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 := norm_cast, simp)]
coe_one [One β] : ⇑(1 : C(α, β)) = 1 :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_one | null |
one_apply [One β] (x : α) : (1 : C(α, β)) x = 1 :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | one_apply | null |
one_comp [One γ] (g : C(α, β)) : (1 : C(β, γ)).comp g = 1 :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | one_comp | null |
comp_one [One β] (g : C(β, γ)) : g.comp (1 : C(α, β)) = const α (g 1) := rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | comp_one | null |
const_one [One β] : const α (1 : β) = 1 := rfl
/-! ### `Nat.cast` -/ | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | const_one | null |
@[simp, norm_cast]
coe_natCast [NatCast β] (n : ℕ) : ((n : C(α, β)) : α → β) = n :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_natCast | null |
natCast_apply [NatCast β] (n : ℕ) (x : α) : (n : C(α, β)) x = n :=
rfl
/-! ### `Int.cast` -/ | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | natCast_apply | null |
@[simp, norm_cast]
coe_intCast [IntCast β] (n : ℤ) : ((n : C(α, β)) : α → β) = n :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_intCast | null |
intCast_apply [IntCast β] (n : ℤ) (x : α) : (n : C(α, β)) x = n :=
rfl
/-! ### `nsmul` and `pow` -/ | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | intCast_apply | null |
instNSMul [AddMonoid β] [ContinuousAdd β] : SMul ℕ C(α, β) :=
⟨fun n f => ⟨n • ⇑f, f.continuous.nsmul n⟩⟩
@[to_additive existing] | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | instNSMul | null |
instPow [Monoid β] [ContinuousMul β] : Pow C(α, β) ℕ :=
⟨fun f n => ⟨(⇑f) ^ n, f.continuous.pow n⟩⟩
@[to_additive (attr := norm_cast) (reorder := 7 8)] | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | instPow | null |
coe_pow [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) : ⇑(f ^ n) = (⇑f) ^ n :=
rfl
@[to_additive (attr := norm_cast)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_pow | null |
pow_apply [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) (x : α) :
(f ^ n) x = f x ^ n :=
rfl
attribute [simp] coe_pow pow_apply
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | pow_apply | null |
pow_comp [Monoid γ] [ContinuousMul γ] (f : C(β, γ)) (n : ℕ) (g : C(α, β)) :
(f ^ n).comp g = f.comp g ^ n :=
rfl
attribute [simp] pow_comp
/-! ### `inv` and `neg` -/
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | pow_comp | null |
@[to_additive (attr := simp)]
coe_inv [Inv β] [ContinuousInv β] (f : C(α, β)) : ⇑f⁻¹ = (⇑f)⁻¹ :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_inv | null |
inv_apply [Inv β] [ContinuousInv β] (f : C(α, β)) (x : α) : f⁻¹ x = (f x)⁻¹ :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | inv_apply | null |
inv_comp [Inv γ] [ContinuousInv γ] (f : C(β, γ)) (g : C(α, β)) :
f⁻¹.comp g = (f.comp g)⁻¹ :=
rfl
/-! ### `div` and `sub` -/
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | inv_comp | null |
@[to_additive (attr := norm_cast, simp)]
coe_div [Div β] [ContinuousDiv β] (f g : C(α, β)) : ⇑(f / g) = f / g :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_div | null |
div_apply [Div β] [ContinuousDiv β] (f g : C(α, β)) (x : α) : (f / g) x = f x / g x :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | div_apply | null |
div_comp [Div γ] [ContinuousDiv γ] (f g : C(β, γ)) (h : C(α, β)) :
(f / g).comp h = f.comp h / g.comp h :=
rfl
/-! ### `zpow` and `zsmul` -/ | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | div_comp | null |
