File: Mathlib/Topology/ContinuousFunction/Algebra.lean

full_name: ContinuousFunctions.instCoeFunElemForallSetOfContinuous
code: instance : CoeFun { f : α → β | Continuous f } fun _ => α → β 
goal: CoeFun ↑{f | Continuous f} fun x => α → β

full_name: ContinuousMap.instAdd
code: @[to_additive]
instance instMul [Mul β] [ContinuousMul β] : Mul C(α, β) 
goal: Add C(α, β)

full_name: ContinuousMap.instMul
code: @[to_additive]
instance instMul [Mul β] [ContinuousMul β] : Mul C(α, β) 
goal: Mul C(α, β)

full_name: ContinuousMap.coe_add
code: @[to_additive (attr := norm_cast, simp)]
theorem coe_mul [Mul β] [ContinuousMul β] (f g : C(α, β)) : ⇑(f * g) = f * g 
goal: ⇑(f + g) = ⇑f + ⇑g

full_name: ContinuousMap.coe_mul
code: @[to_additive (attr := norm_cast, simp)]
theorem coe_mul [Mul β] [ContinuousMul β] (f g : C(α, β)) : ⇑(f * g) = f * g 
goal: ⇑(f * g) = ⇑f * ⇑g

full_name: ContinuousMap.add_apply
code: @[to_additive (attr := simp)]
theorem mul_apply [Mul β] [ContinuousMul β] (f g : C(α, β)) (x : α) : (f * g) x = f x * g x 
goal: (f + g) x = f x + g x

full_name: ContinuousMap.mul_apply
code: @[to_additive (attr := simp)]
theorem mul_apply [Mul β] [ContinuousMul β] (f g : C(α, β)) (x : α) : (f * g) x = f x * g x 
goal: (f * g) x = f x * g x

full_name: ContinuousMap.mul_comp
code: @[to_additive (attr := simp)]
theorem mul_comp [Mul γ] [ContinuousMul γ] (f₁ f₂ : C(β, γ)) (g : C(α, β)) :
    (f₁ * f₂).comp g = f₁.comp g * f₂.comp g 
goal: (f₁ * f₂).comp g = f₁.comp g * f₂.comp g

full_name: ContinuousMap.add_comp
code: @[to_additive (attr := simp)]
theorem mul_comp [Mul γ] [ContinuousMul γ] (f₁ f₂ : C(β, γ)) (g : C(α, β)) :
    (f₁ * f₂).comp g = f₁.comp g * f₂.comp g 
goal: (f₁ + f₂).comp g = f₁.comp g + f₂.comp g

full_name: ContinuousMap.instOne
code: @[to_additive]
instance [One β] : One C(α, β) 
goal: One C(α, β)

full_name: ContinuousMap.instZero
code: @[to_additive]
instance [One β] : One C(α, β) 
goal: Zero C(α, β)

full_name: ContinuousMap.coe_zero
code: @[to_additive (attr := norm_cast, simp)]
theorem coe_one [One β] : ⇑(1 : C(α, β)) = 1 
goal: ⇑0 = 0

full_name: ContinuousMap.coe_one
code: @[to_additive (attr := norm_cast, simp)]
theorem coe_one [One β] : ⇑(1 : C(α, β)) = 1 
goal: ⇑1 = 1

full_name: ContinuousMap.zero_apply
code: @[to_additive (attr := simp)]
theorem one_apply [One β] (x : α) : (1 : C(α, β)) x = 1 
goal: 0 x = 0

full_name: ContinuousMap.one_apply
code: @[to_additive (attr := simp)]
theorem one_apply [One β] (x : α) : (1 : C(α, β)) x = 1 
goal: 1 x = 1

full_name: ContinuousMap.one_comp
code: @[to_additive (attr := simp)]
theorem one_comp [One γ] (g : C(α, β)) : (1 : C(β, γ)).comp g = 1 
goal: ContinuousMap.comp 1 g = 1

full_name: ContinuousMap.zero_comp
code: @[to_additive (attr := simp)]
theorem one_comp [One γ] (g : C(α, β)) : (1 : C(β, γ)).comp g = 1 
goal: ContinuousMap.comp 0 g = 0

full_name: ContinuousMap.instNatCast
code: instance [NatCast β] : NatCast C(α, β) 
goal: NatCast C(α, β)

full_name: ContinuousMap.coe_natCast
code: @[simp, norm_cast]
theorem coe_natCast [NatCast β] (n : ℕ) : ((n : C(α, β)) : α → β) = n 
goal: ⇑↑n = ↑n

full_name: ContinuousMap.natCast_apply
code: @[simp]
theorem natCast_apply [NatCast β] (n : ℕ) (x : α) : (n : C(α, β)) x = n 
goal: ↑n x = ↑n

full_name: ContinuousMap.instIntCast
code: instance [IntCast β] : IntCast C(α, β) 
goal: IntCast C(α, β)

full_name: ContinuousMap.coe_intCast
code: @[simp, norm_cast]
theorem coe_intCast [IntCast β] (n : ℤ) : ((n : C(α, β)) : α → β) = n 
goal: ⇑↑n = ↑n

full_name: ContinuousMap.intCast_apply
code: @[simp]
theorem intCast_apply [IntCast β] (n : ℤ) (x : α) : (n : C(α, β)) x = n 
goal: ↑n x = ↑n

full_name: ContinuousMap.instNSMul
code: instance instNSMul [AddMonoid β] [ContinuousAdd β] : SMul ℕ C(α, β) 
goal: SMul ℕ C(α, β)

full_name: ContinuousMap.instPow
code: @[to_additive existing]
instance instPow [Monoid β] [ContinuousMul β] : Pow C(α, β) ℕ 
goal: Pow C(α, β) ℕ

full_name: ContinuousMap.coe_nsmul
code: @[to_additive (attr := norm_cast) (reorder := 7 8)]
theorem coe_pow [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) : ⇑(f ^ n) = (⇑f) ^ n 
goal: ⇑(n • f) = n • ⇑f

full_name: ContinuousMap.coe_pow
code: @[to_additive (attr := norm_cast) (reorder := 7 8)]
theorem coe_pow [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) : ⇑(f ^ n) = (⇑f) ^ n 
goal: ⇑(f ^ n) = ⇑f ^ n

full_name: ContinuousMap.nsmul_apply
code: @[to_additive (attr := norm_cast)]
theorem pow_apply [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) (x : α) :
    (f ^ n) x = f x ^ n 
goal: (n • f) x = n • f x

full_name: ContinuousMap.pow_apply
code: @[to_additive (attr := norm_cast)]
theorem pow_apply [Monoid β] [ContinuousMul β] (f : C(α, β)) (n : ℕ) (x : α) :
    (f ^ n) x = f x ^ n 
goal: (f ^ n) x = f x ^ n

