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
⌀ |
|---|---|---|---|---|---|---|
@[to_additive (attr := simp)]
one_apply [Π i, One (R i)] [∀ i, OneMemClass (S i) (R i)] (i : ι) :
(1 : Πʳ i, [R i, B i]_[𝓕]) i = 1 :=
rfl
@[to_additive]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
one_apply
| null |
@[to_additive (attr := simp)]
inv_apply [Π i, Inv (R i)] [∀ i, InvMemClass (S i) (R i)]
(x : Πʳ i, [R i, B i]_[𝓕]) (i : ι) : (x⁻¹) i = (x i)⁻¹ :=
rfl
@[to_additive]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
inv_apply
| null |
@[to_additive (attr := simp)]
mul_apply [Π i, Mul (R i)] [∀ i, MulMemClass (S i) (R i)]
(x y : Πʳ i, [R i, B i]_[𝓕]) (i : ι) : (x * y) i = x i * y i :=
rfl
@[to_additive]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
mul_apply
| null |
@[to_additive (attr := simp)]
smul_apply {G : Type*} [Π i, SMul G (R i)] [∀ i, SMulMemClass (S i) G (R i)] (g : G)
(x : Πʳ i, [R i, B i]_[𝓕]) (i : ι) : (g • x) i = g • x i :=
rfl
@[to_additive]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
smul_apply
| null |
@[to_additive (attr := simp)]
div_apply [Π i, DivInvMonoid (R i)] [∀ i, SubgroupClass (S i) (R i)]
(x y : Πʳ i, [R i, B i]_[𝓕]) (i : ι) : (x / y) i = x i / y i :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
div_apply
| null |
instNSMul [Π i, AddMonoid (R i)] [∀ i, AddSubmonoidClass (S i) (R i)] :
SMul ℕ (Πʳ i, [R i, B i]_[𝓕]) where
smul n x := ⟨fun i ↦ n • (x i), x.2.mono fun _ hi ↦ nsmul_mem hi n⟩
@[to_additive existing instNSMul]
|
instance
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
instNSMul
| null |
@[to_additive]
pow_apply [Π i, Monoid (R i)] [∀ i, SubmonoidClass (S i) (R i)]
(x : Πʳ i, [R i, B i]_[𝓕]) (n : ℕ) (i : ι) : (x ^ n) i = x i ^ n :=
rfl
@[to_additive]
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
pow_apply
| null |
instZSMul [Π i, SubNegMonoid (R i)] [∀ i, AddSubgroupClass (S i) (R i)] :
SMul ℤ (Πʳ i, [R i, B i]_[𝓕]) where
smul n x := ⟨fun i ↦ n • x i, x.2.mono fun _ hi ↦ zsmul_mem hi n⟩
@[to_additive existing instZSMul]
|
instance
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
instZSMul
| null |
@[to_additive]
zpow_apply [Π i, DivInvMonoid (R i)] [∀ i, SubgroupClass (S i) (R i)]
(x : Πʳ i, [R i, B i]_[𝓕]) (n : ℤ) (i : ι) : (x ^ n) i = x i ^ n :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
zpow_apply
| null |
@[to_additive /-- The coercion from the restricted product of additive monoids `A i` to the
(normal) product is an additive monoid homomorphism. -/]
coeMonoidHom [∀ i, Monoid (R i)] [∀ i, SubmonoidClass (S i) (R i)] :
Πʳ i, [R i, B i]_[𝓕] →* Π i, R i where
toFun := (↑)
map_one' := rfl
map_mul' _ _ := rfl
|
def
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
coeMonoidHom
|
The coercion from the restricted product of monoids `A i` to the (normal) product
is a monoid homomorphism.
|
evalRingHom (j : ι) [Π i, Ring (R i)] [∀ i, SubringClass (S i) (R i)] :
(Πʳ i, [R i, B i]_[𝓕]) →+* R j where
__ := evalMonoidHom R j
__ := evalAddMonoidHom R j
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
evalRingHom
|
`RestrictedProduct.evalMonoidHom j` is the monoid homomorphism from the restricted
product `Πʳ i, [R i, B i]_[𝓕]` to the component `R j`.
-/
@[to_additive /-- `RestrictedProduct.evalAddMonoidHom j` is the monoid homomorphism from the
restricted product `Πʳ i, [R i, B i]_[𝓕]` to the component `R j`. -/]
def evalMonoidHom (j : ι) [Π i, Monoid (R i)] [∀ i, SubmonoidClass (S i) (R i)] :
(Πʳ i, [R i, B i]_[𝓕]) →* R j where
toFun x := x j
map_one' := rfl
map_mul' _ _ := rfl
@[simp]
lemma evalMonoidHom_apply [Π i, Monoid (R i)] [∀ i, SubmonoidClass (S i) (R i)]
(x : Πʳ i, [R i, B i]_[𝓕]) (j : ι) : evalMonoidHom R j x = x j :=
rfl
/-- `RestrictedProduct.evalRingHom j` is the ring homomorphism from the restricted
product `Πʳ i, [R i, B i]_[𝓕]` to the component `R j`.
|
evalRingHom_apply [Π i, Ring (R i)] [∀ i, SubringClass (S i) (R i)]
(x : Πʳ i, [R i, B i]_[𝓕]) (j : ι) : evalRingHom R j x = x j :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
evalRingHom_apply
| null |
mapAlong (x : Πʳ i, [R₁ i, A₁ i]_[𝓕₁]) : Πʳ j, [R₂ j, A₂ j]_[𝓕₂] :=
⟨fun j ↦ φ j (x (f j)), by
filter_upwards [hf.eventually x.2, hφ] using fun _ h1 h2 ↦ h2 h1⟩
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
mapAlong
|
Given two restricted products `Πʳ (i : ι₁), [R₁ i, A₁ i]_[𝓕₁]` and `Πʳ (j : ι₂), [R₂ j, A₂ j]_[𝓕₂]`,
`RestrictedProduct.mapAlong` gives a function between them. The data needed is a
function `f : ι₂ → ι₁` such that `𝓕₂` tends to `𝓕₁` along `f`, and functions `φ j : R₁ (f j) → R₂ j`
sending `A₁ (f j)` into `A₂ j` for an `𝓕₂`-large set of `j`'s.