instZSMul [AddGroup β] [IsTopologicalAddGroup β] : SMul ℤ C(α, β) where
smul z f := ⟨z • ⇑f, f.continuous.zsmul z⟩
@[to_additive existing] | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | instZSMul | null |
instZPow [Group β] [IsTopologicalGroup β] : Pow C(α, β) ℤ where
pow f z := ⟨(⇑f) ^ z, f.continuous.zpow z⟩
@[to_additive (attr := norm_cast) (reorder := 7 8)] | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | instZPow | null |
coe_zpow [Group β] [IsTopologicalGroup β] (f : C(α, β)) (z : ℤ) : ⇑(f ^ z) = (⇑f) ^ z :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_zpow | null |
zpow_apply [Group β] [IsTopologicalGroup β] (f : C(α, β)) (z : ℤ) (x : α) :
(f ^ z) x = f x ^ z :=
rfl
attribute [simp] coe_zpow zpow_apply
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | zpow_apply | null |
zpow_comp [Group γ] [IsTopologicalGroup γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β)) :
(f ^ z).comp g = f.comp g ^ z :=
rfl
attribute [simp] zpow_comp | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | zpow_comp | null |
@[to_additive /-- The `AddSubmonoid` of continuous maps `α → β`. -/]
continuousSubmonoid (α : Type*) (β : Type*) [TopologicalSpace α] [TopologicalSpace β]
[MulOneClass β] [ContinuousMul β] : Submonoid (α → β) where
carrier := { f : α → β | Continuous f }
one_mem' := @continuous_const _ _ _ _ 1
mul_mem' fc gc := fc.mul gc | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | continuousSubmonoid | The `Submonoid` of continuous maps `α → β`. |
@[to_additive /-- The `AddSubgroup` of continuous maps `α → β`. -/]
continuousSubgroup (α : Type*) (β : Type*) [TopologicalSpace α] [TopologicalSpace β] [Group β]
[IsTopologicalGroup β] : Subgroup (α → β) :=
{ continuousSubmonoid α β with inv_mem' := fun fc => Continuous.inv fc } | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | continuousSubgroup | The subgroup of continuous maps `α → β`. |
@[to_additive (attr := simps)
/-- Coercion to a function as an `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`. -/]
coeFnMonoidHom [Monoid β] [ContinuousMul β] : C(α, β) →* α → β where
toFun f := f
map_one' := coe_one
map_mul' := coe_mul
variable (α) in | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coeFnMonoidHom | Coercion to a function as a `MonoidHom`. Similar to `MonoidHom.coeFn`. |
@[to_additive (attr := simps)
/-- Composition on the left by a (continuous) homomorphism of topological `AddMonoid`s, as an
`AddMonoidHom`. Similar to `AddMonoidHom.comp_left`. -/]
protected _root_.MonoidHom.compLeftContinuous {γ : Type*} [Monoid β] [ContinuousMul β]
[TopologicalSpace γ] [Monoid γ] [ContinuousMul γ] (g : β →* γ) (hg : Continuous g) :
C(α, β) →* C(α, γ) where
toFun f := (⟨g, hg⟩ : C(β, γ)).comp f
map_one' := ext fun _ => g.map_one
map_mul' _ _ := ext fun _ => g.map_mul _ _ | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | _root_.MonoidHom.compLeftContinuous | Composition on the left by a (continuous) homomorphism of topological monoids, as a
`MonoidHom`. Similar to `MonoidHom.compLeft`. |
@[to_additive (attr := simps)
/-- Composition on the right as an `AddMonoidHom`. Similar to `AddMonoidHom.compHom'`. -/]
compMonoidHom' {γ : Type*} [TopologicalSpace γ] [MulOneClass γ] [ContinuousMul γ]
(g : C(α, β)) : C(β, γ) →* C(α, γ) where
toFun f := f.comp g
map_one' := one_comp g
map_mul' f₁ f₂ := mul_comp f₁ f₂ g
@[to_additive (attr := simp)] | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | compMonoidHom' | Composition on the right as a `MonoidHom`. Similar to `MonoidHom.compHom'`. |