full_name: ContinuousMap.nsmul_comp
code: @[to_additive]
theorem pow_comp [Monoid γ] [ContinuousMul γ] (f : C(β, γ)) (n : ℕ) (g : C(α, β)) :
    (f ^ n).comp g = f.comp g ^ n 
goal: (n • f).comp g = n • f.comp g

full_name: ContinuousMap.pow_comp
code: @[to_additive]
theorem pow_comp [Monoid γ] [ContinuousMul γ] (f : C(β, γ)) (n : ℕ) (g : C(α, β)) :
    (f ^ n).comp g = f.comp g ^ n 
goal: (f ^ n).comp g = f.comp g ^ n

full_name: ContinuousMap.instInvOfContinuousInv
code: @[to_additive]
instance [Inv β] [ContinuousInv β] : Inv C(α, β) 
goal: Inv C(α, β)

full_name: ContinuousMap.instNegOfContinuousNeg
code: @[to_additive]
instance [Inv β] [ContinuousInv β] : Inv C(α, β) 
goal: Neg C(α, β)

full_name: ContinuousMap.coe_neg
code: @[to_additive (attr := simp)]
theorem coe_inv [Inv β] [ContinuousInv β] (f : C(α, β)) : ⇑f⁻¹ = (⇑f)⁻¹ 
goal: ⇑(-f) = -⇑f

full_name: ContinuousMap.coe_inv
code: @[to_additive (attr := simp)]
theorem coe_inv [Inv β] [ContinuousInv β] (f : C(α, β)) : ⇑f⁻¹ = (⇑f)⁻¹ 
goal: ⇑f⁻¹ = (⇑f)⁻¹

full_name: ContinuousMap.neg_apply
code: @[to_additive (attr := simp)]
theorem inv_apply [Inv β] [ContinuousInv β] (f : C(α, β)) (x : α) : f⁻¹ x = (f x)⁻¹ 
goal: (-f) x = -f x

full_name: ContinuousMap.inv_apply
code: @[to_additive (attr := simp)]
theorem inv_apply [Inv β] [ContinuousInv β] (f : C(α, β)) (x : α) : f⁻¹ x = (f x)⁻¹ 
goal: f⁻¹ x = (f x)⁻¹

full_name: ContinuousMap.inv_comp
code: @[to_additive (attr := simp)]
theorem inv_comp [Inv γ] [ContinuousInv γ] (f : C(β, γ)) (g : C(α, β)) :
    f⁻¹.comp g = (f.comp g)⁻¹ 
goal: f⁻¹.comp g = (f.comp g)⁻¹

full_name: ContinuousMap.neg_comp
code: @[to_additive (attr := simp)]
theorem inv_comp [Inv γ] [ContinuousInv γ] (f : C(β, γ)) (g : C(α, β)) :
    f⁻¹.comp g = (f.comp g)⁻¹ 
goal: (-f).comp g = -f.comp g

full_name: ContinuousMap.instDivOfContinuousDiv
code: @[to_additive]
instance [Div β] [ContinuousDiv β] : Div C(α, β) 
goal: Div C(α, β)

full_name: ContinuousMap.instSubOfContinuousSub
code: @[to_additive]
instance [Div β] [ContinuousDiv β] : Div C(α, β) 
goal: Sub C(α, β)

full_name: ContinuousMap.coe_div
code: @[to_additive (attr := norm_cast, simp)]
theorem coe_div [Div β] [ContinuousDiv β] (f g : C(α, β)) : ⇑(f / g) = f / g 
goal: ⇑(f / g) = ⇑f / ⇑g

full_name: ContinuousMap.coe_sub
code: @[to_additive (attr := norm_cast, simp)]
theorem coe_div [Div β] [ContinuousDiv β] (f g : C(α, β)) : ⇑(f / g) = f / g 
goal: ⇑(f - g) = ⇑f - ⇑g

full_name: ContinuousMap.div_apply
code: @[to_additive (attr := simp)]
theorem div_apply [Div β] [ContinuousDiv β] (f g : C(α, β)) (x : α) : (f / g) x = f x / g x 
goal: (f / g) x = f x / g x

full_name: ContinuousMap.sub_apply
code: @[to_additive (attr := simp)]
theorem div_apply [Div β] [ContinuousDiv β] (f g : C(α, β)) (x : α) : (f / g) x = f x / g x 
goal: (f - g) x = f x - g x

full_name: ContinuousMap.sub_comp
code: @[to_additive (attr := simp)]
theorem 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 
goal: (f - g).comp h = f.comp h - g.comp h

full_name: ContinuousMap.div_comp
code: @[to_additive (attr := simp)]
theorem 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 
goal: (f / g).comp h = f.comp h / g.comp h

full_name: ContinuousMap.instZSMul
code: instance instZSMul [AddGroup β] [TopologicalAddGroup β] : SMul ℤ C(α, β) 
goal: SMul ℤ C(α, β)

full_name: ContinuousMap.instZPow
code: @[to_additive existing]
instance instZPow [Group β] [TopologicalGroup β] : Pow C(α, β) ℤ 
goal: Pow C(α, β) ℤ

full_name: ContinuousMap.coe_zsmul
code: @[to_additive (attr := norm_cast) (reorder := 7 8)]
theorem coe_zpow [Group β] [TopologicalGroup β] (f : C(α, β)) (z : ℤ) : ⇑(f ^ z) = (⇑f) ^ z 
goal: ⇑(z • f) = z • ⇑f

full_name: ContinuousMap.coe_zpow
code: @[to_additive (attr := norm_cast) (reorder := 7 8)]
theorem coe_zpow [Group β] [TopologicalGroup β] (f : C(α, β)) (z : ℤ) : ⇑(f ^ z) = (⇑f) ^ z 
goal: ⇑(f ^ z) = ⇑f ^ z

full_name: ContinuousMap.zsmul_apply
code: @[to_additive]
theorem zpow_apply [Group β] [TopologicalGroup β] (f : C(α, β)) (z : ℤ) (x : α) :
    (f ^ z) x = f x ^ z 
goal: (z • f) x = z • f x

full_name: ContinuousMap.zpow_apply
code: @[to_additive]
theorem zpow_apply [Group β] [TopologicalGroup β] (f : C(α, β)) (z : ℤ) (x : α) :
    (f ^ z) x = f x ^ z 
goal: (f ^ z) x = f x ^ z

full_name: ContinuousMap.zpow_comp
code: @[to_additive]
theorem zpow_comp [Group γ] [TopologicalGroup γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β)) :
    (f ^ z).comp g = f.comp g ^ z 
goal: (f ^ z).comp g = f.comp g ^ z