See also `mapAlongMonoidHom`, `mapAlongAddMonoidHom` and `mapAlongRingHom` for variants.
|
mapAlong_apply (x : Πʳ i, [R₁ i, A₁ i]_[𝓕₁]) (j : ι₂) :
x.mapAlong R₁ R₂ f hf φ hφ j = φ j (x (f j)) :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
mapAlong_apply
| null |
map {G H : ι → Type*}
{C : (i : ι) → Set (G i)}
{D : (i : ι) → Set (H i)} (φ : (i : ι) → G i → H i)
(hφ : ∀ᶠ i in 𝓕, MapsTo (φ i) (C i) (D i))
(x : Πʳ i, [G i, C i]_[𝓕]) : (Πʳ i, [H i, D i]_[𝓕]) :=
mapAlong G H id Filter.tendsto_id φ hφ x
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
map
|
The maps between restricted products over a fixed index type,
given maps on the factors.
|
map_apply {G H : ι → Type*} {C : (i : ι) → Set (G i)}
{D : (i : ι) → Set (H i)} (φ : (i : ι) → G i → H i)
(hφ : ∀ᶠ i in 𝓕, MapsTo (φ i) (C i) (D i))
(x : Πʳ i, [G i, C i]_[𝓕]) (j : ι) :
x.map φ hφ j = φ j (x j) :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
map_apply
| null |
mapAlongRingHom : Πʳ i, [R₁ i, B₁ i]_[𝓕₁] →+* Πʳ j, [R₂ j, B₂ j]_[𝓕₂] where
__ := mapAlongMonoidHom R₁ R₂ f hf (fun j ↦ φ j) hφ
__ := mapAlongAddMonoidHom R₁ R₂ f hf (fun j ↦ φ j) hφ
@[simp]
|
def
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
mapAlongRingHom
|
Given two restricted products `Πʳ (i : ι₁), [R₁ i, B₁ i]_[𝓕₁]` and `Πʳ (j : ι₂), [R₂ j, B₂ j]_[𝓕₂]`
of monoids, `RestrictedProduct.mapAlongMonoidHom` gives a monoid homomorphism between them.
The data needed is a function `f : ι₂ → ι₁` such that `𝓕₂` tends to `𝓕₁` along `f`, and monoid
homomorphisms `φ j : R₁ (f j) → R₂ j` sending `B₁ (f j)` into `B₂ j` for an `𝓕₂`-large set of `j`'s.
-/
@[to_additive
/-- Given two restricted products `Πʳ (i : ι₁), [R₁ i, B₁ i]_[𝓕₁]` and
`Πʳ (j : ι₂), [R₂ j, B₂ j]_[𝓕₂]` of additive monoids, `RestrictedProduct.mapAlongAddMonoidHom`
gives a additive monoid homomorphism between them. The data needed is a function `f : ι₂ → ι₁` such
that `𝓕₂` tends to `𝓕₁` along `f`, and additive monoid homomorphisms `φ j : R₁ (f j) → R₂ j`
sending `B₁ (f j)` into `B₂ j` for an `𝓕₂`-large set of `j`'s. -/]
def mapAlongMonoidHom : Πʳ i, [R₁ i, B₁ i]_[𝓕₁] →* Πʳ j, [R₂ j, B₂ j]_[𝓕₂] where
toFun := mapAlong R₁ R₂ f hf (fun j r ↦ φ j r) hφ
map_one' := by
ext i
exact map_one (φ i)
map_mul' x y := by
ext i
exact map_mul (φ i) _ _
@[to_additive (attr := simp)]
lemma mapAlongMonoidHom_apply (x : Πʳ i, [R₁ i, B₁ i]_[𝓕₁]) (j : ι₂) :
x.mapAlongMonoidHom R₁ R₂ f hf φ hφ j = φ j (x (f j)) :=
rfl
end monoid
section ring
variable [Π i, Ring (R₁ i)] [Π i, Ring (R₂ i)] [∀ i, SubringClass (S₁ i) (R₁ i)]
[∀ i, SubringClass (S₂ i) (R₂ i)] (φ : ∀ j, R₁ (f j) →+* R₂ j)
(hφ : ∀ᶠ j in 𝓕₂, MapsTo (φ j) (B₁ (f j)) (B₂ j))
/--
Given two restricted products of rings `Πʳ (i : ι₁), [R₁ i, B₁ i]_[𝓕₁]` and
`Πʳ (j : ι₂), [R₂ j, B₂ j]_[𝓕₂]`, `RestrictedProduct.mapAlongRingHom` gives a
ring homomorphism between them. The data needed is a
function `f : ι₂ → ι₁` such that `𝓕₂` tends to `𝓕₁` along `f`, and ring homomorphisms
`φ j : R₁ (f j) → R₂ j` sending `B₁ (f j)` into `B₂ j` for an `𝓕₂`-large set of `j`'s.
|
mapAlongRingHom_apply (x : Πʳ i, [R₁ i, B₁ i]_[𝓕₁]) (j : ι₂) :
x.mapAlongRingHom R₁ R₂ f hf φ hφ j = φ j (x (f j)) :=
rfl
|
lemma
|
Topology
|
[
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Ring.Subring.Defs",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Order.Filter.Cofinite -- for `Πʳ i, [R i, A i]` notation, confuses shake",
"Mathlib.Algebra.Module.Pi"
] |
Mathlib/Topology/Algebra/RestrictedProduct/Basic.lean
|
mapAlongRingHom_apply
| null |
topologicalSpace : TopologicalSpace (Πʳ i, [R i, A i]_[𝓕]) :=
⨆ (S : Set ι) (hS : 𝓕 ≤ 𝓟 S), .coinduced (inclusion R A hS)
(.induced ((↑) : Πʳ i, [R i, A i]_[𝓟 S] → Π i, R i) inferInstance)
@[fun_prop]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
topologicalSpace
| null |
continuous_coe :
Continuous ((↑) : Πʳ i, [R i, A i]_[𝓕] → Π i, R i) :=
continuous_iSup_dom.mpr fun _ ↦ continuous_iSup_dom.mpr fun _ ↦
continuous_coinduced_dom.mpr continuous_induced_dom
@[fun_prop]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_coe
| null |
continuous_eval (i : ι) :
Continuous (fun (x : Πʳ i, [R i, A i]_[𝓕]) ↦ x i) :=
continuous_apply _ |>.comp continuous_coe
@[fun_prop]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_eval
| null |
continuous_inclusion {𝓖 : Filter ι} (h : 𝓕 ≤ 𝓖) :
Continuous (inclusion R A h) := by
simp_rw [continuous_iff_coinduced_le, topologicalSpace, coinduced_iSup, coinduced_compose]
exact iSup₂_le fun S hS ↦ le_iSup₂_of_le S (le_trans h hS) le_rfl
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_inclusion
| null |
topologicalSpace_eq_of_principal :
topologicalSpace R A (𝓟 S) =
.induced ((↑) : Πʳ i, [R i, A i]_[𝓟 S] → Π i, R i) inferInstance :=
le_antisymm (continuous_iff_le_induced.mp continuous_coe) <|
(le_iSup₂_of_le S le_rfl <| by rw [inclusion_eq_id R A (𝓟 S), @coinduced_id])
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
topologicalSpace_eq_of_principal
| null |
topologicalSpace_eq_of_top :
topologicalSpace R A ⊤ =
.induced ((↑) : Πʳ i, [R i, A i]_[⊤] → Π i, R i) inferInstance :=
principal_univ ▸ topologicalSpace_eq_of_principal
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
topologicalSpace_eq_of_top
| null |
topologicalSpace_eq_of_bot :
topologicalSpace R A ⊥ =
