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 ⌀ |
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instNonUnitalAlgHomClass : NonUnitalAlgHomClass (characterSpace 𝕜 A) 𝕜 A 𝕜 :=
{ CharacterSpace.instContinuousLinearMapClass with
map_smulₛₗ := fun φ => map_smul φ
map_zero := fun φ => map_zero φ
map_mul := fun φ => φ.prop.2 } | instance | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | instNonUnitalAlgHomClass | Elements of the character space are non-unital algebra homomorphisms. |
toNonUnitalAlgHom (φ : characterSpace 𝕜 A) : A →ₙₐ[𝕜] 𝕜 where
toFun := (φ : A → 𝕜)
map_mul' := map_mul φ
map_smul' := map_smul φ
map_zero' := map_zero φ
map_add' := map_add φ
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | toNonUnitalAlgHom | An element of the character space, as a non-unital algebra homomorphism. |
coe_toNonUnitalAlgHom (φ : characterSpace 𝕜 A) : ⇑(toNonUnitalAlgHom φ) = φ :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | coe_toNonUnitalAlgHom | null |
instIsEmpty [Subsingleton A] : IsEmpty (characterSpace 𝕜 A) :=
⟨fun φ => φ.prop.1 <|
ContinuousLinearMap.ext fun x => by
rw [show x = 0 from Subsingleton.elim x 0, map_zero, map_zero] ⟩
variable (𝕜 A) | instance | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | instIsEmpty | null |
union_zero :
characterSpace 𝕜 A ∪ {0} = {φ : WeakDual 𝕜 A | ∀ x y : A, φ (x * y) = φ x * φ y} :=
le_antisymm (by
rintro φ (hφ | rfl)
· exact hφ.2
· exact fun _ _ => by exact (zero_mul (0 : 𝕜)).symm)
fun φ hφ => Or.elim (em <| φ = 0) Or.inr fun h₀ => Or.inl ⟨h₀, hφ⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | union_zero | null |
union_zero_isClosed [T2Space 𝕜] [ContinuousMul 𝕜] :
IsClosed (characterSpace 𝕜 A ∪ {0}) := by
simp only [union_zero, Set.setOf_forall]
exact
isClosed_iInter fun x =>
isClosed_iInter fun y =>
isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _) | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | union_zero_isClosed | The `characterSpace 𝕜 A` along with `0` is always a closed set in `WeakDual 𝕜 A`. |
instAlgHomClass : AlgHomClass (characterSpace 𝕜 A) 𝕜 A 𝕜 :=
haveI map_one' : ∀ φ : characterSpace 𝕜 A, φ 1 = 1 := fun φ => by
have h₁ : φ 1 * (1 - φ 1) = 0 := by rw [mul_sub, sub_eq_zero, mul_one, ← map_mul φ, one_mul]
rcases mul_eq_zero.mp h₁ with (h₂ | h₂)
· have : ∀ a, φ (a * 1) = 0 := fun a => by simp only [map_mul φ, h₂, mul_zero]
exact False.elim (φ.prop.1 <| ContinuousLinearMap.ext <| by simpa only [mul_one] using this)
· exact (sub_eq_zero.mp h₂).symm
{ CharacterSpace.instNonUnitalAlgHomClass with
map_one := map_one'
commutes := fun φ r => by
rw [Algebra.algebraMap_eq_smul_one, Algebra.algebraMap_self, RingHom.id_apply]
rw [map_smul, Algebra.id.smul_eq_mul, map_one' φ, mul_one] } | instance | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | instAlgHomClass | In a unital algebra, elements of the character space are algebra homomorphisms. |
@[simps]
toAlgHom (φ : characterSpace 𝕜 A) : A →ₐ[𝕜] 𝕜 :=
{ toNonUnitalAlgHom φ with
map_one' := map_one φ
commutes' := AlgHomClass.commutes φ } | def | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | toAlgHom | An element of the character space of a unital algebra, as an algebra homomorphism. |
eq_set_map_one_map_mul [Nontrivial 𝕜] :
characterSpace 𝕜 A = {φ : WeakDual 𝕜 A | φ 1 = 1 ∧ ∀ x y : A, φ (x * y) = φ x * φ y} := by
ext φ
refine ⟨?_, ?_⟩
· rintro hφ
lift φ to characterSpace 𝕜 A using hφ
exact ⟨map_one φ, map_mul φ⟩
· rintro ⟨hφ₁, hφ₂⟩
refine ⟨?_, hφ₂⟩
rintro rfl
exact zero_ne_one hφ₁ | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | eq_set_map_one_map_mul | null |
protected isClosed [Nontrivial 𝕜] [T2Space 𝕜] [ContinuousMul 𝕜] :
IsClosed (characterSpace 𝕜 A) := by
rw [eq_set_map_one_map_mul, Set.setOf_and]
refine IsClosed.inter (isClosed_eq (eval_continuous _) continuous_const) ?_
simpa only [(union_zero 𝕜 A).symm] using union_zero_isClosed _ _ | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | isClosed | under suitable mild assumptions on `𝕜`, the character space is a closed set in
`WeakDual 𝕜 A`. |
apply_mem_spectrum [Nontrivial 𝕜] (φ : characterSpace 𝕜 A) (a : A) : φ a ∈ spectrum 𝕜 a :=
