mul
map_one' := mk_one
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mkMonoidHom
|
`SeparationQuotient.mk` as a `MonoidHom`.
|
@[to_additive existing instNSmul]
instPow [Monoid M] [ContinuousMul M] : Pow (SeparationQuotient M) ℕ where
pow x n := Quotient.map' (s₁ := inseparableSetoid M) (· ^ n) (fun _ _ h ↦ Inseparable.pow h n) x
@[to_additive, simp] -- `mk_nsmul` is not a `simp` lemma because we have `mk_smul`
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instPow
| null |
mk_pow [Monoid M] [ContinuousMul M] (x : M) (n : ℕ) : mk (x ^ n) = (mk x) ^ n := rfl
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mk_pow
| null |
instMonoid [Monoid M] [ContinuousMul M] : Monoid (SeparationQuotient M) :=
surjective_mk.monoid mk mk_one mk_mul mk_pow
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instMonoid
| null |
instCommMonoid [CommMonoid M] [ContinuousMul M] : CommMonoid (SeparationQuotient M) :=
surjective_mk.commMonoid mk mk_one mk_mul mk_pow
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instCommMonoid
| null |
@[to_additive]
instInv [Inv G] [ContinuousInv G] : Inv (SeparationQuotient G) where
inv := Quotient.map' (·⁻¹) fun _ _ ↦ Inseparable.inv
@[to_additive (attr := simp)]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instInv
| null |
mk_inv [Inv G] [ContinuousInv G] (x : G) : mk x⁻¹ = (mk x)⁻¹ := rfl
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mk_inv
| null |
instContinuousInv [Inv G] [ContinuousInv G] : ContinuousInv (SeparationQuotient G) where
continuous_inv := isQuotientMap_mk.continuous_iff.2 <| continuous_mk.comp continuous_inv
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instContinuousInv
| null |
instInvolutiveInv [InvolutiveInv G] [ContinuousInv G] :
InvolutiveInv (SeparationQuotient G) :=
surjective_mk.involutiveInv mk mk_inv
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instInvolutiveInv
| null |
instInvOneClass [InvOneClass G] [ContinuousInv G] :
InvOneClass (SeparationQuotient G) where
inv_one := congr_arg mk inv_one
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instInvOneClass
| null |
instDiv [Div G] [ContinuousDiv G] : Div (SeparationQuotient G) where
div := Quotient.map₂ (· / ·) fun _ _ h₁ _ _ h₂ ↦ (Inseparable.prod h₁ h₂).map continuous_div'
@[to_additive (attr := simp)]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instDiv
| null |
mk_div [Div G] [ContinuousDiv G] (x y : G) : mk (x / y) = mk x / mk y := rfl
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mk_div
| null |
instContinuousDiv [Div G] [ContinuousDiv G] : ContinuousDiv (SeparationQuotient G) where
continuous_div' := isQuotientMap_prodMap_mk.continuous_iff.2 <| continuous_mk.comp continuous_div'
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instContinuousDiv
| null |
instZSMul [AddGroup G] [IsTopologicalAddGroup G] : SMul ℤ (SeparationQuotient G) :=
inferInstance
@[to_additive existing]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instZSMul
| null |
instZPow [Group G] [IsTopologicalGroup G] : Pow (SeparationQuotient G) ℤ where
pow x n := Quotient.map' (s₁ := inseparableSetoid G) (· ^ n) (fun _ _ h ↦ Inseparable.zpow h n) x
@[to_additive, simp] -- `mk_zsmul` is not a `simp` lemma because we have `mk_smul`
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instZPow
| null |
mk_zpow [Group G] [IsTopologicalGroup G] (x : G) (n : ℤ) : mk (x ^ n) = (mk x) ^ n := rfl
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mk_zpow
| null |
instGroup [Group G] [IsTopologicalGroup G] : Group (SeparationQuotient G) :=
surjective_mk.group mk mk_one mk_mul mk_inv mk_div mk_pow mk_zpow
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instGroup
| null |
instCommGroup [CommGroup G] [IsTopologicalGroup G] : CommGroup (SeparationQuotient G) :=
surjective_mk.commGroup mk mk_one mk_mul mk_inv mk_div mk_pow mk_zpow
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instCommGroup
| null |
instIsTopologicalGroup [Group G] [IsTopologicalGroup G] :
IsTopologicalGroup (SeparationQuotient G) where
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instIsTopologicalGroup
| null |
@[to_additive]
instIsUniformGroup {G : Type*} [Group G] [UniformSpace G] [IsUniformGroup G] :
IsUniformGroup (SeparationQuotient G) where
uniformContinuous_div := by
rw [uniformContinuous_dom₂]
exact uniformContinuous_mk.comp uniformContinuous_div
@[deprecated (since := "2025-03-31")] alias
instUniformAddGroup := SeparationQuotient.instIsUniformAddGroup
@[to_additive existing, deprecated (since := "2025-03-31")] alias