full_name: ContinuousMap.zsmul_comp
code: @[to_additive]
theorem zpow_comp [Group γ] [TopologicalGroup γ] (f : C(β, γ)) (z : ℤ) (g : C(α, β)) :
    (f ^ z).comp g = f.comp g ^ z 
goal: (z • f).comp g = z • f.comp g

full_name: continuousSubmonoid
code: /-- The `Submonoid` of continuous maps `α → β`. -/
@[to_additive "The `AddSubmonoid` of continuous maps `α → β`. "]
def continuousSubmonoid (α : Type*) (β : Type*) [TopologicalSpace α] [TopologicalSpace β]
    [MulOneClass β] [ContinuousMul β] : Submonoid (α → β) 
goal: Submonoid (α → β)

full_name: continuousAddSubmonoid
code: /-- The `Submonoid` of continuous maps `α → β`. -/
@[to_additive "The `AddSubmonoid` of continuous maps `α → β`. "]
def continuousSubmonoid (α : Type*) (β : Type*) [TopologicalSpace α] [TopologicalSpace β]
    [MulOneClass β] [ContinuousMul β] : Submonoid (α → β) 
goal: AddSubmonoid (α → β)

full_name: continuousAddSubgroup
code: /-- The subgroup of continuous maps `α → β`. -/
@[to_additive "The `AddSubgroup` of continuous maps `α → β`. "]
def continuousSubgroup (α : Type*) (β : Type*) [TopologicalSpace α] [TopologicalSpace β] [Group β]
    [TopologicalGroup β] : Subgroup (α → β) 
goal: AddSubgroup (α → β)

full_name: continuousSubgroup
code: /-- The subgroup of continuous maps `α → β`. -/
@[to_additive "The `AddSubgroup` of continuous maps `α → β`. "]
def continuousSubgroup (α : Type*) (β : Type*) [TopologicalSpace α] [TopologicalSpace β] [Group β]
    [TopologicalGroup β] : Subgroup (α → β) 
goal: Subgroup (α → β)

full_name: ContinuousMap.instSemigroupOfContinuousMul
code: @[to_additive]
instance [Semigroup β] [ContinuousMul β] : Semigroup C(α, β) 
goal: Semigroup C(α, β)

full_name: ContinuousMap.instAddSemigroupOfContinuousAdd
code: @[to_additive]
instance [Semigroup β] [ContinuousMul β] : Semigroup C(α, β) 
goal: AddSemigroup C(α, β)

full_name: ContinuousMap.instAddCommSemigroupOfContinuousAdd
code: @[to_additive]
instance [CommSemigroup β] [ContinuousMul β] : CommSemigroup C(α, β) 
goal: AddCommSemigroup C(α, β)

full_name: ContinuousMap.instCommSemigroupOfContinuousMul
code: @[to_additive]
instance [CommSemigroup β] [ContinuousMul β] : CommSemigroup C(α, β) 
goal: CommSemigroup C(α, β)

full_name: ContinuousMap.instAddZeroClassOfContinuousAdd
code: @[to_additive]
instance [MulOneClass β] [ContinuousMul β] : MulOneClass C(α, β) 
goal: AddZeroClass C(α, β)

full_name: ContinuousMap.instMulOneClassOfContinuousMul
code: @[to_additive]
instance [MulOneClass β] [ContinuousMul β] : MulOneClass C(α, β) 
goal: MulOneClass C(α, β)

full_name: ContinuousMap.instMulZeroClassOfContinuousMul
code: instance [MulZeroClass β] [ContinuousMul β] : MulZeroClass C(α, β) 
goal: MulZeroClass C(α, β)

full_name: ContinuousMap.instSemigroupWithZeroOfContinuousMul
code: instance [SemigroupWithZero β] [ContinuousMul β] : SemigroupWithZero C(α, β) 
goal: SemigroupWithZero C(α, β)

full_name: ContinuousMap.instMonoidOfContinuousMul
code: @[to_additive]
instance [Monoid β] [ContinuousMul β] : Monoid C(α, β) 
goal: Monoid C(α, β)

full_name: ContinuousMap.instAddMonoidOfContinuousAdd
code: @[to_additive]
instance [Monoid β] [ContinuousMul β] : Monoid C(α, β) 
goal: AddMonoid C(α, β)

full_name: ContinuousMap.instMonoidWithZeroOfContinuousMul
code: instance [MonoidWithZero β] [ContinuousMul β] : MonoidWithZero C(α, β) 
goal: MonoidWithZero C(α, β)

full_name: ContinuousMap.instCommMonoidOfContinuousMul
code: @[to_additive]
instance [CommMonoid β] [ContinuousMul β] : CommMonoid C(α, β) 
goal: CommMonoid C(α, β)

full_name: ContinuousMap.instAddCommMonoidOfContinuousAdd
code: @[to_additive]
instance [CommMonoid β] [ContinuousMul β] : CommMonoid C(α, β) 
goal: AddCommMonoid C(α, β)

full_name: ContinuousMap.instCommMonoidWithZeroOfContinuousMul
code: instance [CommMonoidWithZero β] [ContinuousMul β] : CommMonoidWithZero C(α, β) 
goal: CommMonoidWithZero C(α, β)

full_name: ContinuousMap.instContinuousMulOfLocallyCompactSpace
code: @[to_additive]
instance [LocallyCompactSpace α] [Mul β] [ContinuousMul β] : ContinuousMul C(α, β) 
goal: ContinuousMul C(α, β)

full_name: ContinuousMap.instContinuousAddOfLocallyCompactSpace
code: @[to_additive]
instance [LocallyCompactSpace α] [Mul β] [ContinuousMul β] : ContinuousMul C(α, β) 
goal: ContinuousAdd C(α, β)

full_name: ContinuousMap.coeFnAddMonoidHom
code: /-- Coercion to a function as a `MonoidHom`. Similar to `MonoidHom.coeFn`. -/
@[to_additive (attr := simps)
  "Coercion to a function as an `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`." ]
def coeFnMonoidHom [Monoid β] [ContinuousMul β] : C(α, β) →* α → β 
goal: C(α, β) →+ α → β

full_name: ContinuousMap.coeFnMonoidHom_apply
code: /-- Coercion to a function as a `MonoidHom`. Similar to `MonoidHom.coeFn`. -/
@[to_additive (attr := simps)
  "Coercion to a function as an `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`." ]
def coeFnMonoidHom [Monoid β] [ContinuousMul β] : C(α, β) →* α → β 
goal: ContinuousMap.coeFnMonoidHom f a = f a

full_name: ContinuousMap.coeFnMonoidHom
code: /-- Coercion to a function as a `MonoidHom`. Similar to `MonoidHom.coeFn`. -/
@[to_additive (attr := simps)
  "Coercion to a function as an `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`." ]
def coeFnMonoidHom [Monoid β] [ContinuousMul β] : C(α, β) →* α → β 
goal: C(α, β) →* α → β