AlgHom.apply_mem_spectrum φ a | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | apply_mem_spectrum | null |
ext_ker {φ ψ : characterSpace 𝕜 A} (h : RingHom.ker φ = RingHom.ker ψ) : φ = ψ := by
ext x
have : x - algebraMap 𝕜 A (ψ x) ∈ RingHom.ker φ := by
simpa only [h, RingHom.mem_ker, map_sub, AlgHomClass.commutes] using sub_self (ψ x)
rwa [RingHom.mem_ker, map_sub, AlgHomClass.commutes, sub_eq_zero] at this | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | ext_ker | null |
ker_isMaximal (φ : characterSpace 𝕜 A) : (RingHom.ker φ).IsMaximal :=
RingHom.ker_isMaximal_of_surjective φ fun z ↦ ⟨algebraMap 𝕜 A z, by simp [AlgHomClass.commutes]⟩ | instance | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | ker_isMaximal | The `RingHom.ker` of `φ : characterSpace 𝕜 A` is maximal. |
@[simps]
gelfandTransform : A →ₐ[𝕜] C(characterSpace 𝕜 A, 𝕜) where
toFun a :=
{ toFun := fun φ => φ a
continuous_toFun := (eval_continuous a).comp continuous_induced_dom }
map_one' := by ext a; simp only [coe_mk, coe_one, Pi.one_apply, map_one a]
map_mul' a b := by ext; simp only [map_mul, coe_mk, coe_mul, Pi.mul_apply]
map_zero' := by ext; simp only [map_zero, coe_mk, coe_zero, Pi.zero_apply]
map_add' a b := by ext; simp only [map_add, coe_mk, coe_add, Pi.add_apply]
commutes' k := by ext; simp [AlgHomClass.commutes] | def | Topology | [
"Mathlib.Topology.Algebra.Module.WeakDual",
"Mathlib.Algebra.Algebra.Spectrum.Basic",
"Mathlib.Topology.ContinuousMap.Algebra",
"Mathlib.Data.Set.Lattice"
] | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | gelfandTransform | The **Gelfand transform** is an algebra homomorphism (over `𝕜`) from a topological `𝕜`-algebra
`A` into the `𝕜`-algebra of continuous `𝕜`-valued functions on the `characterSpace 𝕜 A`.
The character space itself consists of all algebra homomorphisms from `A` to `𝕜`. |
@[ext]
ClosedSubmodule extends Submodule R M, Closeds M where | structure | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | ClosedSubmodule | The type of closed submodules of a topological module. |
@[simps!]
comap (f : M →L[R] N) (s : ClosedSubmodule R N) : ClosedSubmodule R M where
toSubmodule := .comap f s
isClosed' := by simpa using s.isClosed.preimage f.continuous
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | comap | Reinterpret a closed submodule as a submodule. -/
add_decl_doc toSubmodule
/-- Reinterpret a closed submodule as a closed set. -/
add_decl_doc toCloseds
lemma toSubmodule_injective : Injective (toSubmodule : ClosedSubmodule R M → Submodule R M) :=
fun s t h ↦ by cases s; congr!
instance : SetLike (ClosedSubmodule R M) M where
coe s := s.1
coe_injective' _ _ h := toSubmodule_injective <| SetLike.coe_injective h
lemma toCloseds_injective : Injective (toCloseds : ClosedSubmodule R M → Closeds M) :=
fun _s _t h ↦ SetLike.coe_injective congr(($h : Set M))
instance : AddSubmonoidClass (ClosedSubmodule R M) M where
zero_mem s := s.zero_mem
add_mem {s} := s.add_mem
instance : SMulMemClass (ClosedSubmodule R M) R M where
smul_mem {s} r := s.smul_mem r
instance : Coe (ClosedSubmodule R M) (Submodule R M) where
coe := toSubmodule
@[simp] lemma carrier_eq_coe (s : ClosedSubmodule R M) : s.carrier = s := rfl
@[simp] lemma mem_mk {s : Submodule R M} {hs} : x ∈ mk s hs ↔ x ∈ s := .rfl
@[simp, norm_cast]
lemma coe_toSubmodule (s : ClosedSubmodule R M) : (s.toSubmodule : Set M) = s := rfl
@[simp]
lemma coe_toCloseds (s : ClosedSubmodule R M) : (s.toCloseds : Set M) = s := rfl
lemma isClosed (s : ClosedSubmodule R M) : IsClosed (s : Set M) := s.isClosed'
initialize_simps_projections ClosedSubmodule (carrier → coe, as_prefix coe)
instance : CanLift (Submodule R M) (ClosedSubmodule R M) toSubmodule (IsClosed (X := M) ·) where
prf s hs := ⟨⟨s, hs⟩, rfl⟩
@[simp, norm_cast] lemma toSubmodule_le_toSubmodule {s t : ClosedSubmodule R M} :
s.toSubmodule ≤ t.toSubmodule ↔ s ≤ t := .rfl
/-- The preimage of a closed submodule under a continuous linear map as a closed submodule. |
mem_comap {f : M →L[R] N} {s : ClosedSubmodule R N} {x : M} : x ∈ s.comap f ↔ f x ∈ s := .rfl
@[simp] lemma toSubmodule_comap (f : M →L[R] N) (s : ClosedSubmodule R N) :
(s.comap f).toSubmodule = s.toSubmodule.comap f := rfl