coe_prod [CommMonoid β] [ContinuousMul β] {ι : Type*} (s : Finset ι) (f : ι → C(α, β)) :
⇑(∏ i ∈ s, f i) = ∏ i ∈ s, (f i : α → β) :=
map_prod coeFnMonoidHom f s
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_prod | null |
prod_apply [CommMonoid β] [ContinuousMul β] {ι : Type*} (s : Finset ι) (f : ι → C(α, β))
(a : α) : (∏ i ∈ s, f i) a = ∏ i ∈ s, f i a := by simp
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | prod_apply | null |
@[to_additive]
instCommGroupContinuousMap [CommGroup β] [IsTopologicalGroup β] : CommGroup C(α, β) :=
coe_injective.commGroup _ coe_one coe_mul coe_inv coe_div coe_pow coe_zpow
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | instCommGroupContinuousMap | null |
@[to_additive
/-- If an infinite sum of functions in `C(α, β)` converges to `g` (for the compact-open topology),
then the pointwise sum converges to `g x` for all `x ∈ α`. -/]
hasProd_apply {γ : Type*} [CommMonoid β] [ContinuousMul β]
{f : γ → C(α, β)} {g : C(α, β)} (hf : HasProd f g) (x : α) :
HasProd (fun i : γ => f i x) (g x) := by
let ev : C(α, β) →* β := (Pi.evalMonoidHom _ x).comp coeFnMonoidHom
exact hf.map ev (continuous_eval_const x)
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | hasProd_apply | If an infinite product of functions in `C(α, β)` converges to `g`
(for the compact-open topology), then the pointwise product converges to `g x` for all `x ∈ α`. |
multipliable_apply [CommMonoid β] [ContinuousMul β] {γ : Type*} {f : γ → C(α, β)}
(hf : Multipliable f) (x : α) : Multipliable fun i : γ => f i x :=
(hasProd_apply hf.hasProd x).multipliable
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | multipliable_apply | null |
tprod_apply [T2Space β] [CommMonoid β] [ContinuousMul β] {γ : Type*} {f : γ → C(α, β)}
(hf : Multipliable f) (x : α) :
∏' i : γ, f i x = (∏' i : γ, f i) x :=
(hasProd_apply hf.hasProd x).tprod_eq | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | tprod_apply | null |
continuousSubsemiring (α : Type*) (R : Type*) [TopologicalSpace α] [TopologicalSpace R]
[NonAssocSemiring R] [IsTopologicalSemiring R] : Subsemiring (α → R) :=
{ continuousAddSubmonoid α R, continuousSubmonoid α R with } | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | continuousSubsemiring | The subsemiring of continuous maps `α → β`. |
continuousSubring (α : Type*) (R : Type*) [TopologicalSpace α] [TopologicalSpace R] [Ring R]
[IsTopologicalRing R] : Subring (α → R) :=
{ continuousAddSubgroup α R, continuousSubsemiring α R with } | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | continuousSubring | The subring of continuous maps `α → β`. |
instRing {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [Ring β]
[IsTopologicalRing β] : Ring C(α, β) :=
coe_injective.ring _ coe_zero coe_one coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul coe_pow
coe_natCast coe_intCast | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | instRing | null |
@[simps!]
protected _root_.RingHom.compLeftContinuous (α : Type*) {β : Type*} {γ : Type*}
[TopologicalSpace α]
[TopologicalSpace β] [Semiring β] [IsTopologicalSemiring β] [TopologicalSpace γ] [Semiring γ]
[IsTopologicalSemiring γ] (g : β →+* γ) (hg : Continuous g) : C(α, β) →+* C(α, γ) :=
{ g.toMonoidHom.compLeftContinuous α hg, g.toAddMonoidHom.compLeftContinuous α hg with } | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | _root_.RingHom.compLeftContinuous | Composition on the left by a (continuous) homomorphism of topological semirings, as a
`RingHom`. Similar to `RingHom.compLeft`. |
@[simps!]