full_name: ContinuousMap.coeFnAddMonoidHom_apply
code: /-- Coercion to a function as a `MonoidHom`. Similar to `MonoidHom.coeFn`. -/
@[to_additive (attr := simps)
  "Coercion to a function as an `AddMonoidHom`. Similar to `AddMonoidHom.coeFn`." ]
def coeFnMonoidHom [Monoid β] [ContinuousMul β] : C(α, β) →* α → β 
goal: ContinuousMap.coeFnAddMonoidHom f a = f a

full_name: MonoidHom.compLeftContinuous_apply
code: /-- 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 left by a (continuous) homomorphism of topological `AddMonoid`s, as an
`AddMonoidHom`. Similar to `AddMonoidHom.comp_left`."]
protected def _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) 
goal: (MonoidHom.compLeftContinuous α g hg) f = { toFun := ⇑g, continuous_toFun := hg }.comp f

full_name: AddMonoidHom.compLeftContinuous_apply
code: /-- 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 left by a (continuous) homomorphism of topological `AddMonoid`s, as an
`AddMonoidHom`. Similar to `AddMonoidHom.comp_left`."]
protected def _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) 
goal: (AddMonoidHom.compLeftContinuous α g hg) f = { toFun := ⇑g, continuous_toFun := hg }.comp f

full_name: AddMonoidHom.compLeftContinuous
code: /-- 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 left by a (continuous) homomorphism of topological `AddMonoid`s, as an
`AddMonoidHom`. Similar to `AddMonoidHom.comp_left`."]
protected def _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) 
goal: C(α, β) →+ C(α, γ)

full_name: MonoidHom.compLeftContinuous
code: /-- 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 left by a (continuous) homomorphism of topological `AddMonoid`s, as an
`AddMonoidHom`. Similar to `AddMonoidHom.comp_left`."]
protected def _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) 
goal: C(α, β) →* C(α, γ)

full_name: ContinuousMap.compMonoidHom'
code: /-- Composition on the right as a `MonoidHom`. Similar to `MonoidHom.compHom'`. -/
@[to_additive (attr := simps)
      "Composition on the right as an `AddMonoidHom`. Similar to `AddMonoidHom.compHom'`."]
def compMonoidHom' {γ : Type*} [TopologicalSpace γ] [MulOneClass γ] [ContinuousMul γ]
    (g : C(α, β)) : C(β, γ) →* C(α, γ) 
goal: C(β, γ) →* C(α, γ)

full_name: ContinuousMap.compAddMonoidHom'_apply
code: /-- Composition on the right as a `MonoidHom`. Similar to `MonoidHom.compHom'`. -/
@[to_additive (attr := simps)
      "Composition on the right as an `AddMonoidHom`. Similar to `AddMonoidHom.compHom'`."]
def compMonoidHom' {γ : Type*} [TopologicalSpace γ] [MulOneClass γ] [ContinuousMul γ]
    (g : C(α, β)) : C(β, γ) →* C(α, γ) 
goal: g.compAddMonoidHom' f = f.comp g

full_name: ContinuousMap.compAddMonoidHom'
code: /-- Composition on the right as a `MonoidHom`. Similar to `MonoidHom.compHom'`. -/
@[to_additive (attr := simps)
      "Composition on the right as an `AddMonoidHom`. Similar to `AddMonoidHom.compHom'`."]
def compMonoidHom' {γ : Type*} [TopologicalSpace γ] [MulOneClass γ] [ContinuousMul γ]
    (g : C(α, β)) : C(β, γ) →* C(α, γ) 
goal: C(β, γ) →+ C(α, γ)

full_name: ContinuousMap.compMonoidHom'_apply
code: /-- Composition on the right as a `MonoidHom`. Similar to `MonoidHom.compHom'`. -/
@[to_additive (attr := simps)
      "Composition on the right as an `AddMonoidHom`. Similar to `AddMonoidHom.compHom'`."]
def compMonoidHom' {γ : Type*} [TopologicalSpace γ] [MulOneClass γ] [ContinuousMul γ]
    (g : C(α, β)) : C(β, γ) →* C(α, γ) 
goal: g.compMonoidHom' f = f.comp g

full_name: ContinuousMap.coe_prod
code: @[to_additive (attr := simp)]
theorem 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) 
goal: ⇑(∏ i ∈ s, f i) = ∏ i ∈ s, ⇑(f i)

full_name: ContinuousMap.coe_sum
code: @[to_additive (attr := simp)]
theorem 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) 
goal: ⇑(∑ i ∈ s, f i) = ∑ i ∈ s, ⇑(f i)

full_name: ContinuousMap.prod_apply
code: @[to_additive]
theorem 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 
goal: (∏ i ∈ s, f i) a = ∏ i ∈ s, (f i) a

full_name: ContinuousMap.sum_apply
code: @[to_additive]
theorem 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 
goal: (∑ i ∈ s, f i) a = ∑ i ∈ s, (f i) a

full_name: ContinuousMap.instGroupOfTopologicalGroup
code: @[to_additive]
instance [Group β] [TopologicalGroup β] : Group C(α, β) 
goal: Group C(α, β)

full_name: ContinuousMap.instAddGroupOfTopologicalAddGroup
code: @[to_additive]
instance [Group β] [TopologicalGroup β] : Group C(α, β) 
goal: AddGroup C(α, β)

full_name: ContinuousMap.instAddCommGroupContinuousMap
code: @[to_additive]
instance instCommGroupContinuousMap [CommGroup β] [TopologicalGroup β] : CommGroup C(α, β) 
goal: AddCommGroup C(α, β)

full_name: ContinuousMap.instCommGroupContinuousMap
code: @[to_additive]
instance instCommGroupContinuousMap [CommGroup β] [TopologicalGroup β] : CommGroup C(α, β) 
goal: CommGroup C(α, β)

full_name: ContinuousMap.instTopologicalGroup
code: @[to_additive]
instance [CommGroup β] [TopologicalGroup β] : TopologicalGroup C(α, β) 
goal: TopologicalGroup C(α, β)

full_name: ContinuousMap.instTopologicalAddGroup
code: @[to_additive]
instance [CommGroup β] [TopologicalGroup β] : TopologicalGroup C(α, β) 
goal: TopologicalAddGroup C(α, β)

full_name: ContinuousMap.hasSum_apply
code: /-- If `α` is locally compact, and 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`. -/
theorem hasSum_apply {γ : Type*} [AddCommMonoid β] [ContinuousAdd β]
    {f : γ → C(α, β)} {g : C(α, β)} (hf : HasSum f g) (x : α) :
    HasSum (fun i : γ => f i x) (g x) 
goal: HasSum (fun i => (f i) x) (g x)