instUniformGroup := SeparationQuotient.instIsUniformGroup
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instIsUniformGroup
| null |
instMulZeroClass [MulZeroClass M₀] [ContinuousMul M₀] :
MulZeroClass (SeparationQuotient M₀) :=
surjective_mk.mulZeroClass mk mk_zero mk_mul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instMulZeroClass
| null |
instSemigroupWithZero [SemigroupWithZero M₀] [ContinuousMul M₀] :
SemigroupWithZero (SeparationQuotient M₀) :=
surjective_mk.semigroupWithZero mk mk_zero mk_mul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instSemigroupWithZero
| null |
instMulZeroOneClass [MulZeroOneClass M₀] [ContinuousMul M₀] :
MulZeroOneClass (SeparationQuotient M₀) :=
surjective_mk.mulZeroOneClass mk mk_zero mk_one mk_mul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instMulZeroOneClass
| null |
instMonoidWithZero [MonoidWithZero M₀] [ContinuousMul M₀] :
MonoidWithZero (SeparationQuotient M₀) :=
surjective_mk.monoidWithZero mk mk_zero mk_one mk_mul mk_pow
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instMonoidWithZero
| null |
instCommMonoidWithZero [CommMonoidWithZero M₀] [ContinuousMul M₀] :
CommMonoidWithZero (SeparationQuotient M₀) :=
surjective_mk.commMonoidWithZero mk mk_zero mk_one mk_mul mk_pow
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instCommMonoidWithZero
| null |
instDistrib [Distrib R] [ContinuousMul R] [ContinuousAdd R] :
Distrib (SeparationQuotient R) :=
surjective_mk.distrib mk mk_add mk_mul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instDistrib
| null |
instLeftDistribClass [Mul R] [Add R] [LeftDistribClass R]
[ContinuousMul R] [ContinuousAdd R] :
LeftDistribClass (SeparationQuotient R) :=
surjective_mk.leftDistribClass mk mk_add mk_mul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instLeftDistribClass
| null |
instRightDistribClass [Mul R] [Add R] [RightDistribClass R]
[ContinuousMul R] [ContinuousAdd R] :
RightDistribClass (SeparationQuotient R) :=
surjective_mk.rightDistribClass mk mk_add mk_mul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instRightDistribClass
| null |
instNonUnitalnonAssocSemiring [NonUnitalNonAssocSemiring R]
[IsTopologicalSemiring R] : NonUnitalNonAssocSemiring (SeparationQuotient R) :=
surjective_mk.nonUnitalNonAssocSemiring mk mk_zero mk_add mk_mul mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNonUnitalnonAssocSemiring
| null |
instNonUnitalSemiring [NonUnitalSemiring R] [IsTopologicalSemiring R] :
NonUnitalSemiring (SeparationQuotient R) :=
surjective_mk.nonUnitalSemiring mk mk_zero mk_add mk_mul mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNonUnitalSemiring
| null |
instNatCast [NatCast R] : NatCast (SeparationQuotient R) where
natCast n := mk n
@[simp, norm_cast]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNatCast
| null |
mk_natCast [NatCast R] (n : ℕ) : mk (n : R) = n := rfl
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mk_natCast
| null |
mk_ofNat [NatCast R] (n : ℕ) [n.AtLeastTwo] :
mk (ofNat(n) : R) = OfNat.ofNat n :=
rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mk_ofNat
| null |
instIntCast [IntCast R] : IntCast (SeparationQuotient R) where
intCast n := mk n
@[simp, norm_cast]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instIntCast
| null |
mk_intCast [IntCast R] (n : ℤ) : mk (n : R) = n := rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mk_intCast
| null |
instNonAssocSemiring [NonAssocSemiring R] [IsTopologicalSemiring R] :
NonAssocSemiring (SeparationQuotient R) :=
surjective_mk.nonAssocSemiring mk mk_zero mk_one mk_add mk_mul mk_smul mk_natCast
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNonAssocSemiring
| null |
instNonUnitalNonAssocRing [NonUnitalNonAssocRing R] [IsTopologicalRing R] :
NonUnitalNonAssocRing (SeparationQuotient R) :=
surjective_mk.nonUnitalNonAssocRing mk mk_zero mk_add mk_mul mk_neg mk_sub mk_smul mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNonUnitalNonAssocRing
| null |
instNonUnitalRing [NonUnitalRing R] [IsTopologicalRing R] :
NonUnitalRing (SeparationQuotient R) :=
surjective_mk.nonUnitalRing mk mk_zero mk_add mk_mul mk_neg mk_sub mk_smul mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNonUnitalRing
| null |
instNonAssocRing [NonAssocRing R] [IsTopologicalRing R] :
NonAssocRing (SeparationQuotient R) :=
surjective_mk.nonAssocRing mk mk_zero mk_one mk_add mk_mul mk_neg mk_sub mk_smul mk_smul
mk_natCast mk_intCast
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNonAssocRing
| null |
instSemiring [Semiring R] [IsTopologicalSemiring R] :