@[simp] lemma comap_id (s : ClosedSubmodule R M) : s.comap (.id _ _) = s := rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | mem_comap | null |
comap_comap (g : N →L[R] O) (f : M →L[R] N) (s : ClosedSubmodule R O) :
(s.comap g).comap f = s.comap (g.comp f) := rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | comap_comap | null |
instInf : Min (ClosedSubmodule R M) where
min s t := ⟨s ⊓ t, s.isClosed.inter t.isClosed⟩ | instance | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | instInf | null |
instInfSet : InfSet (ClosedSubmodule R M) where
sInf S := ⟨⨅ s ∈ S, s, by simpa using isClosed_biInter fun x hx ↦ x.isClosed⟩
@[simp, norm_cast] | instance | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | instInfSet | null |
toSubmodule_sInf (S : Set (ClosedSubmodule R M)) :
toSubmodule (sInf S) = ⨅ s ∈ S, s.toSubmodule := rfl
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | toSubmodule_sInf | null |
toSubmodule_iInf (f : ι → ClosedSubmodule R M) :
toSubmodule (⨅ i, f i) = ⨅ i, (f i).toSubmodule := by rw [iInf, toSubmodule_sInf, iInf_range]
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | toSubmodule_iInf | null |
coe_sInf (S : Set (ClosedSubmodule R M)) : ↑(sInf S) = ⨅ s ∈ S, (s : Set M) := by
simp [← coe_toSubmodule]
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | coe_sInf | null |
coe_iInf (f : ι → ClosedSubmodule R M) : ↑(⨅ i, f i) = ⨅ i, (f i : Set M) := by
simp [← coe_toSubmodule]
@[simp] lemma mem_sInf {S : Set (ClosedSubmodule R M)} : x ∈ sInf S ↔ ∀ s ∈ S, x ∈ s := by
simp [← SetLike.mem_coe]
@[simp] lemma mem_iInf {f : ι → ClosedSubmodule R M} : x ∈ ⨅ i, f i ↔ ∀ i, x ∈ f i := by
simp [← SetLike.mem_coe] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | coe_iInf | null |
instSemilatticeInf : SemilatticeInf (ClosedSubmodule R M) :=
toSubmodule_injective.semilatticeInf _ fun _ _ ↦ rfl
@[simp, norm_cast] | instance | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | instSemilatticeInf | null |
toSubmodule_inf (s t : ClosedSubmodule R M) :
toSubmodule (s ⊓ t) = s.toSubmodule ⊓ t.toSubmodule := rfl
@[simp, norm_cast] lemma coe_inf (s t : ClosedSubmodule R M) : ↑(s ⊓ t) = (s ⊓ t : Set M) := rfl
@[simp] lemma mem_inf : x ∈ s ⊓ t ↔ x ∈ s ∧ x ∈ t := .rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | toSubmodule_inf | null |
@[simp, norm_cast] toSubmodule_top : toSubmodule (⊤ : ClosedSubmodule R M) = ⊤ := rfl
@[simp, norm_cast] lemma coe_top : ((⊤ : ClosedSubmodule R M) : Set M) = .univ := rfl
@[simp] lemma mem_top : x ∈ (⊤ : ClosedSubmodule R M) := trivial | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | toSubmodule_top | null |
instOrderBot : OrderBot (ClosedSubmodule R M) where
bot := ⟨⊥, isClosed_singleton⟩
bot_le s := bot_le (a := s.toSubmodule)
@[simp, norm_cast] lemma toSubmodule_bot : toSubmodule (⊥ : ClosedSubmodule R M) = ⊥ := rfl
@[simp, norm_cast] lemma coe_bot : ((⊥ : ClosedSubmodule R M) : Set M) = {0} := rfl
@[simp] lemma mem_bot : x ∈ (⊥ : ClosedSubmodule R M) ↔ x = 0 := .rfl | instance | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | instOrderBot | null |
@[simps!]
protected closure (s : Submodule R M) : ClosedSubmodule R M where
toSubmodule := s.topologicalClosure
isClosed' := isClosed_closure
@[simp] lemma closure_le {s : Submodule R M} {t : ClosedSubmodule R M} : s.closure ≤ t ↔ s ≤ t :=
t.isClosed.closure_subset_iff
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | closure | The closure of a submodule as a closed submodule. |
mem_closure_iff {x : M} {s : Submodule R M} : x ∈ s.closure ↔ x ∈ s.topologicalClosure :=
Iff.rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | mem_closure_iff | null |
@[simp]
closure_toSubmodule_eq {s : ClosedSubmodule R N} : s.toSubmodule.closure = s := by
ext x
simp [closure_eq_iff_isClosed.mpr (ClosedSubmodule.isClosed s)] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | closure_toSubmodule_eq | null |
map (f : M →L[R] N) (s : ClosedSubmodule R M) : ClosedSubmodule R N :=
(s.toSubmodule.map f).closure
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | map | The closure of the image of a closed submodule under a continuous linear map is a closed
submodule.