full_name: ContinuousMap.summable_apply
code: theorem summable_apply [AddCommMonoid β] [ContinuousAdd β] {γ : Type*} {f : γ → C(α, β)}
    (hf : Summable f) (x : α) : Summable fun i : γ => f i x 
goal: Summable fun i => (f i) x

full_name: ContinuousMap.tsum_apply
code: theorem tsum_apply [T2Space β] [AddCommMonoid β] [ContinuousAdd β] {γ : Type*} {f : γ → C(α, β)}
    (hf : Summable f) (x : α) :
    ∑' i : γ, f i x = (∑' i : γ, f i) x 
goal: ∑' (i : γ), (f i) x = (∑' (i : γ), f i) x

full_name: continuousSubsemiring
code: /-- The subsemiring of continuous maps `α → β`. -/
def continuousSubsemiring (α : Type*) (R : Type*) [TopologicalSpace α] [TopologicalSpace R]
    [NonAssocSemiring R] [TopologicalSemiring R] : Subsemiring (α → R) 
goal: Subsemiring (α → R)

full_name: continuousSubring
code: /-- The subring of continuous maps `α → β`. -/
def continuousSubring (α : Type*) (R : Type*) [TopologicalSpace α] [TopologicalSpace R] [Ring R]
    [TopologicalRing R] : Subring (α → R) 
goal: Subring (α → R)

full_name: ContinuousMap.instNonUnitalNonAssocSemiringOfTopologicalSemiring
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β]
    [NonUnitalNonAssocSemiring β] [TopologicalSemiring β] : NonUnitalNonAssocSemiring C(α, β) 
goal: NonUnitalNonAssocSemiring C(α, β)

full_name: ContinuousMap.instNonUnitalSemiringOfTopologicalSemiring
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [NonUnitalSemiring β]
    [TopologicalSemiring β] : NonUnitalSemiring C(α, β) 
goal: NonUnitalSemiring C(α, β)

full_name: ContinuousMap.instAddMonoidWithOneOfContinuousAdd
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [AddMonoidWithOne β]
    [ContinuousAdd β] : AddMonoidWithOne C(α, β) 
goal: AddMonoidWithOne C(α, β)

full_name: ContinuousMap.instNonAssocSemiringOfTopologicalSemiring
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [NonAssocSemiring β]
    [TopologicalSemiring β] : NonAssocSemiring C(α, β) 
goal: NonAssocSemiring C(α, β)

full_name: ContinuousMap.instSemiringOfTopologicalSemiring
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [Semiring β]
    [TopologicalSemiring β] : Semiring C(α, β) 
goal: Semiring C(α, β)

full_name: ContinuousMap.instNonUnitalNonAssocRingOfTopologicalRing
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β]
    [NonUnitalNonAssocRing β] [TopologicalRing β] : NonUnitalNonAssocRing C(α, β) 
goal: NonUnitalNonAssocRing C(α, β)

full_name: ContinuousMap.instNonUnitalRingOfTopologicalRing
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [NonUnitalRing β]
    [TopologicalRing β] : NonUnitalRing C(α, β) 
goal: NonUnitalRing C(α, β)

full_name: ContinuousMap.instNonAssocRingOfTopologicalRing
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [NonAssocRing β]
    [TopologicalRing β] : NonAssocRing C(α, β) 
goal: NonAssocRing C(α, β)

full_name: ContinuousMap.instRing
code: instance instRing {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [Ring β]
    [TopologicalRing β] : Ring C(α, β) 
goal: Ring C(α, β)

full_name: ContinuousMap.instNonUnitalCommSemiringOfTopologicalSemiring
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β]
    [NonUnitalCommSemiring β] [TopologicalSemiring β] : NonUnitalCommSemiring C(α, β) 
goal: NonUnitalCommSemiring C(α, β)

full_name: ContinuousMap.instCommSemiringOfTopologicalSemiring
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [CommSemiring β]
    [TopologicalSemiring β] : CommSemiring C(α, β) 
goal: CommSemiring C(α, β)

full_name: ContinuousMap.instNonUnitalCommRingOfTopologicalRing
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [NonUnitalCommRing β]
    [TopologicalRing β] : NonUnitalCommRing C(α, β) 
goal: NonUnitalCommRing C(α, β)

full_name: ContinuousMap.instCommRingOfTopologicalRing
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [CommRing β]
    [TopologicalRing β] : CommRing C(α, β) 
goal: CommRing C(α, β)

full_name: ContinuousMap.instTopologicalSemiringOfLocallyCompactSpace
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [LocallyCompactSpace α]
    [NonUnitalSemiring β] [TopologicalSemiring β] : TopologicalSemiring C(α, β) 
goal: TopologicalSemiring C(α, β)

full_name: ContinuousMap.instTopologicalRingOfLocallyCompactSpace
code: instance {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [LocallyCompactSpace α]
    [NonUnitalRing β] [TopologicalRing β] : TopologicalRing C(α, β) 
goal: TopologicalRing C(α, β)

full_name: RingHom.compLeftContinuous_apply_apply
code: /-- Composition on the left by a (continuous) homomorphism of topological semirings, as a
`RingHom`.  Similar to `RingHom.compLeft`. -/
@[simps!]
protected def _root_.RingHom.compLeftContinuous (α : Type*) {β : Type*} {γ : Type*}
    [TopologicalSpace α]
    [TopologicalSpace β] [Semiring β] [TopologicalSemiring β] [TopologicalSpace γ] [Semiring γ]
    [TopologicalSemiring γ] (g : β →+* γ) (hg : Continuous g) : C(α, β) →+* C(α, γ) 
goal: ∀ (a : α), ((RingHom.compLeftContinuous α g hg) f) a = g (f a)

full_name: RingHom.compLeftContinuous
code: /-- Composition on the left by a (continuous) homomorphism of topological semirings, as a
`RingHom`.  Similar to `RingHom.compLeft`. -/
@[simps!]
protected def _root_.RingHom.compLeftContinuous (α : Type*) {β : Type*} {γ : Type*}
    [TopologicalSpace α]
    [TopologicalSpace β] [Semiring β] [TopologicalSemiring β] [TopologicalSpace γ] [Semiring γ]
    [TopologicalSemiring γ] (g : β →+* γ) (hg : Continuous g) : C(α, β) →+* C(α, γ) 
goal: C(α, β) →+* C(α, γ)

full_name: ContinuousMap.coeFnRingHom_apply
code: /-- Coercion to a function as a `RingHom`. -/
@[simps!]
def coeFnRingHom {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [Semiring β]
    [TopologicalSemiring β] : C(α, β) →+* α → β 
goal: ContinuousMap.coeFnRingHom f a = f a