.induced ((↑) : Πʳ i, [R i, A i]_[⊥] → Π i, R i) inferInstance :=
principal_empty ▸ topologicalSpace_eq_of_principal
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
topologicalSpace_eq_of_bot
| null |
isEmbedding_coe_of_principal :
IsEmbedding ((↑) : Πʳ i, [R i, A i]_[𝓟 S] → Π i, R i) where
eq_induced := topologicalSpace_eq_of_principal
injective := DFunLike.coe_injective
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isEmbedding_coe_of_principal
| null |
isEmbedding_coe_of_top :
IsEmbedding ((↑) : Πʳ i, [R i, A i]_[⊤] → Π i, R i) :=
principal_univ ▸ isEmbedding_coe_of_principal
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isEmbedding_coe_of_top
| null |
isEmbedding_coe_of_bot :
IsEmbedding ((↑) : Πʳ i, [R i, A i]_[⊥] → Π i, R i) :=
principal_empty ▸ isEmbedding_coe_of_principal
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isEmbedding_coe_of_bot
| null |
continuous_rng_of_principal {X : Type*} [TopologicalSpace X]
{f : X → Πʳ i, [R i, A i]_[𝓟 S]} :
Continuous f ↔ Continuous ((↑) ∘ f : X → Π i, R i) :=
isEmbedding_coe_of_principal.continuous_iff
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_rng_of_principal
| null |
continuous_rng_of_top {X : Type*} [TopologicalSpace X]
{f : X → Πʳ i, [R i, A i]_[⊤]} :
Continuous f ↔ Continuous ((↑) ∘ f : X → Π i, R i) :=
isEmbedding_coe_of_top.continuous_iff
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_rng_of_top
| null |
continuous_rng_of_bot {X : Type*} [TopologicalSpace X]
{f : X → Πʳ i, [R i, A i]_[⊥]} :
Continuous f ↔ Continuous ((↑) ∘ f : X → Π i, R i) :=
isEmbedding_coe_of_bot.continuous_iff
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_rng_of_bot
| null |
continuous_rng_of_principal_iff_forall {X : Type*} [TopologicalSpace X]
{f : X → Πʳ (i : ι), [R i, A i]_[𝓟 S]} :
Continuous f ↔ ∀ i : ι, Continuous ((fun x ↦ x i) ∘ f) :=
continuous_rng_of_principal.trans continuous_pi_iff
|
lemma
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_rng_of_principal_iff_forall
| null |
homeoTop : (Π i, A i) ≃ₜ (Πʳ i, [R i, A i]_[⊤]) where
toFun f := ⟨fun i ↦ f i, fun i ↦ (f i).2⟩
invFun f i := ⟨f i, f.2 i⟩
continuous_toFun := continuous_rng_of_top.mpr <| continuous_pi fun i ↦
continuous_subtype_val.comp <| continuous_apply i
continuous_invFun := continuous_pi fun i ↦ continuous_induced_rng.mpr <| continuous_eval i
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
homeoTop
|
The obvious bijection between `Πʳ i, [R i, A i]_[⊤]` and `Π i, A i` is a homeomorphism.
|
homeoBot : (Π i, R i) ≃ₜ (Πʳ i, [R i, A i]_[⊥]) where
toFun f := ⟨fun i ↦ f i, eventually_bot⟩
invFun f i := f i
continuous_toFun := continuous_rng_of_bot.mpr <| continuous_pi fun i ↦ continuous_apply i
continuous_invFun := continuous_pi continuous_eval
|
def
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
homeoBot
|
The obvious bijection between `Πʳ i, [R i, A i]_[⊥]` and `Π i, R i` is a homeomorphism.
|
weaklyLocallyCompactSpace_of_principal [∀ i, WeaklyLocallyCompactSpace (R i)]
(hS : cofinite ≤ 𝓟 S) (hAcompact : ∀ i ∈ S, IsCompact (A i)) :
WeaklyLocallyCompactSpace (Πʳ i, [R i, A i]_[𝓟 S]) where
exists_compact_mem_nhds := fun x ↦ by
rw [le_principal_iff, mem_cofinite] at hS
classical
have : ∀ i, ∃ K, IsCompact K ∧ K ∈ 𝓝 (x i) := fun i ↦ exists_compact_mem_nhds (x i)
choose K K_compact hK using this
set Q : Set (Π i, R i) := univ.pi (fun i ↦ if i ∈ S then A i else K i) with Q_def
have Q_compact : IsCompact Q := isCompact_univ_pi fun i ↦ by
split_ifs with his
· exact hAcompact i his
· exact K_compact i
set U : Set (Π i, R i) := Sᶜ.pi K
have U_nhds : U ∈ 𝓝 (x : Π i, R i) := set_pi_mem_nhds hS fun i _ ↦ hK i
have QU : (↑) ⁻¹' U ⊆ ((↑) ⁻¹' Q : Set (Πʳ i, [R i, A i]_[𝓟 S])) := fun y H i _ ↦ by
dsimp only
split_ifs with hi
· exact y.2 hi
· exact H i hi
refine ⟨((↑) ⁻¹' Q), ?_, mem_of_superset ?_ QU⟩
· refine isEmbedding_coe_of_principal.isCompact_preimage_iff ?_ |>.mpr Q_compact
simp_rw [range_coe_principal, Q_def, pi_if, mem_univ, true_and]
exact inter_subset_left
· simpa only [isEmbedding_coe_of_principal.nhds_eq_comap] using preimage_mem_comap U_nhds
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
weaklyLocallyCompactSpace_of_principal
|
Assume that `S` is a subset of `ι` with finite complement, that each `R i` is weakly locally
compact, and that `A i` is *compact* for all `i ∈ S`. Then the restricted product
`Πʳ i, [R i, A i]_[𝓟 S]` is locally compact.
Note: we spell "`S` has finite complement" as `cofinite ≤ 𝓟 S`.
|
topologicalSpace_eq_iSup :
topologicalSpace R A 𝓕 = ⨆ (S : Set ι) (hS : 𝓕 ≤ 𝓟 S),
.coinduced (inclusion R A hS) (topologicalSpace R A (𝓟 S)) := by
simp_rw [topologicalSpace_eq_of_principal, topologicalSpace]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
topologicalSpace_eq_iSup
| null |
continuous_dom {X : Type*} [TopologicalSpace X]
{f : Πʳ i, [R i, A i]_[𝓕] → X} :
Continuous f ↔ ∀ (S : Set ι) (hS : 𝓕 ≤ 𝓟 S), Continuous (f ∘ inclusion R A hS) := by
simp_rw [topologicalSpace_eq_of_principal, continuous_iSup_dom, continuous_coinduced_dom]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_dom
|
The **universal property** of the topology on the restricted product: a map from
`Πʳ i, [R i, A i]_[𝓕]` is continuous *iff* its restriction to each `Πʳ i, [R i, A i]_[𝓟 s]`
(with `𝓕 ≤ 𝓟 s`) is continuous.