coeFnRingHom {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [Semiring β]
[IsTopologicalSemiring β] : C(α, β) →+* α → β :=
{ (coeFnMonoidHom : C(α, β) →* _),
(coeFnAddMonoidHom : C(α, β) →+ _) with } | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coeFnRingHom | Coercion to a function as a `RingHom`. |
continuousSubmodule : Submodule R (α → M) :=
{ continuousAddSubgroup α M with
carrier := { f : α → M | Continuous f }
smul_mem' := fun c _ hf => hf.const_smul c } | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | continuousSubmodule | The `R`-submodule of continuous maps `α → M`. |
@[to_additive]
instSMul [SMul R M] [ContinuousConstSMul R M] : SMul R C(α, M) :=
⟨fun r f => ⟨r • ⇑f, f.continuous.const_smul r⟩⟩
@[to_additive] | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | instSMul | null |
@[to_additive (attr := simp, norm_cast)]
coe_smul [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) : ⇑(c • f) = c • ⇑f :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_smul | null |
smul_apply [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) (a : α) :
(c • f) a = c • f a :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | smul_apply | null |
smul_comp [SMul R M] [ContinuousConstSMul R M] (r : R) (f : C(β, M)) (g : C(α, β)) :
(r • f).comp g = r • f.comp g :=
rfl
@[to_additive] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | smul_comp | null |
module : Module R C(α, M) :=
Function.Injective.module R coeFnAddMonoidHom coe_injective coe_smul
variable (R) | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | module | null |
@[simps]
protected _root_.ContinuousLinearMap.compLeftContinuous (α : Type*) [TopologicalSpace α]
(g : M →L[R] M₂) : C(α, M) →L[R] C(α, M₂) where
__ := g.toLinearMap.toAddMonoidHom.compLeftContinuous α g.continuous
map_smul' := fun c _ => ext fun _ => g.map_smul' c _
cont := ContinuousMap.continuous_postcomp _ | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | _root_.ContinuousLinearMap.compLeftContinuous | Composition on the left by a continuous linear map, as a `ContinuousLinearMap`.
Similar to `LinearMap.compLeft`. |
@[simps!]
_root_.ContinuousLinearMap.const (α : Type*) [TopologicalSpace α] : M →L[R] C(α, M) where
toFun m := .const α m
map_add' _ _ := rfl
map_smul' _ _ := rfl
cont := ContinuousMap.continuous_const' | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | _root_.ContinuousLinearMap.const | The constant map `x ↦ y ↦ x` as a `ContinuousLinearMap`. |
@[simps]
coeFnLinearMap : C(α, M) →ₗ[R] α → M :=
{ (coeFnAddMonoidHom : C(α, M) →+ _) with
map_smul' := coe_smul } | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coeFnLinearMap | Coercion to a function as a `LinearMap`. |
@[simps apply]
evalCLM (x : α) : C(α, M) →L[R] M where
toFun f := f x
map_add' _ _ := add_apply _ _ x
map_smul' _ _ := smul_apply _ _ x | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | evalCLM | Evaluation at a point, as a continuous linear map. |
continuousSubalgebra : Subalgebra R (α → A) :=
{ continuousSubsemiring α A with
carrier := { f : α → A | Continuous f }
algebraMap_mem' := fun r => (continuous_const : Continuous fun _ : α => algebraMap R A r) } | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | continuousSubalgebra | The `R`-subalgebra of continuous maps `α → A`. |
ContinuousMap.C : R →+* C(α, A) where
toFun := fun c : R => ⟨fun _ : α => (algebraMap R A) c, continuous_const⟩
map_one' := by ext _; exact (algebraMap R A).map_one
map_mul' c₁ c₂ := by ext _; exact (algebraMap R A).map_mul _ _
map_zero' := by ext _; exact (algebraMap R A).map_zero
map_add' c₁ c₂ := by ext _; exact (algebraMap R A).map_add _ _
@[simp] | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | ContinuousMap.C | Continuous constant functions as a `RingHom`. |
ContinuousMap.C_apply (r : R) (a : α) : ContinuousMap.C r a = algebraMap R A r :=
rfl | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | ContinuousMap.C_apply | null |
ContinuousMap.algebra : Algebra R C(α, A) where
algebraMap := ContinuousMap.C
commutes' c f := by ext x; exact Algebra.commutes' _ _
smul_def' c f := by ext x; exact Algebra.smul_def' _ _
variable (R) | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | ContinuousMap.algebra | null |
@[simps!]