`ClosedSubmodule.map f` is left-adjoint to `ClosedSubmodule.comap f`.
See `ClosedSubmodule.gc_map_comap`. |
map_id [ContinuousAdd M] [ContinuousConstSMul R M] (s : ClosedSubmodule R M) :
s.map (.id _ _) = s := SetLike.coe_injective <| by simpa [map] using s.isClosed.closure_eq | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | map_id | null |
map_le_iff_le_comap {s : ClosedSubmodule R M} {t : ClosedSubmodule R N} :
map f s ≤ t ↔ s ≤ comap f t := by
simp [map, Submodule.map_le_iff_le_comap]; simp [← toSubmodule_le_toSubmodule] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | map_le_iff_le_comap | null |
gc_map_comap : GaloisConnection (map f) (comap f) := fun _ _ ↦ map_le_iff_le_comap
variable {s t : ClosedSubmodule R N} {x : N} | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | gc_map_comap | null |
@[simp]
toSubmodule_sup :
toSubmodule (s ⊔ t) = (s.toSubmodule ⊔ t.toSubmodule).closure := rfl
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | toSubmodule_sup | null |
coe_sup :
↑(s ⊔ t) = closure (s.toSubmodule ⊔ t.toSubmodule).carrier := by
simp only [← coe_toSubmodule, toSubmodule_sup]
simp only [coe_toSubmodule, Submodule.coe_closure, Submodule.carrier_eq_coe]
@[simp] lemma mem_sup :
x ∈ s ⊔ t ↔ x ∈ closure (s.toSubmodule ⊔ t.toSubmodule).carrier := Iff.rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | coe_sup | null |
@[simp]
toSubmodule_sSup (S : Set (ClosedSubmodule R N)) :
toSubmodule (sSup S) = (⨆ s ∈ S, s.toSubmodule).closure := rfl
@[simp] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | toSubmodule_sSup | null |
toSubmodule_iSup (f : ι → ClosedSubmodule R N) :
toSubmodule (⨆ i, f i) = (⨆ i, (f i).toSubmodule).closure := by
rw [iSup, toSubmodule_sSup, iSup_range]
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | toSubmodule_iSup | null |
coe_sSup (S : Set (ClosedSubmodule R N)) :
↑(sSup S) = closure (⨆ s ∈ S, s.toSubmodule).carrier := by
simp only [← coe_toSubmodule, toSubmodule_sSup]
simp only [coe_toSubmodule, Submodule.coe_closure, Submodule.carrier_eq_coe]
@[simp, norm_cast] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | coe_sSup | null |
coe_iSup (f : ι → ClosedSubmodule R N) :
↑(⨆ i, f i) = closure (⨆ i, (f i).toSubmodule).carrier := by
simp only [← coe_toSubmodule, toSubmodule_iSup, Submodule.carrier_eq_coe]
rfl
@[simp] lemma mem_sSup {S : Set (ClosedSubmodule R N)} :
x ∈ sSup S ↔ x ∈ closure (⨆ s ∈ S, s.toSubmodule).carrier := Iff.rfl
@[simp] lemma mem_iSup {f : ι → ClosedSubmodule R N} :
x ∈ ⨆ i, f i ↔ x ∈ closure (⨆ i, (f i).toSubmodule).carrier := by
simp [← SetLike.mem_coe] | lemma | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Sets.Closeds"
] | Mathlib/Topology/Algebra/Module/ClosedSubmodule.lean | coe_iSup | null |
Submodule.isCompact_of_fg [CompactSpace R] {N : Submodule R M} (hN : N.FG) :
IsCompact (X := M) N := by
obtain ⟨s, hs⟩ := hN
have : LinearMap.range (Fintype.linearCombination R (α := s) Subtype.val) = N := by
simp [hs]
rw [← this]
refine isCompact_range ?_
simp only [Fintype.linearCombination, Finset.univ_eq_attach, LinearMap.coe_mk,
AddHom.coe_mk]
fun_prop | lemma | Topology | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.Finiteness.Defs",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.RingTheory.Noetherian.Defs"
] | Mathlib/Topology/Algebra/Module/Compact.lean | Submodule.isCompact_of_fg | null |
Ideal.isCompact_of_fg [IsTopologicalSemiring R] [CompactSpace R]
{I : Ideal R} (hI : I.FG) : IsCompact (X := R) I :=
Submodule.isCompact_of_fg hI
variable (R M) in | lemma | Topology | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.Finiteness.Defs",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.RingTheory.Noetherian.Defs"
] | Mathlib/Topology/Algebra/Module/Compact.lean | Ideal.isCompact_of_fg | null |
Module.Finite.compactSpace [CompactSpace R] [Module.Finite R M] : CompactSpace M :=
⟨Submodule.isCompact_of_fg (Module.Finite.fg_top (R := R))⟩ | lemma | Topology | [
"Mathlib.LinearAlgebra.Finsupp.LinearCombination",
"Mathlib.RingTheory.Finiteness.Defs",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.RingTheory.Noetherian.Defs"