See also `RestrictedProduct.continuous_dom_prod_left`.
|
isEmbedding_inclusion_principal {S : Set ι} (hS : 𝓕 ≤ 𝓟 S) :
IsEmbedding (inclusion R A hS) :=
.of_comp (continuous_inclusion hS) continuous_coe isEmbedding_coe_of_principal
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isEmbedding_inclusion_principal
| null |
isEmbedding_inclusion_top :
IsEmbedding (inclusion R A (le_top : 𝓕 ≤ ⊤)) :=
.of_comp (continuous_inclusion _) continuous_coe isEmbedding_coe_of_top
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isEmbedding_inclusion_top
| null |
isEmbedding_structureMap :
IsEmbedding (structureMap R A 𝓕) :=
isEmbedding_inclusion_top.comp homeoTop.isEmbedding
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isEmbedding_structureMap
|
`Π i, A i` has the subset topology from the restricted product.
|
isOpen_forall_imp_mem_of_principal {S : Set ι} (hS : cofinite ≤ 𝓟 S) {p : ι → Prop} :
IsOpen {f : Πʳ i, [R i, A i]_[𝓟 S] | ∀ i, p i → f.1 i ∈ A i} := by
rw [le_principal_iff] at hS
convert isOpen_set_pi (hS.inter_of_left {i | p i}) (fun i _ ↦ hAopen i) |>.preimage continuous_coe
ext f
refine ⟨fun H i hi ↦ H i hi.2, fun H i hiT ↦ ?_⟩
by_cases hiS : i ∈ S
· exact f.2 hiS
· exact H i ⟨hiS, hiT⟩
include hAopen in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isOpen_forall_imp_mem_of_principal
| null |
isOpen_forall_mem_of_principal {S : Set ι} (hS : cofinite ≤ 𝓟 S) :
IsOpen {f : Πʳ i, [R i, A i]_[𝓟 S] | ∀ i, f.1 i ∈ A i} := by
convert isOpen_forall_imp_mem_of_principal hAopen hS (p := fun _ ↦ True)
simp
include hAopen in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isOpen_forall_mem_of_principal
| null |
isOpen_forall_imp_mem {p : ι → Prop} :
IsOpen {f : Πʳ i, [R i, A i] | ∀ i, p i → f.1 i ∈ A i} := by
simp_rw [topologicalSpace_eq_iSup cofinite, isOpen_iSup_iff, isOpen_coinduced]
exact fun S hS ↦ isOpen_forall_imp_mem_of_principal hAopen hS
include hAopen in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isOpen_forall_imp_mem
| null |
isOpen_forall_mem :
IsOpen {f : Πʳ i, [R i, A i] | ∀ i, f.1 i ∈ A i} := by
simp_rw [topologicalSpace_eq_iSup cofinite, isOpen_iSup_iff, isOpen_coinduced]
exact fun S hS ↦ isOpen_forall_mem_of_principal hAopen hS
include hAopen in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isOpen_forall_mem
| null |
isOpenEmbedding_inclusion_principal {S : Set ι} (hS : cofinite ≤ 𝓟 S) :
IsOpenEmbedding (inclusion R A hS) where
toIsEmbedding := isEmbedding_inclusion_principal hS
isOpen_range := by
rw [range_inclusion]
exact isOpen_forall_imp_mem hAopen
include hAopen in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isOpenEmbedding_inclusion_principal
| null |
isOpenEmbedding_structureMap :
IsOpenEmbedding (structureMap R A cofinite) where
toIsEmbedding := isEmbedding_structureMap
isOpen_range := by
rw [range_structureMap]
exact isOpen_forall_mem hAopen
include hAopen in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isOpenEmbedding_structureMap
|
`Π i, A i` is homeomorphic to an open subset of the restricted product.
|
nhds_eq_map_inclusion {S : Set ι} (hS : cofinite ≤ 𝓟 S)
(x : Πʳ i, [R i, A i]_[𝓟 S]) :
(𝓝 (inclusion R A hS x)) = .map (inclusion R A hS) (𝓝 x) := by
rw [isOpenEmbedding_inclusion_principal hAopen hS |>.map_nhds_eq x]
include hAopen in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
nhds_eq_map_inclusion
| null |
nhds_eq_map_structureMap
(x : Π i, A i) :
(𝓝 (structureMap R A cofinite x)) = .map (structureMap R A cofinite) (𝓝 x) := by
rw [isOpenEmbedding_structureMap hAopen |>.map_nhds_eq x]
include hAopen in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
nhds_eq_map_structureMap
| null |
weaklyLocallyCompactSpace_of_cofinite [∀ i, WeaklyLocallyCompactSpace (R i)]
(hAcompact : ∀ᶠ i in cofinite, IsCompact (A i)) :
WeaklyLocallyCompactSpace (Πʳ i, [R i, A i]) where
exists_compact_mem_nhds := fun x ↦ by
set S := {i | IsCompact (A i) ∧ x i ∈ A i}
have hS : cofinite ≤ 𝓟 S := le_principal_iff.mpr (hAcompact.and x.2)
have hSx : ∀ i ∈ S, x i ∈ A i := fun i hi ↦ hi.2
have hSA : ∀ i ∈ S, IsCompact (A i) := fun i hi ↦ hi.1
haveI := weaklyLocallyCompactSpace_of_principal hS hSA
rcases exists_inclusion_eq_of_eventually R A hS hSx with ⟨x', hxx'⟩
rw [← hxx', nhds_eq_map_inclusion hAopen]
rcases exists_compact_mem_nhds x' with ⟨K, K_compact, hK⟩
exact ⟨inclusion R A hS '' K, K_compact.image (continuous_inclusion hS), image_mem_map hK⟩
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
weaklyLocallyCompactSpace_of_cofinite
|
If each `R i` is weakly locally compact, each `A i` is open, and all but finitely many `A i`s
are also compact, then the restricted product `Πʳ i, [R i, A i]` is weakly locally compact.
|
continuous_dom_prod_right {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
{f : Πʳ i, [R i, A i] × Y → X} :
Continuous f ↔ ∀ (S : Set ι) (hS : cofinite ≤ 𝓟 S),
Continuous (f ∘ (Prod.map (inclusion R A hS) id)) := by
refine ⟨fun H S hS ↦ H.comp ((continuous_inclusion hS).prodMap continuous_id),
fun H ↦ ?_⟩
simp_rw [continuous_iff_continuousAt, ContinuousAt]
rintro ⟨x, y⟩
set S : Set ι := {i | x i ∈ A i}
have hS : cofinite ≤ 𝓟 S := le_principal_iff.mpr x.2
have hxS : ∀ i ∈ S, x i ∈ A i := fun i hi ↦ hi
rcases exists_inclusion_eq_of_eventually R A hS hxS with ⟨x', hxx'⟩
rw [← hxx', nhds_prod_eq, nhds_eq_map_inclusion hAopen hS x',
← Filter.map_id (f := 𝓝 y), prod_map_map_eq, ← nhds_prod_eq, tendsto_map'_iff]
exact H S hS |>.tendsto ⟨x', y⟩
include hAopen in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_dom_prod_right
|
The **universal property with parameters** of the topology on the restricted product:
for any topological space `Y` of "parameters", a map from `(Πʳ i, [R i, A i]) × Y` is continuous
*iff* its restriction to each `(Πʳ i, [R i, A i]_[𝓟 S]) × Y` (with `S` cofinite) is continuous.