protected AlgHom.compLeftContinuous {α : Type*} [TopologicalSpace α] (g : A →ₐ[R] A₂)
(hg : Continuous g) : C(α, A) →ₐ[R] C(α, A₂) :=
{ g.toRingHom.compLeftContinuous α hg with
commutes' := fun _ => ContinuousMap.ext fun _ => g.commutes' _ }
variable (A) | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | AlgHom.compLeftContinuous | Composition on the left by a (continuous) homomorphism of topological `R`-algebras, as an
`AlgHom`. Similar to `AlgHom.compLeft`. |
@[simps]
ContinuousMap.compRightAlgHom {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
(f : C(α, β)) : C(β, A) →ₐ[R] C(α, A) where
toFun g := g.comp f
map_zero' := ext fun _ ↦ rfl
map_add' _ _ := ext fun _ ↦ rfl
map_one' := ext fun _ ↦ rfl
map_mul' _ _ := ext fun _ ↦ rfl
commutes' _ := ext fun _ ↦ rfl | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | ContinuousMap.compRightAlgHom | Precomposition of functions into a topological semiring by a continuous map is an algebra
homomorphism. |
ContinuousMap.compRightAlgHom_continuous {α β : Type*} [TopologicalSpace α]
[TopologicalSpace β] (f : C(α, β)) : Continuous (compRightAlgHom R A f) :=
continuous_precomp f
variable {A} | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | ContinuousMap.compRightAlgHom_continuous | null |
@[simps!]
ContinuousMap.coeFnAlgHom : C(α, A) →ₐ[R] α → A :=
{ (ContinuousMap.coeFnRingHom : C(α, A) →+* _) with
commutes' := fun _ => rfl }
variable {R} | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | ContinuousMap.coeFnAlgHom | Coercion to a function as an `AlgHom`. |
Subalgebra.SeparatesPoints (s : Subalgebra R C(α, A)) : Prop :=
Set.SeparatesPoints ((fun f : C(α, A) => (f : α → A)) '' (s : Set C(α, A))) | abbrev | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | Subalgebra.SeparatesPoints | A version of `Set.SeparatesPoints` for subalgebras of the continuous functions,
used for stating the Stone-Weierstrass theorem. |
Subalgebra.separatesPoints_monotone :
Monotone fun s : Subalgebra R C(α, A) => s.SeparatesPoints := fun s s' r h x y n => by
obtain ⟨f, m, w⟩ := h n
rcases m with ⟨f, ⟨m, rfl⟩⟩
exact ⟨_, ⟨f, ⟨r m, rfl⟩⟩, w⟩
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | Subalgebra.separatesPoints_monotone | null |
algebraMap_apply (k : R) (a : α) : algebraMap R C(α, A) k a = k • (1 : A) := by
rw [Algebra.algebraMap_eq_smul_one]
rfl
variable {𝕜 : Type*} [TopologicalSpace 𝕜]
variable (s : Set C(α, 𝕜)) (f : s) (x : α) | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | algebraMap_apply | null |
Set.SeparatesPointsStrongly (s : Set C(α, 𝕜)) : Prop :=
∀ (v : α → 𝕜) (x y : α), ∃ f ∈ s, (f x : 𝕜) = v x ∧ f y = v y
variable [Field 𝕜] [IsTopologicalRing 𝕜] | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | Set.SeparatesPointsStrongly | A set of continuous maps "separates points strongly"
if for each pair of distinct points there is a function with specified values on them.
We give a slightly unusual formulation, where the specified values are given by some
function `v`, and we ask `f x = v x ∧ f y = v y`. This avoids needing a hypothesis `x ≠ y`.
In fact, this definition would work perfectly well for a set of non-continuous functions,
but as the only current use case is in the Stone-Weierstrass theorem,
writing it this way avoids having to deal with casts inside the set.