] | Mathlib/Topology/Algebra/Module/Compact.lean | Module.Finite.compactSpace | null |
noncomputable det {R : Type*} [CommRing R] {M : Type*} [TopologicalSpace M] [AddCommGroup M]
[Module R M] (A : M →L[R] M) : R :=
LinearMap.det (A : M →ₗ[R] M) | abbrev | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.LinearAlgebra.Determinant"
] | Mathlib/Topology/Algebra/Module/Determinant.lean | det | The determinant of a continuous linear map, mainly as a convenience device to be able to
write `A.det` instead of `(A : M →ₗ[R] M).det`. |
det_pi {ι R M : Type*} [Fintype ι] [CommRing R] [AddCommGroup M]
[TopologicalSpace M] [Module R M] [Module.Free R M] [Module.Finite R M]
(f : ι → M →L[R] M) :
(pi (fun i ↦ (f i).comp (proj i))).det = ∏ i, (f i).det :=
LinearMap.det_pi _ | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.LinearAlgebra.Determinant"
] | Mathlib/Topology/Algebra/Module/Determinant.lean | det_pi | null |
det_one_smulRight {𝕜 : Type*} [CommRing 𝕜] [TopologicalSpace 𝕜] [ContinuousMul 𝕜] (v : 𝕜) :
((1 : 𝕜 →L[𝕜] 𝕜).smulRight v).det = v := by
simp | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.LinearAlgebra.Determinant"
] | Mathlib/Topology/Algebra/Module/Determinant.lean | det_one_smulRight | null |
@[simp]
det_coe_symm {R : Type*} [Field R] {M : Type*} [TopologicalSpace M] [AddCommGroup M]
[Module R M] (A : M ≃L[R] M) : (A.symm : M →L[R] M).det = (A : M →L[R] M).det⁻¹ :=
LinearEquiv.det_coe_symm A.toLinearEquiv | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.Equiv",
"Mathlib.LinearAlgebra.Determinant"
] | Mathlib/Topology/Algebra/Module/Determinant.lean | det_coe_symm | null |
ContinuousLinearEquiv {R : Type*} {S : Type*} [Semiring R] [Semiring S] (σ : R →+* S)
{σ' : S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : Type*) [TopologicalSpace M]
[AddCommMonoid M] (M₂ : Type*) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M]
[Module S M₂] extends M ≃ₛₗ[σ] M₂ where
continuous_toFun : Continuous toFun := by continuity
continuous_invFun : Continuous invFun := by continuity
attribute [inherit_doc ContinuousLinearEquiv] ContinuousLinearEquiv.continuous_toFun
ContinuousLinearEquiv.continuous_invFun
@[inherit_doc]
notation:50 M " ≃SL[" σ "] " M₂ => ContinuousLinearEquiv σ M M₂
@[inherit_doc]
notation:50 M " ≃L[" R "] " M₂ => ContinuousLinearEquiv (RingHom.id R) M M₂ | structure | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | ContinuousLinearEquiv | Continuous linear equivalences between modules. We only put the type classes that are necessary
for the definition, although in applications `M` and `M₂` will be topological modules over the
topological semiring `R`. |
ContinuousSemilinearEquivClass (F : Type*) {R : outParam Type*} {S : outParam Type*}
[Semiring R] [Semiring S] (σ : outParam <| R →+* S) {σ' : outParam <| S →+* R}
[RingHomInvPair σ σ'] [RingHomInvPair σ' σ] (M : outParam Type*) [TopologicalSpace M]
[AddCommMonoid M] (M₂ : outParam Type*) [TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M]
[Module S M₂] [EquivLike F M M₂] : Prop extends SemilinearEquivClass F σ M M₂ where
map_continuous : ∀ f : F, Continuous f := by continuity
inv_continuous : ∀ f : F, Continuous (EquivLike.inv f) := by continuity
attribute [inherit_doc ContinuousSemilinearEquivClass]
ContinuousSemilinearEquivClass.map_continuous
ContinuousSemilinearEquivClass.inv_continuous | class | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | ContinuousSemilinearEquivClass | `ContinuousSemilinearEquivClass F σ M M₂` asserts `F` is a type of bundled continuous
`σ`-semilinear equivs `M → M₂`. See also `ContinuousLinearEquivClass F R M M₂` for the case
where `σ` is the identity map on `R`. A map `f` between an `R`-module and an `S`-module over a ring
homomorphism `σ : R →+* S` is semilinear if it satisfies the two properties `f (x + y) = f x + f y`
and `f (c • x) = (σ c) • f x`. |
ContinuousLinearEquivClass (F : Type*) (R : outParam Type*) [Semiring R]
(M : outParam Type*) [TopologicalSpace M] [AddCommMonoid M] (M₂ : outParam Type*)