|
continuous_dom_prod_left {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y]
{f : Y × Πʳ i, [R i, A i] → X} :
Continuous f ↔ ∀ (S : Set ι) (hS : cofinite ≤ 𝓟 S),
Continuous (f ∘ (Prod.map id (inclusion R A hS))) := by
refine ⟨fun H S hS ↦ H.comp (continuous_id.prodMap (continuous_inclusion hS)),
fun H ↦ ?_⟩
simp_rw [continuous_iff_continuousAt, ContinuousAt]
rintro ⟨y, x⟩
set S : Set ι := {i | x i ∈ A i}
have hS : cofinite ≤ 𝓟 S := le_principal_iff.mpr x.2
have hxS : ∀ i ∈ S, x i ∈ A i := fun i hi ↦ hi
rcases exists_inclusion_eq_of_eventually R A hS hxS with ⟨x', hxx'⟩
rw [← hxx', nhds_prod_eq, nhds_eq_map_inclusion hAopen hS x',
← Filter.map_id (f := 𝓝 y), prod_map_map_eq, ← nhds_prod_eq, tendsto_map'_iff]
exact H S hS |>.tendsto ⟨y, x'⟩
include hAopen in
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_dom_prod_left
|
The **universal property with parameters** of the topology on the restricted product:
for any topological space `Y` of "parameters", a map from `Y × Πʳ i, [R i, A i]` is continuous
*iff* its restriction to each `Y × Πʳ i, [R i, A i]_[𝓟 S]` (with `S` cofinite) is continuous.
|
continuous_dom_prod {R' : ι → Type*} {A' : (i : ι) → Set (R' i)}
[∀ i, TopologicalSpace (R' i)] (hAopen' : ∀ i, IsOpen (A' i))
{X : Type*} [TopologicalSpace X]
{f : Πʳ i, [R i, A i] × Πʳ i, [R' i, A' i] → X} :
Continuous f ↔ ∀ (S : Set ι) (hS : cofinite ≤ 𝓟 S),
Continuous (f ∘ (Prod.map (inclusion R A hS) (inclusion R' A' hS))) := by
simp_rw [continuous_dom_prod_right hAopen, continuous_dom_prod_left hAopen']
refine ⟨fun H S hS ↦ H S hS S hS, fun H S hS T hT ↦ ?_⟩
set U := S ∩ T
have hU : cofinite ≤ 𝓟 (S ∩ T) := inf_principal ▸ le_inf hS hT
have hSU : 𝓟 U ≤ 𝓟 S := principal_mono.mpr inter_subset_left
have hTU : 𝓟 U ≤ 𝓟 T := principal_mono.mpr inter_subset_right
exact (H U hU).comp ((continuous_inclusion hSU).prodMap (continuous_inclusion hTU))
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_dom_prod
|
A map from `Πʳ i, [R i, A i] × Πʳ i, [R' i, A' i]` is continuous
*iff* its restriction to each `Πʳ i, [R i, A i]_[𝓟 S] × Πʳ i, [R' i, A' i]_[𝓟 S]`
(with `S` cofinite) is continuous.
This is the key result for continuity of multiplication and addition.
|
continuous_dom_pi {n : Type*} [Finite n] {X : Type*}
[TopologicalSpace X] {A : n → ι → Type*}
[∀ j i, TopologicalSpace (A j i)]
{C : (j : n) → (i : ι) → Set (A j i)}
(hCopen : ∀ j i, IsOpen (C j i))
{f : (Π j : n, Πʳ i : ι, [A j i, C j i]) → X} :
Continuous f ↔
∀ (S : Set ι) (hS : cofinite ≤ 𝓟 S), Continuous (f ∘ Pi.map fun _ ↦ inclusion _ _ hS) := by
refine ⟨by fun_prop, fun H ↦ ?_⟩
simp_rw [continuous_iff_continuousAt, ContinuousAt]
intro x
set S : Set ι := {i | ∀ j, x j i ∈ C j i}
have hS : cofinite ≤ 𝓟 S := by
rw [le_principal_iff]
change ∀ᶠ i in cofinite, ∀ j : n, x j i ∈ C j i
simp [-eventually_cofinite]
let x' (j : n) : Πʳ i : ι, [A j i, C j i]_[𝓟 S] := .mk (fun i ↦ x j i) (fun i hi ↦ hi _)
have hxx' : Pi.map (fun j ↦ inclusion _ _ hS) x' = x := rfl
simp_rw [← hxx', nhds_pi, Pi.map_apply, nhds_eq_map_inclusion (hCopen _), ← map_piMap_pi_finite,
tendsto_map'_iff, ← nhds_pi]
exact (H _ _).tendsto _
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuous_dom_pi
|
A finitary (instead of binary) version of `continuous_dom_prod`.
|
nhds_zero_eq_map_ofPre [Π i, Zero (R i)] [∀ i, ZeroMemClass (S i) (R i)]
(hBopen : ∀ i, IsOpen (B i : Set (R i))) (hT : cofinite ≤ 𝓟 T) :
(𝓝 (inclusion R (fun i ↦ B i) hT 0)) = .map (inclusion R (fun i ↦ B i) hT) (𝓝 0) :=
nhds_eq_map_inclusion hBopen hT 0
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
nhds_zero_eq_map_ofPre
| null |
nhds_zero_eq_map_structureMap [Π i, Zero (R i)] [∀ i, ZeroMemClass (S i) (R i)]
(hBopen : ∀ i, IsOpen (B i : Set (R i))) :
(𝓝 (structureMap R (fun i ↦ B i) cofinite 0)) =
.map (structureMap R (fun i ↦ B i) cofinite) (𝓝 0) :=
nhds_eq_map_structureMap hBopen 0
variable [hBopen : Fact (∀ i, IsOpen (B i : Set (R i)))]
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
nhds_zero_eq_map_structureMap
| null |
@[to_additive]
continuousSMul {G : Type*} [TopologicalSpace G] [Π i, SMul G (R i)]
[∀ i, SMulMemClass (S i) G (R i)] [∀ i, ContinuousSMul G (R i)] :
ContinuousSMul G (Πʳ i, [R i, B i]) where
continuous_smul := by
rw [continuous_dom_prod_left hBopen.out]
exact fun S hS ↦ (continuous_inclusion hS).comp continuous_smul
@[to_additive]
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
continuousSMul
| null |
isTopologicalGroup [Π i, Group (R i)] [∀ i, SubgroupClass (S i) (R i)]
[∀ i, IsTopologicalGroup (R i)] :
IsTopologicalGroup (Πʳ i, [R i, B i]) where
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isTopologicalGroup
| null |
isTopologicalRing [Π i, Ring (R i)] [∀ i, SubringClass (S i) (R i)]
[∀ i, IsTopologicalRing (R i)] :
IsTopologicalRing (Πʳ i, [R i, B i]) where
|
instance
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
isTopologicalRing
| null |
locallyCompactSpace_of_group [Π i, Group (R i)] [∀ i, SubgroupClass (S i) (R i)]
[∀ i, IsTopologicalGroup (R i)] [∀ i, LocallyCompactSpace (R i)]
(hBcompact : ∀ᶠ i in cofinite, IsCompact (B i : Set (R i))) :
LocallyCompactSpace (Πʳ i, [R i, B i]) :=
haveI : WeaklyLocallyCompactSpace (Πʳ i, [R i, B i]) :=
weaklyLocallyCompactSpace_of_cofinite hBopen.out hBcompact
inferInstance
open Pointwise in
@[to_additive]
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
locallyCompactSpace_of_group
| null |
mapAlong_continuous (φ_cont : ∀ j, Continuous (φ j)) :
Continuous (mapAlong R₁ R₂ f hf φ hφ) := by
rw [continuous_dom]
intro S hS
set T := f ⁻¹' S ∩ {j | MapsTo (φ j) (A₁ (f j)) (A₂ j)}
have hT : 𝓕₂ ≤ 𝓟 T := by
rw [le_principal_iff] at hS ⊢
exact inter_mem (hf hS) hφ
have hf' : Tendsto f (𝓟 T) (𝓟 S) := by aesop
have hφ' : ∀ᶠ j in 𝓟 T, MapsTo (φ j) (A₁ (f j)) (A₂ j) := by aesop
have key : mapAlong R₁ R₂ f hf φ hφ ∘ inclusion R₁ A₁ hS =
inclusion R₂ A₂ hT ∘ mapAlong R₁ R₂ f hf' φ hφ' := rfl
rw [key]
exact continuous_inclusion _ |>.comp <|
continuous_rng_of_principal.mpr <|
continuous_pi fun j ↦ φ_cont j |>.comp <| continuous_eval (f j)
|
theorem
|
Topology
|
[
"Mathlib.Topology.Algebra.Group.Pointwise",
"Mathlib.Topology.Algebra.RestrictedProduct.Basic",
"Mathlib.Topology.Algebra.Ring.Basic"
] |
Mathlib/Topology/Algebra/RestrictedProduct/TopologicalSpace.lean
|
mapAlong_continuous
| null |
IsTopologicalSemiring [TopologicalSpace R] [NonUnitalNonAssocSemiring R] : Prop
extends ContinuousAdd R, ContinuousMul R
|
class
|
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
|
IsTopologicalSemiring
|
a topological semiring is a semiring `R` where addition and multiplication are continuous.