(This may need to change if we do Stone-Weierstrass on non-compact spaces,
where the functions would be continuous functions vanishing at infinity.) |
Subalgebra.SeparatesPoints.strongly {s : Subalgebra 𝕜 C(α, 𝕜)} (h : s.SeparatesPoints) :
(s : Set C(α, 𝕜)).SeparatesPointsStrongly := fun v x y => by
by_cases n : x = y
· subst n
exact ⟨_, (v x • (1 : s) : s).prop, mul_one _, mul_one _⟩
obtain ⟨_, ⟨f, hf, rfl⟩, hxy⟩ := h n
replace hxy : f x - f y ≠ 0 := sub_ne_zero_of_ne hxy
let a := v x
let b := v y
let f' : s :=
((b - a) * (f x - f y)⁻¹) • (algebraMap _ s (f x) - (⟨f, hf⟩ : s)) + algebraMap _ s a
refine ⟨f', f'.prop, ?_, ?_⟩
· simp [a, b, f']
· simp [a, b, f', inv_mul_cancel_right₀ hxy] | theorem | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | Subalgebra.SeparatesPoints.strongly | Working in continuous functions into a topological field,
a subalgebra of functions that separates points also separates points strongly.
By the hypothesis, we can find a function `f` so `f x ≠ f y`.
By an affine transformation in the field we can arrange so that `f x = a` and `f x = b`. |
ContinuousMap.subsingleton_subalgebra (α : Type*) [TopologicalSpace α] (R : Type*)
[CommSemiring R] [TopologicalSpace R] [IsTopologicalSemiring R] [Subsingleton α] :
Subsingleton (Subalgebra R C(α, R)) :=
⟨fun s₁ s₂ => by
cases isEmpty_or_nonempty α
· have : Subsingleton C(α, R) := DFunLike.coe_injective.subsingleton
subsingleton
· inhabit α
ext f
have h : f = algebraMap R C(α, R) (f default) := by
ext x'
simp only [mul_one, Algebra.id.smul_eq_mul, algebraMap_apply]
congr
simp [eq_iff_true_of_subsingleton]
rw [h]
simp only [Subalgebra.algebraMap_mem]⟩ | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | ContinuousMap.subsingleton_subalgebra | null |
instSMul' : SMul C(α, R) C(α, M) :=
⟨fun f g => ⟨fun x => f x • g x, Continuous.smul f.2 g.2⟩⟩ | instance | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | instSMul' | null |
@[simp] coe_smul' (f : C(α, R)) (g : C(α, M)) :
⇑(f • g) = ⇑f • ⇑g :=
rfl | lemma | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | coe_smul' | Coercion to a function for a scalar-valued continuous map multiplying a vector-valued one
(as opposed to `ContinuousMap.coe_smul` which is multiplication by a constant scalar). |
@[simps]
ContinuousMap.evalAlgHom (x : X) : C(X, R) →ₐ[S] R where
toFun f := f x
map_zero' := rfl
map_one' := rfl
map_add' _ _ := rfl
map_mul' _ _ := rfl
commutes' _ := rfl | def | Topology | [
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Topology.Algebra.Module.LinearMap",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/ContinuousMap/Algebra.lean | ContinuousMap.evalAlgHom | Evaluation of a scalar-valued continuous map multiplying a vector-valued one
(as opposed to `ContinuousMap.smul_apply` which is multiplication by a constant scalar). -/
-- (this doesn't need to be @[simp] since it can be derived from `coe_smul'` and `Pi.smul_apply'`)
lemma smul_apply' (f : C(α, R)) (g : C(α, M)) (x : α) :
(f • g) x = f x • g x :=
rfl
instance module' [IsTopologicalSemiring R] [ContinuousAdd M] :
Module C(α, R) C(α, M) where
smul := (· • ·)
smul_add c f g := by ext x; exact smul_add (c x) (f x) (g x)
add_smul c₁ c₂ f := by ext x; exact add_smul (c₁ x) (c₂ x) (f x)
mul_smul c₁ c₂ f := by ext x; exact mul_smul (c₁ x) (c₂ x) (f x)
one_smul f := by ext x; exact one_smul R (f x)
zero_smul f := by ext x; exact zero_smul _ _
smul_zero r := by ext x; exact smul_zero _
end ContinuousMap
end ModuleOverContinuousFunctions
/-! ### Evaluation as a bundled map -/
variable {X : Type*} (S R : Type*) [TopologicalSpace X] [CommSemiring S] [CommSemiring R]