[TopologicalSpace M₂] [AddCommMonoid M₂] [Module R M] [Module R M₂] [EquivLike F M M₂] :=
ContinuousSemilinearEquivClass F (RingHom.id R) M M₂ | abbrev | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | ContinuousLinearEquivClass | `ContinuousLinearEquivClass F σ M M₂` asserts `F` is a type of bundled continuous
`R`-linear equivs `M → M₂`. This is an abbreviation for
`ContinuousSemilinearEquivClass F (RingHom.id R) M M₂`. |
iInfKerProjEquiv {I J : Set ι} [DecidablePred fun i => i ∈ I] (hd : Disjoint I J)
(hu : Set.univ ⊆ I ∪ J) :
(⨅ i ∈ J, ker (proj i : (∀ i, φ i) →L[R] φ i) :
Submodule R (∀ i, φ i)) ≃L[R] ∀ i : I, φ i where
toLinearEquiv := LinearMap.iInfKerProjEquiv R φ hd hu
continuous_toFun :=
continuous_pi fun i =>
Continuous.comp (continuous_apply (A := φ) i) <|
@continuous_subtype_val _ _ fun x =>
x ∈ (⨅ i ∈ J, ker (proj i : (∀ i, φ i) →L[R] φ i) : Submodule R (∀ i, φ i))
continuous_invFun :=
Continuous.subtype_mk
(continuous_pi fun i => by
dsimp
split_ifs <;> [apply continuous_apply; exact continuous_zero])
_ | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | iInfKerProjEquiv | If `I` and `J` are complementary index sets, the product of the kernels of the `J`th projections
of `φ` is linearly equivalent to the product over `I`. |
@[coe]
toContinuousLinearMap (e : M₁ ≃SL[σ₁₂] M₂) : M₁ →SL[σ₁₂] M₂ :=
{ e.toLinearEquiv.toLinearMap with cont := e.continuous_toFun } | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | toContinuousLinearMap | A continuous linear equivalence induces a continuous linear map. |
ContinuousLinearMap.coe : Coe (M₁ ≃SL[σ₁₂] M₂) (M₁ →SL[σ₁₂] M₂) :=
⟨toContinuousLinearMap⟩ | instance | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | ContinuousLinearMap.coe | Coerce continuous linear equivs to continuous linear maps. |
equivLike :
EquivLike (M₁ ≃SL[σ₁₂] M₂) M₁ M₂ where
coe f := f.toFun
inv f := f.invFun
coe_injective' f g h₁ h₂ := by
obtain ⟨f', _⟩ := f
obtain ⟨g', _⟩ := g
rcases f' with ⟨⟨⟨_, _⟩, _⟩, _⟩
rcases g' with ⟨⟨⟨_, _⟩, _⟩, _⟩
congr
left_inv f := f.left_inv
right_inv f := f.right_inv | instance | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | equivLike | null |
continuousSemilinearEquivClass :
ContinuousSemilinearEquivClass (M₁ ≃SL[σ₁₂] M₂) σ₁₂ M₁ M₂ where
map_add f := f.map_add'
map_smulₛₗ f := f.map_smul'
map_continuous := continuous_toFun
inv_continuous := continuous_invFun
@[simp] | instance | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | continuousSemilinearEquivClass | null |
coe_mk (e : M₁ ≃ₛₗ[σ₁₂] M₂) (a b) : ⇑(ContinuousLinearEquiv.mk e a b) = e := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_mk | null |
coe_apply (e : M₁ ≃SL[σ₁₂] M₂) (b : M₁) : (e : M₁ →SL[σ₁₂] M₂) b = e b :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_apply | null |
coe_toLinearEquiv (f : M₁ ≃SL[σ₁₂] M₂) : ⇑f.toLinearEquiv = f :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_toLinearEquiv | null |
coe_coe (e : M₁ ≃SL[σ₁₂] M₂) : ⇑(e : M₁ →SL[σ₁₂] M₂) = e :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_coe | null |
toLinearEquiv_injective :
Function.Injective (toLinearEquiv : (M₁ ≃SL[σ₁₂] M₂) → M₁ ≃ₛₗ[σ₁₂] M₂) := by
rintro ⟨e, _, _⟩ ⟨e', _, _⟩ rfl
rfl
@[ext] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | toLinearEquiv_injective | null |
ext {f g : M₁ ≃SL[σ₁₂] M₂} (h : (f : M₁ → M₂) = g) : f = g :=
toLinearEquiv_injective <| LinearEquiv.ext <| congr_fun h | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | ext | null |
coe_injective : Function.Injective ((↑) : (M₁ ≃SL[σ₁₂] M₂) → M₁ →SL[σ₁₂] M₂) :=
fun _e _e' h => ext <| funext <| ContinuousLinearMap.ext_iff.1 h
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_injective | null |
coe_inj {e e' : M₁ ≃SL[σ₁₂] M₂} : (e : M₁ →SL[σ₁₂] M₂) = e' ↔ e = e' :=
coe_injective.eq_iff | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_inj | null |
toHomeomorph (e : M₁ ≃SL[σ₁₂] M₂) : M₁ ≃ₜ M₂ :=
{ e with toEquiv := e.toLinearEquiv.toEquiv }
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | toHomeomorph | A continuous linear equivalence induces a homeomorphism. |