We allow for non-unital and non-associative semirings as well.
The `IsTopologicalSemiring` class should *only* be instantiated in the presence of a
`NonUnitalNonAssocSemiring` instance; if there is an instance of `NonUnitalNonAssocRing`,
then `IsTopologicalRing` should be used. Note: in the presence of `NonAssocRing`, these classes are
mathematically equivalent (see `IsTopologicalSemiring.continuousNeg_of_mul` or
`IsTopologicalSemiring.toIsTopologicalRing`).
|
IsTopologicalRing [TopologicalSpace R] [NonUnitalNonAssocRing R] : Prop
extends IsTopologicalSemiring R, ContinuousNeg R
variable {R}
|
class
|
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
|
IsTopologicalRing
|
A topological ring is a ring `R` where addition, multiplication and negation are continuous.
If `R` is a (unital) ring, then continuity of negation can be derived from continuity of
multiplication as it is multiplication with `-1`. (See
`IsTopologicalSemiring.continuousNeg_of_mul` and
`topological_semiring.to_topological_add_group`)
|
IsTopologicalSemiring.continuousNeg_of_mul [TopologicalSpace R] [NonAssocRing R]
[ContinuousMul R] : ContinuousNeg R where
continuous_neg := by
simpa using (continuous_const.mul continuous_id : Continuous fun x : R => -1 * x)
|
theorem
|
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
|
IsTopologicalSemiring.continuousNeg_of_mul
|
If `R` is a ring with a continuous multiplication, then negation is continuous as well since it
is just multiplication with `-1`.
|
IsTopologicalSemiring.toIsTopologicalRing [TopologicalSpace R] [NonAssocRing R]
(_ : IsTopologicalSemiring R) : IsTopologicalRing R where
toContinuousNeg := IsTopologicalSemiring.continuousNeg_of_mul
|
theorem
|
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
|
IsTopologicalSemiring.toIsTopologicalRing
|
If `R` is a ring which is a topological semiring, then it is automatically a topological
ring. This exists so that one can place a topological ring structure on `R` without explicitly
proving `continuous_neg`.
|
instIsTopologicalSemiring (S : NonUnitalSubsemiring R) : IsTopologicalSemiring S :=
{ S.toSubsemigroup.continuousMul, S.toAddSubmonoid.continuousAdd with }
|
instance
|
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
|
instIsTopologicalSemiring
| null |
topologicalClosure (s : NonUnitalSubsemiring R) : NonUnitalSubsemiring R :=
{ s.toSubsemigroup.topologicalClosure, s.toAddSubmonoid.topologicalClosure with
carrier := _root_.closure (s : Set R) }
@[simp]
|
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
|
topologicalClosure
|
The (topological) closure of a non-unital subsemiring of a non-unital topological semiring is
itself a non-unital subsemiring.
|
topologicalClosure_coe (s : NonUnitalSubsemiring R) :
(s.topologicalClosure : Set R) = _root_.closure (s : Set R) :=
rfl
|
theorem
|
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
|
topologicalClosure_coe
| null |
le_topologicalClosure (s : NonUnitalSubsemiring R) : s ≤ s.topologicalClosure :=
_root_.subset_closure
|
theorem
|
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
|
le_topologicalClosure
| null |
isClosed_topologicalClosure (s : NonUnitalSubsemiring R) :
IsClosed (s.topologicalClosure : Set R) := isClosed_closure
|
theorem
|
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
|
isClosed_topologicalClosure
| null |
topologicalClosure_minimal (s : NonUnitalSubsemiring R) {t : NonUnitalSubsemiring R}
(h : s ≤ t) (ht : IsClosed (t : Set R)) : s.topologicalClosure ≤ t :=
closure_minimal h ht
|
theorem
|
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
|
topologicalClosure_minimal
| null |
nonUnitalCommSemiringTopologicalClosure [T2Space R] (s : NonUnitalSubsemiring R)
(hs : ∀ x y : s, x * y = y * x) : NonUnitalCommSemiring s.topologicalClosure :=
{ NonUnitalSubsemiringClass.toNonUnitalSemiring s.topologicalClosure,
s.toSubsemigroup.commSemigroupTopologicalClosure hs with }
|
abbrev
|
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
|
nonUnitalCommSemiringTopologicalClosure
|
If a non-unital subsemiring of a non-unital topological semiring is commutative, then so is its
topological closure.
See note [reducible non-instances]
|
topologicalSemiring (S : Subsemiring R) : IsTopologicalSemiring S :=
{ S.toSubmonoid.continuousMul, S.toAddSubmonoid.continuousAdd with }
|
instance
|
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
|
topologicalSemiring
| null |
continuousSMul (s : Subsemiring R) (X) [TopologicalSpace X] [MulAction R X]
[ContinuousSMul R X] : ContinuousSMul s X :=
Submonoid.continuousSMul
|
instance
|
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
|
continuousSMul
| null |
Subsemiring.topologicalClosure (s : Subsemiring R) : Subsemiring R :=
{ s.toSubmonoid.topologicalClosure, s.toAddSubmonoid.topologicalClosure with
carrier := _root_.closure (s : Set R) }
@[simp]
|
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
|
Subsemiring.topologicalClosure
|
The (topological-space) closure of a subsemiring of a topological semiring is
itself a subsemiring.