full_name: ContinuousMap.coeFnRingHom
code: /-- Coercion to a function as a `RingHom`. -/
@[simps!]
def coeFnRingHom {α : Type*} {β : Type*} [TopologicalSpace α] [TopologicalSpace β] [Semiring β]
    [TopologicalSemiring β] : C(α, β) →+* α → β 
goal: C(α, β) →+* α → β

full_name: continuousSubmodule
code: /-- The `R`-submodule of continuous maps `α → M`. -/
def continuousSubmodule : Submodule R (α → M) 
goal: Submodule R (α → M)

full_name: ContinuousMap.instVAdd
code: @[to_additive]
instance instSMul [SMul R M] [ContinuousConstSMul R M] : SMul R C(α, M) 
goal: VAdd R C(α, M)

full_name: ContinuousMap.instSMul
code: @[to_additive]
instance instSMul [SMul R M] [ContinuousConstSMul R M] : SMul R C(α, M) 
goal: SMul R C(α, M)

full_name: ContinuousMap.instContinuousConstSMulOfLocallyCompactSpace
code: @[to_additive]
instance [LocallyCompactSpace α] [SMul R M] [ContinuousConstSMul R M] :
    ContinuousConstSMul R C(α, M) 
goal: ContinuousConstSMul R C(α, M)

full_name: ContinuousMap.instContinuousConstVAddOfLocallyCompactSpace
code: @[to_additive]
instance [LocallyCompactSpace α] [SMul R M] [ContinuousConstSMul R M] :
    ContinuousConstSMul R C(α, M) 
goal: ContinuousConstVAdd R C(α, M)

full_name: ContinuousMap.instContinuousSMulOfLocallyCompactSpace
code: @[to_additive]
instance [LocallyCompactSpace α] [TopologicalSpace R] [SMul R M] [ContinuousSMul R M] :
    ContinuousSMul R C(α, M) 
goal: ContinuousSMul R C(α, M)

full_name: ContinuousMap.instContinuousVAddOfLocallyCompactSpace
code: @[to_additive]
instance [LocallyCompactSpace α] [TopologicalSpace R] [SMul R M] [ContinuousSMul R M] :
    ContinuousSMul R C(α, M) 
goal: ContinuousVAdd R C(α, M)

full_name: ContinuousMap.coe_smul
code: @[to_additive (attr := simp, norm_cast)]
theorem coe_smul [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) : ⇑(c • f) = c • ⇑f 
goal: ⇑(c • f) = c • ⇑f

full_name: ContinuousMap.coe_vadd
code: @[to_additive (attr := simp, norm_cast)]
theorem coe_smul [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) : ⇑(c • f) = c • ⇑f 
goal: ⇑(c +ᵥ f) = c +ᵥ ⇑f

full_name: ContinuousMap.vadd_apply
code: @[to_additive]
theorem smul_apply [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) (a : α) :
    (c • f) a = c • f a 
goal: (c +ᵥ f) a = c +ᵥ f a

full_name: ContinuousMap.smul_apply
code: @[to_additive]
theorem smul_apply [SMul R M] [ContinuousConstSMul R M] (c : R) (f : C(α, M)) (a : α) :
    (c • f) a = c • f a 
goal: (c • f) a = c • f a

full_name: ContinuousMap.smul_comp
code: @[to_additive (attr := simp)]
theorem 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 
goal: (r • f).comp g = r • f.comp g

full_name: ContinuousMap.vadd_comp
code: @[to_additive (attr := simp)]
theorem 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 
goal: (r +ᵥ f).comp g = r +ᵥ f.comp g

full_name: ContinuousMap.instVAddCommClass
code: @[to_additive]
instance [SMul R M] [ContinuousConstSMul R M] [SMul R₁ M] [ContinuousConstSMul R₁ M]
    [SMulCommClass R R₁ M] : SMulCommClass R R₁ C(α, M) 
goal: VAddCommClass R R₁ C(α, M)

full_name: ContinuousMap.instSMulCommClass
code: @[to_additive]
instance [SMul R M] [ContinuousConstSMul R M] [SMul R₁ M] [ContinuousConstSMul R₁ M]
    [SMulCommClass R R₁ M] : SMulCommClass R R₁ C(α, M) 
goal: SMulCommClass R R₁ C(α, M)

full_name: ContinuousMap.instIsScalarTower
code: instance [SMul R M] [ContinuousConstSMul R M] [SMul R₁ M] [ContinuousConstSMul R₁ M] [SMul R R₁]
    [IsScalarTower R R₁ M] : IsScalarTower R R₁ C(α, M) 
goal: IsScalarTower R R₁ C(α, M)

full_name: ContinuousMap.instIsCentralScalar
code: instance [SMul R M] [SMul Rᵐᵒᵖ M] [ContinuousConstSMul R M] [IsCentralScalar R M] :
    IsCentralScalar R C(α, M) 
goal: IsCentralScalar R C(α, M)

full_name: ContinuousMap.instMulActionOfContinuousConstSMul
code: instance [Monoid R] [MulAction R M] [ContinuousConstSMul R M] : MulAction R C(α, M) 
goal: MulAction R C(α, M)

full_name: ContinuousMap.instDistribMulActionOfContinuousConstSMul
code: instance [Monoid R] [AddMonoid M] [DistribMulAction R M] [ContinuousAdd M]
    [ContinuousConstSMul R M] : DistribMulAction R C(α, M) 
goal: DistribMulAction R C(α, M)

full_name: ContinuousMap.module
code: instance module : Module R C(α, M) 
goal: Module R C(α, M)

full_name: ContinuousLinearMap.compLeftContinuous
code: /-- Composition on the left by a continuous linear map, as a `LinearMap`.
Similar to `LinearMap.compLeft`. -/
@[simps]
protected def _root_.ContinuousLinearMap.compLeftContinuous (α : Type*) [TopologicalSpace α]
    (g : M →L[R] M₂) : C(α, M) →ₗ[R] C(α, M₂) 
goal: C(α, M) →ₗ[R] C(α, M₂)

full_name: ContinuousLinearMap.compLeftContinuous_apply
code: /-- Composition on the left by a continuous linear map, as a `LinearMap`.
Similar to `LinearMap.compLeft`. -/
@[simps]
protected def _root_.ContinuousLinearMap.compLeftContinuous (α : Type*) [TopologicalSpace α]
    (g : M →L[R] M₂) : C(α, M) →ₗ[R] C(α, M₂) 
goal: ∀ (a : C(α, M)),
  (ContinuousLinearMap.compLeftContinuous R α g) a =
    (↑(AddMonoidHom.compLeftContinuous α (↑g).toAddMonoidHom ⋯)).toFun a

full_name: ContinuousMap.coeFnLinearMap
code: /-- Coercion to a function as a `LinearMap`. -/
@[simps]
def coeFnLinearMap : C(α, M) →ₗ[R] α → M 
goal: C(α, M) →ₗ[R] α → M