variable [Algebra S R] [TopologicalSpace R] [IsTopologicalSemiring R]
/-- Evaluation of continuous maps at a point, bundled as an algebra homomorphism. |
map_continuousAt (f : F) (a : α) : ContinuousAt f a :=
(map_continuous f).continuousAt | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | map_continuousAt | null |
map_continuousWithinAt (f : F) (s : Set α) (a : α) : ContinuousWithinAt f s a :=
(map_continuous f).continuousWithinAt | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | map_continuousWithinAt | null |
protected continuousAt (f : C(α, β)) (x : α) : ContinuousAt f x :=
map_continuousAt f x | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | continuousAt | Deprecated. Use `map_continuousAt` instead. |
map_specializes (f : C(α, β)) {x y : α} (h : x ⤳ y) : f x ⤳ f y :=
h.map f.2 | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | map_specializes | null |
@[simps]
equivFnOfDiscrete : C(α, β) ≃ (α → β) :=
⟨fun f => f,
fun f => ⟨f, continuous_of_discreteTopology⟩,
fun _ => by ext; rfl,
fun _ => by ext; rfl⟩
@[simp] lemma coe_equivFnOfDiscrete : ⇑equivFnOfDiscrete = (DFunLike.coe : C(α, β) → α → β) := rfl
@[simp] lemma equivFnOfDiscrete_symm_apply (f : α → β) : equivFnOfDiscrete.symm f = f := rfl | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | equivFnOfDiscrete | The continuous functions from `α` to `β` are the same as the plain functions when `α` is discrete. |
protected id : C(α, α) where
toFun := id
@[simp, norm_cast] | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | id | The identity as a continuous map. |
coe_id : ⇑(ContinuousMap.id α) = id :=
rfl | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | coe_id | null |
const (b : β) : C(α, β) where
toFun := fun _ : α => b
@[simp] | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | const | The constant map as a continuous map. |
coe_const (b : β) : ⇑(const α b) = Function.const α b :=
rfl | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | coe_const | null |
@[simps -fullyApplied]
constPi : C(β, α → β) where
toFun b := Function.const α b | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | constPi | `Function.const α b` as a bundled continuous function of `b`. |
@[simp]
id_apply (a : α) : ContinuousMap.id α a = a :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | id_apply | null |
const_apply (b : β) (a : α) : const α b a = b :=
rfl | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | const_apply | null |
comp (f : C(β, γ)) (g : C(α, β)) : C(α, γ) where
toFun := f ∘ g
@[simp] | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | comp | The composition of continuous maps, as a continuous map. |
coe_comp (f : C(β, γ)) (g : C(α, β)) : ⇑(comp f g) = f ∘ g :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | coe_comp | null |
comp_apply (f : C(β, γ)) (g : C(α, β)) (a : α) : comp f g a = f (g a) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | comp_apply | null |
comp_assoc (f : C(γ, δ)) (g : C(β, γ)) (h : C(α, β)) :
(f.comp g).comp h = f.comp (g.comp h) :=
rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | comp_assoc | null |
id_comp (f : C(α, β)) : (ContinuousMap.id _).comp f = f :=
ext fun _ => rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | id_comp | null |
comp_id (f : C(α, β)) : f.comp (ContinuousMap.id _) = f :=
ext fun _ => rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | comp_id | null |
const_comp (c : γ) (f : C(α, β)) : (const β c).comp f = const α c :=
ext fun _ => rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | const_comp | null |