|
Subsemiring.topologicalClosure_coe (s : Subsemiring R) :
(s.topologicalClosure : Set R) = _root_.closure (s : Set R) :=
rfl
|
theorem
|
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
|
Subsemiring.topologicalClosure_coe
| null |
Subsemiring.le_topologicalClosure (s : Subsemiring R) : s ≤ s.topologicalClosure :=
_root_.subset_closure
|
theorem
|
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
|
Subsemiring.le_topologicalClosure
| null |
Subsemiring.isClosed_topologicalClosure (s : Subsemiring R) :
IsClosed (s.topologicalClosure : Set R) := isClosed_closure
|
theorem
|
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
|
Subsemiring.isClosed_topologicalClosure
| null |
Subsemiring.topologicalClosure_minimal (s : Subsemiring R) {t : Subsemiring R} (h : s ≤ t)
(ht : IsClosed (t : Set R)) : s.topologicalClosure ≤ t :=
closure_minimal h ht
|
theorem
|
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
|
Subsemiring.topologicalClosure_minimal
| null |
Subsemiring.commSemiringTopologicalClosure [T2Space R] (s : Subsemiring R)
(hs : ∀ x y : s, x * y = y * x) : CommSemiring s.topologicalClosure :=
{ s.topologicalClosure.toSemiring, s.toSubmonoid.commMonoidTopologicalClosure hs with }
|
abbrev
|
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
|
Subsemiring.commSemiringTopologicalClosure
|
If a subsemiring of a topological semiring is commutative, then so is its
topological closure.
See note [reducible non-instances].
|
mulLeft_continuous (x : R) : Continuous (AddMonoidHom.mulLeft x) :=
continuous_const.mul continuous_id
|
theorem
|
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
|
mulLeft_continuous
|
The product topology on the Cartesian product of two topological semirings
makes the product into a topological semiring. -/
instance [NonUnitalNonAssocSemiring R] [NonUnitalNonAssocSemiring S] [IsTopologicalSemiring R]
[IsTopologicalSemiring S] : IsTopologicalSemiring (R × S) where
/-- The product topology on the Cartesian product of two topological rings
makes the product into a topological ring. -/
instance [NonUnitalNonAssocRing R] [NonUnitalNonAssocRing S] [IsTopologicalRing R]
[IsTopologicalRing S] : IsTopologicalRing (R × S) where
end
#adaptation_note /-- nightly-2024-04-08
needed to help `Pi.instIsTopologicalSemiring` -/
instance {ι : Type*} {R : ι → Type*} [∀ i, TopologicalSpace (R i)]
[∀ i, NonUnitalNonAssocSemiring (R i)] [∀ i, IsTopologicalSemiring (R i)] :
ContinuousAdd ((i : ι) → R i) :=
inferInstance
instance Pi.instIsTopologicalSemiring {ι : Type*} {R : ι → Type*} [∀ i, TopologicalSpace (R i)]
[∀ i, NonUnitalNonAssocSemiring (R i)] [∀ i, IsTopologicalSemiring (R i)] :
IsTopologicalSemiring (∀ i, R i) where
instance Pi.instIsTopologicalRing {ι : Type*} {R : ι → Type*} [∀ i, TopologicalSpace (R i)]
[∀ i, NonUnitalNonAssocRing (R i)] [∀ i, IsTopologicalRing (R i)] :
IsTopologicalRing (∀ i, R i) := ⟨⟩
section MulOpposite
open MulOpposite
instance [NonUnitalNonAssocSemiring R] [TopologicalSpace R] [ContinuousAdd R] :
ContinuousAdd Rᵐᵒᵖ :=
continuousAdd_induced opAddEquiv.symm
instance [NonUnitalNonAssocSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] :
IsTopologicalSemiring Rᵐᵒᵖ := ⟨⟩
instance [NonUnitalNonAssocRing R] [TopologicalSpace R] [ContinuousNeg R] : ContinuousNeg Rᵐᵒᵖ :=
opHomeomorph.symm.isInducing.continuousNeg fun _ => rfl
instance [NonUnitalNonAssocRing R] [TopologicalSpace R] [IsTopologicalRing R] :
IsTopologicalRing Rᵐᵒᵖ := ⟨⟩
end MulOpposite
section AddOpposite
open AddOpposite
instance [NonUnitalNonAssocSemiring R] [TopologicalSpace R] [ContinuousMul R] :
ContinuousMul Rᵃᵒᵖ :=
continuousMul_induced opMulEquiv.symm
instance [NonUnitalNonAssocSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] :
IsTopologicalSemiring Rᵃᵒᵖ := ⟨⟩
instance [NonUnitalNonAssocRing R] [TopologicalSpace R] [IsTopologicalRing R] :
IsTopologicalRing Rᵃᵒᵖ := ⟨⟩
end AddOpposite
section
variable {R : Type*} [NonUnitalNonAssocRing R] [TopologicalSpace R]
theorem IsTopologicalRing.of_addGroup_of_nhds_zero [IsTopologicalAddGroup R]
(hmul : Tendsto (uncurry ((· * ·) : R → R → R)) (𝓝 0 ×ˢ 𝓝 0) <| 𝓝 0)
(hmul_left : ∀ x₀ : R, Tendsto (fun x : R => x₀ * x) (𝓝 0) <| 𝓝 0)
(hmul_right : ∀ x₀ : R, Tendsto (fun x : R => x * x₀) (𝓝 0) <| 𝓝 0) : IsTopologicalRing R where
continuous_mul := by
refine continuous_of_continuousAt_zero₂ (AddMonoidHom.mul (R := R)) ?_ ?_ ?_ <;>
simpa only [ContinuousAt, mul_zero, zero_mul, nhds_prod_eq, AddMonoidHom.mul_apply]
theorem IsTopologicalRing.of_nhds_zero
(hadd : Tendsto (uncurry ((· + ·) : R → R → R)) (𝓝 0 ×ˢ 𝓝 0) <| 𝓝 0)
(hneg : Tendsto (fun x => -x : R → R) (𝓝 0) (𝓝 0))
(hmul : Tendsto (uncurry ((· * ·) : R → R → R)) (𝓝 0 ×ˢ 𝓝 0) <| 𝓝 0)
(hmul_left : ∀ x₀ : R, Tendsto (fun x : R => x₀ * x) (𝓝 0) <| 𝓝 0)
(hmul_right : ∀ x₀ : R, Tendsto (fun x : R => x * x₀) (𝓝 0) <| 𝓝 0)
(hleft : ∀ x₀ : R, 𝓝 x₀ = map (fun x => x₀ + x) (𝓝 0)) : IsTopologicalRing R :=
have := IsTopologicalAddGroup.of_comm_of_nhds_zero hadd hneg hleft
IsTopologicalRing.of_addGroup_of_nhds_zero hmul hmul_left hmul_right
end
variable [TopologicalSpace R]
section
variable [NonUnitalNonAssocRing R] [IsTopologicalRing R]
instance : IsTopologicalRing (ULift R) where
/-- In a topological semiring, the left-multiplication `AddMonoidHom` is continuous.
|
mulRight_continuous (x : R) : Continuous (AddMonoidHom.mulRight x) :=
continuous_id.mul continuous_const
|
theorem
|
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
|
mulRight_continuous
|
In a topological semiring, the right-multiplication `AddMonoidHom` is continuous.