full_name: ContinuousMap.coeFnLinearMap_apply
code: /-- Coercion to a function as a `LinearMap`. -/
@[simps]
def coeFnLinearMap : C(α, M) →ₗ[R] α → M 
goal: ∀ (a : C(α, M)) (a_1 : α), (ContinuousMap.coeFnLinearMap R) a a_1 = (↑ContinuousMap.coeFnAddMonoidHom).toFun a a_1

full_name: continuousSubalgebra
code: /-- The `R`-subalgebra of continuous maps `α → A`. -/
def continuousSubalgebra : Subalgebra R (α → A) 
goal: Subalgebra R (α → A)

full_name: ContinuousMap.C
code: /-- Continuous constant functions as a `RingHom`. -/
def ContinuousMap.C : R →+* C(α, A) 
goal: R →+* C(α, A)

full_name: ContinuousMap.C_apply
code: @[simp]
theorem ContinuousMap.C_apply (r : R) (a : α) : ContinuousMap.C r a = algebraMap R A r 
goal: (ContinuousMap.C r) a = (algebraMap R A) r

full_name: ContinuousMap.algebra
code: instance ContinuousMap.algebra : Algebra R C(α, A) 
goal: Algebra R C(α, A)

full_name: AlgHom.compLeftContinuous
code: /-- Composition on the left by a (continuous) homomorphism of topological `R`-algebras, as an
`AlgHom`. Similar to `AlgHom.compLeft`. -/
@[simps!]
protected def AlgHom.compLeftContinuous {α : Type*} [TopologicalSpace α] (g : A →ₐ[R] A₂)
    (hg : Continuous g) : C(α, A) →ₐ[R] C(α, A₂) 
goal: C(α, A) →ₐ[R] C(α, A₂)

full_name: AlgHom.compLeftContinuous_apply_apply
code: /-- Composition on the left by a (continuous) homomorphism of topological `R`-algebras, as an
`AlgHom`. Similar to `AlgHom.compLeft`. -/
@[simps!]
protected def AlgHom.compLeftContinuous {α : Type*} [TopologicalSpace α] (g : A →ₐ[R] A₂)
    (hg : Continuous g) : C(α, A) →ₐ[R] C(α, A₂) 
goal: ∀ (a : α), ((AlgHom.compLeftContinuous R g hg) f) a = g (f a)

full_name: ContinuousMap.compRightAlgHom
code: /-- Precomposition of functions into a topological semiring by a continuous map is an algebra
homomorphism. -/
@[simps]
def ContinuousMap.compRightAlgHom {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
    (f : C(α, β)) : C(β, A) →ₐ[R] C(α, A) 
goal: C(β, A) →ₐ[R] C(α, A)

full_name: ContinuousMap.compRightAlgHom_apply
code: /-- Precomposition of functions into a topological semiring by a continuous map is an algebra
homomorphism. -/
@[simps]
def ContinuousMap.compRightAlgHom {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
    (f : C(α, β)) : C(β, A) →ₐ[R] C(α, A) 
goal: (ContinuousMap.compRightAlgHom R A f) g = g.comp f

full_name: ContinuousMap.coeFnAlgHom
code: /-- Coercion to a function as an `AlgHom`. -/
@[simps!]
def ContinuousMap.coeFnAlgHom : C(α, A) →ₐ[R] α → A 
goal: C(α, A) →ₐ[R] α → A

full_name: ContinuousMap.coeFnAlgHom_apply
code: /-- Coercion to a function as an `AlgHom`. -/
@[simps!]
def ContinuousMap.coeFnAlgHom : C(α, A) →ₐ[R] α → A 
goal: (ContinuousMap.coeFnAlgHom R) f a = f a

full_name: Subalgebra.SeparatesPoints
code: /-- A version of `Set.SeparatesPoints` for subalgebras of the continuous functions,
used for stating the Stone-Weierstrass theorem.
-/
abbrev Subalgebra.SeparatesPoints (s : Subalgebra R C(α, A)) : Prop 
goal: Prop

full_name: Subalgebra.separatesPoints_monotone
code: theorem Subalgebra.separatesPoints_monotone :
    Monotone fun s : Subalgebra R C(α, A) => s.SeparatesPoints 
goal: Monotone fun s => s.SeparatesPoints

full_name: algebraMap_apply
code: @[simp]
theorem algebraMap_apply (k : R) (a : α) : algebraMap R C(α, A) k a = k • (1 : A) 
goal: ((algebraMap R C(α, A)) k) a = k • 1

full_name: Set.SeparatesPointsStrongly
code: /-- 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.)
-/
def Set.SeparatesPointsStrongly (s : Set C(α, 𝕜)) : Prop 
goal: Prop

full_name: Subalgebra.SeparatesPoints.strongly
code: /-- 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`.
-/
theorem Subalgebra.SeparatesPoints.strongly {s : Subalgebra 𝕜 C(α, 𝕜)} (h : s.SeparatesPoints) :
    (s : Set C(α, 𝕜)).SeparatesPointsStrongly 
goal: (↑s).SeparatesPointsStrongly

full_name: ContinuousMap.subsingleton_subalgebra
code: instance ContinuousMap.subsingleton_subalgebra (α : Type*) [TopologicalSpace α] (R : Type*)
    [CommSemiring R] [TopologicalSpace R] [TopologicalSemiring R] [Subsingleton α] :
    Subsingleton (Subalgebra R C(α, R)) 
goal: Subsingleton (Subalgebra R C(α, R))

full_name: ContinuousMap.instSMul'
code: instance instSMul' {α : Type*} [TopologicalSpace α] {R : Type*} [Semiring R] [TopologicalSpace R]
    {M : Type*} [TopologicalSpace M] [AddCommMonoid M] [Module R M] [ContinuousSMul R M] :
    SMul C(α, R) C(α, M) 
goal: SMul C(α, R) C(α, M)

full_name: ContinuousMap.module'
code: instance module' {α : Type*} [TopologicalSpace α] (R : Type*) [Semiring R] [TopologicalSpace R]
    [TopologicalSemiring R] (M : Type*) [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M]
    [Module R M] [ContinuousSMul R M] : Module C(α, R) C(α, M) 
goal: Module C(α, R) C(α, M)

full_name: ContinuousMap.instCovariantClass_mul_le_left
code: @[to_additive]
instance instCovariantClass_mul_le_left [PartialOrder β] [Mul β] [ContinuousMul β]
    [CovariantClass β β (· * ·) (· ≤ ·)] :
    CovariantClass C(α, β) C(α, β) (· * ·) (· ≤ ·) 
goal: CovariantClass C(α, β) C(α, β) (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1