coe_toHomeomorph (e : M₁ ≃SL[σ₁₂] M₂) : ⇑e.toHomeomorph = e :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_toHomeomorph | null |
isOpenMap (e : M₁ ≃SL[σ₁₂] M₂) : IsOpenMap e :=
(ContinuousLinearEquiv.toHomeomorph e).isOpenMap | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | isOpenMap | null |
image_closure (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₁) : e '' closure s = closure (e '' s) :=
e.toHomeomorph.image_closure s | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | image_closure | null |
preimage_closure (e : M₁ ≃SL[σ₁₂] M₂) (s : Set M₂) : e ⁻¹' closure s = closure (e ⁻¹' s) :=
e.toHomeomorph.preimage_closure s
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | preimage_closure | null |
isClosed_image (e : M₁ ≃SL[σ₁₂] M₂) {s : Set M₁} : IsClosed (e '' s) ↔ IsClosed s :=
e.toHomeomorph.isClosed_image | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | isClosed_image | null |
map_nhds_eq (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : map e (𝓝 x) = 𝓝 (e x) :=
e.toHomeomorph.map_nhds_eq x | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | map_nhds_eq | null |
map_zero (e : M₁ ≃SL[σ₁₂] M₂) : e (0 : M₁) = 0 :=
(e : M₁ →SL[σ₁₂] M₂).map_zero | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | map_zero | null |
map_add (e : M₁ ≃SL[σ₁₂] M₂) (x y : M₁) : e (x + y) = e x + e y :=
(e : M₁ →SL[σ₁₂] M₂).map_add x y
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | map_add | null |
map_smulₛₗ (e : M₁ ≃SL[σ₁₂] M₂) (c : R₁) (x : M₁) : e (c • x) = σ₁₂ c • e x :=
(e : M₁ →SL[σ₁₂] M₂).map_smulₛₗ c x | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | map_smulₛₗ | null |
map_smul [Module R₁ M₂] (e : M₁ ≃L[R₁] M₂) (c : R₁) (x : M₁) : e (c • x) = c • e x :=
(e : M₁ →L[R₁] M₂).map_smul c x | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | map_smul | null |
map_eq_zero_iff (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : e x = 0 ↔ x = 0 :=
e.toLinearEquiv.map_eq_zero_iff
attribute [continuity]
ContinuousLinearEquiv.continuous_toFun ContinuousLinearEquiv.continuous_invFun
@[continuity] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | map_eq_zero_iff | null |
protected continuous (e : M₁ ≃SL[σ₁₂] M₂) : Continuous (e : M₁ → M₂) :=
e.continuous_toFun | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | continuous | null |
protected continuousOn (e : M₁ ≃SL[σ₁₂] M₂) {s : Set M₁} : ContinuousOn (e : M₁ → M₂) s :=
e.continuous.continuousOn | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | continuousOn | null |
protected continuousAt (e : M₁ ≃SL[σ₁₂] M₂) {x : M₁} : ContinuousAt (e : M₁ → M₂) x :=
e.continuous.continuousAt | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | continuousAt | null |
protected continuousWithinAt (e : M₁ ≃SL[σ₁₂] M₂) {s : Set M₁} {x : M₁} :
ContinuousWithinAt (e : M₁ → M₂) s x :=
e.continuous.continuousWithinAt | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | continuousWithinAt | null |
comp_continuousOn_iff {α : Type*} [TopologicalSpace α] (e : M₁ ≃SL[σ₁₂] M₂) {f : α → M₁}
{s : Set α} : ContinuousOn (e ∘ f) s ↔ ContinuousOn f s :=
e.toHomeomorph.comp_continuousOn_iff _ _ | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | comp_continuousOn_iff | null |
comp_continuous_iff {α : Type*} [TopologicalSpace α] (e : M₁ ≃SL[σ₁₂] M₂) {f : α → M₁} :
Continuous (e ∘ f) ↔ Continuous f :=
e.toHomeomorph.comp_continuous_iff | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | comp_continuous_iff | null |
ext₁ [TopologicalSpace R₁] {f g : R₁ ≃L[R₁] M₁} (h : f 1 = g 1) : f = g :=
ext <| funext fun x => mul_one x ▸ by rw [← smul_eq_mul, map_smul, h, map_smul] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | ext₁ | An extensionality lemma for `R ≃L[R] M`. |
@[refl]
protected refl : M₁ ≃L[R₁] M₁ :=
{ LinearEquiv.refl R₁ M₁ with
continuous_toFun := continuous_id
continuous_invFun := continuous_id }
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | refl | The identity map as a continuous linear equivalence. |
refl_apply (x : M₁) :