|
instIsTopologicalRing (S : NonUnitalSubring R) : IsTopologicalRing S :=
{ S.toSubsemigroup.continuousMul, inferInstanceAs (IsTopologicalAddGroup S.toAddSubgroup) with }
|
instance
|
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
|
instIsTopologicalRing
| null |
topologicalClosure (S : NonUnitalSubring R) : NonUnitalSubring R :=
{ S.toSubsemigroup.topologicalClosure, S.toAddSubgroup.topologicalClosure with
carrier := _root_.closure (S : Set R) }
|
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
|
topologicalClosure
|
The (topological) closure of a non-unital subring of a non-unital topological ring is
itself a non-unital subring.
|
le_topologicalClosure (s : NonUnitalSubring R) : s ≤ s.topologicalClosure :=
_root_.subset_closure
|
theorem
|
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
|
le_topologicalClosure
| null |
isClosed_topologicalClosure (s : NonUnitalSubring R) :
IsClosed (s.topologicalClosure : Set R) := isClosed_closure
|
theorem
|
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
|
isClosed_topologicalClosure
| null |
topologicalClosure_minimal (s : NonUnitalSubring R) {t : NonUnitalSubring R} (h : s ≤ t)
(ht : IsClosed (t : Set R)) : s.topologicalClosure ≤ t :=
closure_minimal h ht
|
theorem
|
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
|
topologicalClosure_minimal
| null |
nonUnitalCommRingTopologicalClosure [T2Space R] (s : NonUnitalSubring R)
(hs : ∀ x y : s, x * y = y * x) : NonUnitalCommRing s.topologicalClosure :=
{ s.topologicalClosure.toNonUnitalRing, s.toSubsemigroup.commSemigroupTopologicalClosure hs with }
|
abbrev
|
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
|
nonUnitalCommRingTopologicalClosure
|
If a non-unital subring of a non-unital topological ring is commutative, then so is its
topological closure.
See note [reducible non-instances]
|
Subring.instIsTopologicalRing (S : Subring R) : IsTopologicalRing S :=
{ S.toSubmonoid.continuousMul, inferInstanceAs (IsTopologicalAddGroup S.toAddSubgroup) with }
|
instance
|
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
|
Subring.instIsTopologicalRing
| null |
Subring.continuousSMul (s : Subring R) (X) [TopologicalSpace X] [MulAction R X]
[ContinuousSMul R X] : ContinuousSMul s X :=
Subsemiring.continuousSMul s.toSubsemiring X
|
instance
|
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
|
Subring.continuousSMul
| null |
Subring.topologicalClosure (S : Subring R) : Subring R :=
{ S.toSubmonoid.topologicalClosure, S.toAddSubgroup.topologicalClosure with
carrier := _root_.closure (S : Set R) }
|
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
|
Subring.topologicalClosure
|
The (topological-space) closure of a subring of a topological ring is
itself a subring.
|
Subring.le_topologicalClosure (s : Subring R) : s ≤ s.topologicalClosure :=
_root_.subset_closure
|
theorem
|
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
|
Subring.le_topologicalClosure
| null |
Subring.isClosed_topologicalClosure (s : Subring R) :
IsClosed (s.topologicalClosure : Set R) := isClosed_closure
|
theorem
|
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
|
Subring.isClosed_topologicalClosure
| null |
Subring.topologicalClosure_minimal (s : Subring R) {t : Subring R} (h : s ≤ t)
(ht : IsClosed (t : Set R)) : s.topologicalClosure ≤ t :=
closure_minimal h ht
|
theorem
|
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
|
Subring.topologicalClosure_minimal
| null |
Subring.commRingTopologicalClosure [T2Space R] (s : Subring R)
(hs : ∀ x y : s, x * y = y * x) : CommRing s.topologicalClosure :=
{ s.topologicalClosure.toRing, s.toSubmonoid.commMonoidTopologicalClosure hs with }
|
abbrev
|
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
|
Subring.commRingTopologicalClosure
|
If a subring of a topological ring is commutative, then so is its topological closure.
See note [reducible non-instances].
|
RingTopology (R : Type u) [Ring R] : Type u
extends TopologicalSpace R, IsTopologicalRing R
|
structure
|
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
|
RingTopology
|
A ring topology on a ring `R` is a topology for which addition, negation and multiplication
are continuous.
|
inhabited {R : Type u} [Ring R] : Inhabited (RingTopology R) :=
⟨let _ : TopologicalSpace R := ⊤;
{ continuous_add := continuous_top
continuous_mul := continuous_top
continuous_neg := continuous_top }⟩
|
instance
|
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
|
inhabited
| null |
toTopologicalSpace_injective :
Injective (toTopologicalSpace : RingTopology R → TopologicalSpace R) := by
intro f g _; cases f; cases g; congr
@[ext]
|
theorem
|
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
|
toTopologicalSpace_injective
| null |
ext {f g : RingTopology R} (h : f.IsOpen = g.IsOpen) : f = g :=
toTopologicalSpace_injective <| TopologicalSpace.ext h
|
theorem
|
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
|
ext
| null |
coinduced {R S : Type*} [t : TopologicalSpace R] [Ring S] (f : R → S) : RingTopology S :=
sInf { b : RingTopology S | t.coinduced f ≤ b.toTopologicalSpace }
|
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
|
coinduced
|
The ordering on ring topologies on the ring `R`.
`t ≤ s` if every set open in `s` is also open in `t` (`t` is finer than `s`). -/
instance : PartialOrder (RingTopology R) :=
PartialOrder.lift RingTopology.toTopologicalSpace toTopologicalSpace_injective
private def def_sInf (S : Set (RingTopology R)) : RingTopology R :=
let _ := sInf (toTopologicalSpace '' S)
{ toContinuousAdd := continuousAdd_sInf <| forall_mem_image.2 fun t _ =>
let _ := t.1; t.toContinuousAdd
toContinuousMul := continuousMul_sInf <| forall_mem_image.2 fun t _ =>
let _ := t.1; t.toContinuousMul
toContinuousNeg := continuousNeg_sInf <| forall_mem_image.2 fun t _ =>
let _ := t.1; t.toContinuousNeg }
/-- Ring topologies on `R` form a complete lattice, with `⊥` the discrete topology and `⊤` the
indiscrete topology.
The infimum of a collection of ring topologies is the topology generated by all their open sets
(which is a ring topology).
The supremum of two ring topologies `s` and `t` is the infimum of the family of all ring topologies
contained in the intersection of `s` and `t`. -/
instance : CompleteSemilatticeInf (RingTopology R) where
sInf := def_sInf
sInf_le := fun _ a haS => sInf_le (α := TopologicalSpace R) ⟨a, ⟨haS, rfl⟩⟩
le_sInf := fun _ _ h => le_sInf (α := TopologicalSpace R) <| forall_mem_image.2 h
instance : CompleteLattice (RingTopology R) :=
completeLatticeOfCompleteSemilatticeInf _
/-- Given `f : R → S` and a topology on `R`, the coinduced ring topology on `S` is the finest
topology such that `f` is continuous and `S` is a topological ring.
|
coinduced_continuous {R S : Type*} [t : TopologicalSpace R] [Ring S] (f : R → S) :
Continuous[t, (coinduced f).toTopologicalSpace] f :=
continuous_sInf_rng.2 <| forall_mem_image.2 fun _ => continuous_iff_coinduced_le.2
|
theorem
|
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
|
coinduced_continuous
| null |
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