Semiring (SeparationQuotient R) :=
surjective_mk.semiring mk mk_zero mk_one mk_add mk_mul mk_smul mk_pow mk_natCast
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instSemiring
| null |
instRing [Ring R] [IsTopologicalRing R] :
Ring (SeparationQuotient R) :=
surjective_mk.ring mk mk_zero mk_one mk_add mk_mul mk_neg mk_sub mk_smul mk_smul mk_pow
mk_natCast mk_intCast
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instRing
| null |
instNonUnitalNonAssocCommSemiring [NonUnitalNonAssocCommSemiring R]
[IsTopologicalSemiring R] :
NonUnitalNonAssocCommSemiring (SeparationQuotient R) :=
surjective_mk.nonUnitalNonAssocCommSemiring mk mk_zero mk_add mk_mul mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNonUnitalNonAssocCommSemiring
| null |
instNonUnitalCommSemiring [NonUnitalCommSemiring R] [IsTopologicalSemiring R] :
NonUnitalCommSemiring (SeparationQuotient R) :=
surjective_mk.nonUnitalCommSemiring mk mk_zero mk_add mk_mul mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNonUnitalCommSemiring
| null |
instCommSemiring [CommSemiring R] [IsTopologicalSemiring R] :
CommSemiring (SeparationQuotient R) :=
surjective_mk.commSemiring mk mk_zero mk_one mk_add mk_mul mk_smul mk_pow mk_natCast
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instCommSemiring
| null |
instHasDistribNeg [Mul R] [HasDistribNeg R] [ContinuousMul R] [ContinuousNeg R] :
HasDistribNeg (SeparationQuotient R) :=
surjective_mk.hasDistribNeg mk mk_neg mk_mul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instHasDistribNeg
| null |
instNonUnitalNonAssocCommRing [NonUnitalNonAssocCommRing R] [IsTopologicalRing R] :
NonUnitalNonAssocCommRing (SeparationQuotient R) :=
surjective_mk.nonUnitalNonAssocCommRing mk mk_zero mk_add mk_mul mk_neg mk_sub mk_smul mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNonUnitalNonAssocCommRing
| null |
instNonUnitalCommRing [NonUnitalCommRing R] [IsTopologicalRing R] :
NonUnitalCommRing (SeparationQuotient R) :=
surjective_mk.nonUnitalCommRing mk mk_zero mk_add mk_mul mk_neg mk_sub mk_smul mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instNonUnitalCommRing
| null |
instCommRing [CommRing R] [IsTopologicalRing R] :
CommRing (SeparationQuotient R) :=
surjective_mk.commRing mk mk_zero mk_one mk_add mk_mul mk_neg mk_sub mk_smul mk_smul mk_pow
mk_natCast mk_intCast
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instCommRing
| null |
@[simps]
mkRingHom [NonAssocSemiring R] [IsTopologicalSemiring R] : R →+* SeparationQuotient R where
toFun := mk
map_one' := mk_one; map_zero' := mk_zero; map_add' := mk_add; map_mul' := mk_mul
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mkRingHom
|
`SeparationQuotient.mk` as a `RingHom`.
|
instDistribSMul [AddZeroClass A] [DistribSMul M A]
[ContinuousAdd A] [ContinuousConstSMul M A] :
DistribSMul M (SeparationQuotient A) :=
surjective_mk.distribSMul mkAddMonoidHom mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instDistribSMul
| null |
instDistribMulAction [Monoid M] [AddMonoid A] [DistribMulAction M A]
[ContinuousAdd A] [ContinuousConstSMul M A] :
DistribMulAction M (SeparationQuotient A) :=
surjective_mk.distribMulAction mkAddMonoidHom mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instDistribMulAction
| null |
instMulDistribMulAction [Monoid M] [Monoid A] [MulDistribMulAction M A]
[ContinuousMul A] [ContinuousConstSMul M A] :
MulDistribMulAction M (SeparationQuotient A) :=
surjective_mk.mulDistribMulAction mkMonoidHom mk_smul
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instMulDistribMulAction
| null |
instModule : Module R (SeparationQuotient M) :=
surjective_mk.module R mkAddMonoidHom mk_smul
variable (R M)
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
instModule
| null |
@[simps]
mkCLM : M →L[R] SeparationQuotient M where
toFun := mk
map_add' := mk_add
map_smul' := mk_smul
variable {R M}
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
mkCLM
|
`SeparationQuotient.mk` as a continuous linear map.
|
@[simps]
noncomputable liftCLM {σ : R →+* S} (f : M →SL[σ] N) (hf : ∀ x y, Inseparable x y → f x = f y) :
SeparationQuotient M →SL[σ] N where
toFun := SeparationQuotient.lift f hf
map_add' := Quotient.ind₂ <| map_add f
map_smul' {r} := Quotient.ind <| map_smulₛₗ f r
@[simp]
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Module.LinearMap"
] |
Mathlib/Topology/Algebra/SeparationQuotient/Basic.lean
|
liftCLM
|
The lift (as a continuous linear map) of `f` with `f x = f y` for `Inseparable x y`.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.