ContinuousLinearEquiv.refl R₁ M₁ x = x := rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | refl_apply | null |
@[simp, norm_cast]
coe_refl : ↑(ContinuousLinearEquiv.refl R₁ M₁) = ContinuousLinearMap.id R₁ M₁ :=
rfl
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_refl | null |
coe_refl' : ⇑(ContinuousLinearEquiv.refl R₁ M₁) = id :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_refl' | null |
@[symm]
protected symm (e : M₁ ≃SL[σ₁₂] M₂) : M₂ ≃SL[σ₂₁] M₁ :=
{ e.toLinearEquiv.symm with
continuous_toFun := e.continuous_invFun
continuous_invFun := e.continuous_toFun }
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | symm | The inverse of a continuous linear equivalence as a continuous linear equivalence |
toLinearEquiv_symm (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.toLinearEquiv = e.toLinearEquiv.symm :=
rfl
@[deprecated (since := "2025-06-08")] alias symm_toLinearEquiv := toLinearEquiv_symm
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | toLinearEquiv_symm | null |
coe_symm_toLinearEquiv (e : M₁ ≃SL[σ₁₂] M₂) : ⇑e.toLinearEquiv.symm = e.symm :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_symm_toLinearEquiv | null |
toHomeomorph_symm (e : M₁ ≃SL[σ₁₂] M₂) : e.symm.toHomeomorph = e.toHomeomorph.symm :=
rfl
@[deprecated "use instead `toHomeomorph_symm`, in the reverse direction" (since := "2025-06-08")] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | toHomeomorph_symm | null |
symm_toHomeomorph (e : M₁ ≃SL[σ₁₂] M₂) : e.toHomeomorph.symm = e.symm.toHomeomorph :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | symm_toHomeomorph | null |
coe_symm_toHomeomorph (e : M₁ ≃SL[σ₁₂] M₂) : ⇑e.toHomeomorph.symm = e.symm :=
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | coe_symm_toHomeomorph | null |
Simps.apply (h : M₁ ≃SL[σ₁₂] M₂) : M₁ → M₂ :=
h | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | Simps.apply | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. |
Simps.symm_apply (h : M₁ ≃SL[σ₁₂] M₂) : M₂ → M₁ :=
h.symm
initialize_simps_projections ContinuousLinearEquiv (toFun → apply, invFun → symm_apply) | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | Simps.symm_apply | See Note [custom simps projection] |
symm_map_nhds_eq (e : M₁ ≃SL[σ₁₂] M₂) (x : M₁) : map e.symm (𝓝 (e x)) = 𝓝 x :=
e.toHomeomorph.symm_map_nhds_eq x | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | symm_map_nhds_eq | null |
@[trans]
protected trans (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) : M₁ ≃SL[σ₁₃] M₃ :=
{ e₁.toLinearEquiv.trans e₂.toLinearEquiv with
continuous_toFun := e₂.continuous_toFun.comp e₁.continuous_toFun
continuous_invFun := e₁.continuous_invFun.comp e₂.continuous_invFun }
@[simp] | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | trans | The composition of two continuous linear equivalences as a continuous linear equivalence. |
trans_toLinearEquiv (e₁ : M₁ ≃SL[σ₁₂] M₂) (e₂ : M₂ ≃SL[σ₂₃] M₃) :
(e₁.trans e₂).toLinearEquiv = e₁.toLinearEquiv.trans e₂.toLinearEquiv := by
ext
rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | trans_toLinearEquiv | null |
prodCongr [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂) (e' : M₃ ≃L[R₁] M₄) :
(M₁ × M₃) ≃L[R₁] M₂ × M₄ :=
{ e.toLinearEquiv.prodCongr e'.toLinearEquiv with
continuous_toFun := e.continuous_toFun.prodMap e'.continuous_toFun
continuous_invFun := e.continuous_invFun.prodMap e'.continuous_invFun }
@[deprecated (since := "2025-06-06")] alias prod := prodCongr
@[simp, norm_cast] | def | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | prodCongr | Product of two continuous linear equivalences. The map comes from `Equiv.prodCongr`. |
prodCongr_apply [Module R₁ M₂] [Module R₁ M₃] [Module R₁ M₄] (e : M₁ ≃L[R₁] M₂)
(e' : M₃ ≃L[R₁] M₄) (x) : e.prodCongr e' x = (e x.1, e' x.2) :=
rfl
@[deprecated (since := "2025-06-06")] alias prod_apply := prodCongr_apply
@[simp, norm_cast] | theorem | Topology | [
"Mathlib.Topology.Algebra.Module.LinearMapPiProd"
] | Mathlib/Topology/Algebra/Module/Equiv.lean | prodCongr_apply | null |
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