comp_const (f : C(β, γ)) (b : β) : f.comp (const α b) = const α (f b) :=
ext fun _ => rfl
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | comp_const | null |
cancel_right {f₁ f₂ : C(β, γ)} {g : C(α, β)} (hg : Surjective g) :
f₁.comp g = f₂.comp g ↔ f₁ = f₂ :=
⟨fun h => ext <| hg.forall.2 <| DFunLike.ext_iff.1 h, congr_arg (ContinuousMap.comp · g)⟩
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | cancel_right | null |
cancel_left {f : C(β, γ)} {g₁ g₂ : C(α, β)} (hf : Injective f) :
f.comp g₁ = f.comp g₂ ↔ g₁ = g₂ :=
⟨fun h => ext fun a => hf <| by rw [← comp_apply, h, comp_apply], congr_arg _⟩ | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | cancel_left | null |
@[simps]
_root_.Homeomorph.continuousMapCongr {X₁ X₂ Y₁ Y₂ : Type*}
[TopologicalSpace X₁] [TopologicalSpace X₂]
[TopologicalSpace Y₁] [TopologicalSpace Y₂]
(e : X₁ ≃ₜ X₂) (e' : Y₁ ≃ₜ Y₂) :
C(X₁, Y₁) ≃ C(X₂, Y₂) where
toFun f := ContinuousMap.comp ⟨_, e'.continuous⟩ (f.comp ⟨_, e.symm.continuous⟩)
invFun g := ContinuousMap.comp ⟨_, e'.symm.continuous⟩ (g.comp ⟨_, e.continuous⟩)
left_inv _ := by aesop
right_inv _ := by aesop | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | _root_.Homeomorph.continuousMapCongr | The bijection `C(X₁, Y₁) ≃ C(X₂, Y₂)` induced by homeomorphisms
`e : X₁ ≃ₜ X₂` and `e' : Y₁ ≃ₜ Y₂`. |
@[simps -fullyApplied]
fst : C(α × β, α) where
toFun := Prod.fst | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | fst | `Prod.fst : (x, y) ↦ x` as a bundled continuous map. |
@[simps -fullyApplied]
snd : C(α × β, β) where
toFun := Prod.snd | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | snd | `Prod.snd : (x, y) ↦ y` as a bundled continuous map. |
prodMk (f : C(α, β₁)) (g : C(α, β₂)) : C(α, β₁ × β₂) where
toFun x := (f x, g x) | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | prodMk | Given two continuous maps `f` and `g`, this is the continuous map `x ↦ (f x, g x)`. |
@[simps]
prodMap (f : C(α₁, α₂)) (g : C(β₁, β₂)) : C(α₁ × β₁, α₂ × β₂) where
toFun := Prod.map f g
@[simp] | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | prodMap | Given two continuous maps `f` and `g`, this is the continuous map `(x, y) ↦ (f x, g y)`. |
prod_eval (f : C(α, β₁)) (g : C(α, β₂)) (a : α) : (prodMk f g) a = (f a, g a) :=
rfl | theorem | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | prod_eval | null |
@[simps!]
prodSwap : C(α × β, β × α) := .prodMk .snd .fst | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | prodSwap | `Prod.swap` bundled as a `ContinuousMap`. |
@[simps apply]
sigmaMk (i : I) : C(X i, Σ i, X i) where
toFun := Sigma.mk i | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | sigmaMk | `Sigma.mk i` as a bundled continuous map. |
@[simps]
sigma (f : ∀ i, C(X i, A)) : C((Σ i, X i), A) where
toFun ig := f ig.fst ig.snd
variable (A X) in | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | sigma | To give a continuous map out of a disjoint union, it suffices to give a continuous map out of
each term. This is `Sigma.uncurry` for continuous maps. |
@[simps]
sigmaEquiv : (∀ i, C(X i, A)) ≃ C((Σ i, X i), A) where
toFun := sigma
invFun f i := f.comp (sigmaMk i) | def | Topology | [
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousMap.Defs",
"Mathlib.Topology.Homeomorph.Defs",
"Mathlib.Topology.Separation.Hausdorff"
] | Mathlib/Topology/ContinuousMap/Basic.lean | sigmaEquiv | Giving a continuous map out of a disjoint union is the same as giving a continuous map out of
each term. This is a version of `Equiv.piCurry` for continuous maps. |
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