full_name: ContinuousMap.instCovariantClass_add_le_left
code: @[to_additive]
instance instCovariantClass_mul_le_left [PartialOrder β] [Mul β] [ContinuousMul β]
    [CovariantClass β β (· * ·) (· ≤ ·)] :
    CovariantClass C(α, β) C(α, β) (· * ·) (· ≤ ·) 
goal: CovariantClass C(α, β) C(α, β) (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

full_name: ContinuousMap.instCovariantClass_add_le_right
code: @[to_additive]
instance instCovariantClass_mul_le_right [PartialOrder β] [Mul β] [ContinuousMul β]
    [CovariantClass β β (Function.swap (· * ·)) (· ≤ ·)] :
    CovariantClass C(α, β) C(α, β) (Function.swap (· * ·)) (· ≤ ·) 
goal: CovariantClass C(α, β) C(α, β) (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

full_name: ContinuousMap.instCovariantClass_mul_le_right
code: @[to_additive]
instance instCovariantClass_mul_le_right [PartialOrder β] [Mul β] [ContinuousMul β]
    [CovariantClass β β (Function.swap (· * ·)) (· ≤ ·)] :
    CovariantClass C(α, β) C(α, β) (Function.swap (· * ·)) (· ≤ ·) 
goal: CovariantClass C(α, β) C(α, β) (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1

full_name: ContinuousMap.coe_mabs
code: @[to_additive (attr := simp, norm_cast)]
lemma coe_mabs (f : C(α, β)) : ⇑|f|ₘ = |⇑f|ₘ 
goal: ⇑(mabs f) = mabs ⇑f

full_name: ContinuousMap.coe_abs
code: @[to_additive (attr := simp, norm_cast)]
lemma coe_mabs (f : C(α, β)) : ⇑|f|ₘ = |⇑f|ₘ 
goal: ⇑|f| = |⇑f|

full_name: ContinuousMap.abs_apply
code: @[to_additive (attr := simp)]
lemma mabs_apply (f : C(α, β)) (x : α) : |f|ₘ x = |f x|ₘ 
goal: |f| x = |f x|

full_name: ContinuousMap.mabs_apply
code: @[to_additive (attr := simp)]
lemma mabs_apply (f : C(α, β)) (x : α) : |f|ₘ x = |f x|ₘ 
goal: (mabs f) x = mabs (f x)

full_name: ContinuousMap.instStar
code: instance : Star C(α, β) 
goal: Star C(α, β)

full_name: ContinuousMap.coe_star
code: @[simp]
theorem coe_star (f : C(α, β)) : ⇑(star f) = star (⇑f) 
goal: ⇑(star f) = star ⇑f

full_name: ContinuousMap.star_apply
code: @[simp]
theorem star_apply (f : C(α, β)) (x : α) : star f x = star (f x) 
goal: (star f) x = star (f x)

full_name: ContinuousMap.instTrivialStar
code: instance instTrivialStar [TrivialStar β] : TrivialStar C(α, β) 
goal: TrivialStar C(α, β)

full_name: ContinuousMap.instInvolutiveStarOfContinuousStar
code: instance [InvolutiveStar β] [ContinuousStar β] : InvolutiveStar C(α, β) 
goal: InvolutiveStar C(α, β)

full_name: ContinuousMap.starAddMonoid
code: instance starAddMonoid [AddMonoid β] [ContinuousAdd β] [StarAddMonoid β] [ContinuousStar β] :
    StarAddMonoid C(α, β) 
goal: StarAddMonoid C(α, β)

full_name: ContinuousMap.starMul
code: instance starMul [Mul β] [ContinuousMul β] [StarMul β] [ContinuousStar β] :
    StarMul C(α, β) 
goal: StarMul C(α, β)

full_name: ContinuousMap.instStarRingOfContinuousStar
code: instance [NonUnitalNonAssocSemiring β] [TopologicalSemiring β] [StarRing β] [ContinuousStar β] :
    StarRing C(α, β) 
goal: StarRing C(α, β)

full_name: ContinuousMap.instStarModule
code: instance [Star R] [Star β] [SMul R β] [StarModule R β] [ContinuousStar β]
    [ContinuousConstSMul R β] : StarModule R C(α, β) 
goal: StarModule R C(α, β)

full_name: ContinuousMap.compStarAlgHom'
code: /-- The functorial map taking `f : C(X, Y)` to `C(Y, A) →⋆ₐ[𝕜] C(X, A)` given by pre-composition
with the continuous function `f`. See `ContinuousMap.compMonoidHom'` and
`ContinuousMap.compAddMonoidHom'`, `ContinuousMap.compRightAlgHom` for bundlings of
pre-composition into a `MonoidHom`, an `AddMonoidHom` and an `AlgHom`, respectively, under
suitable assumptions on `A`. -/
@[simps]
def compStarAlgHom' (f : C(X, Y)) : C(Y, A) →⋆ₐ[𝕜] C(X, A) 
goal: C(Y, A) →⋆ₐ[𝕜] C(X, A)

full_name: ContinuousMap.compStarAlgHom'_apply
code: /-- The functorial map taking `f : C(X, Y)` to `C(Y, A) →⋆ₐ[𝕜] C(X, A)` given by pre-composition
with the continuous function `f`. See `ContinuousMap.compMonoidHom'` and
`ContinuousMap.compAddMonoidHom'`, `ContinuousMap.compRightAlgHom` for bundlings of
pre-composition into a `MonoidHom`, an `AddMonoidHom` and an `AlgHom`, respectively, under
suitable assumptions on `A`. -/
@[simps]
def compStarAlgHom' (f : C(X, Y)) : C(Y, A) →⋆ₐ[𝕜] C(X, A) 
goal: (ContinuousMap.compStarAlgHom' 𝕜 A f) g = g.comp f

full_name: ContinuousMap.compStarAlgHom'_id
code: /-- `ContinuousMap.compStarAlgHom'` sends the identity continuous map to the identity
`StarAlgHom` -/
theorem compStarAlgHom'_id : compStarAlgHom' 𝕜 A (ContinuousMap.id X) = StarAlgHom.id 𝕜 C(X, A) 
goal: ContinuousMap.compStarAlgHom' 𝕜 A (ContinuousMap.id X) = StarAlgHom.id 𝕜 C(X, A)

full_name: ContinuousMap.compStarAlgHom'_comp
code: /-- `ContinuousMap.compStarAlgHom'` is functorial. -/
theorem compStarAlgHom'_comp (g : C(Y, Z)) (f : C(X, Y)) :
    compStarAlgHom' 𝕜 A (g.comp f) = (compStarAlgHom' 𝕜 A f).comp (compStarAlgHom' 𝕜 A g) 
goal: ContinuousMap.compStarAlgHom' 𝕜 A (g.comp f) =
  (ContinuousMap.compStarAlgHom' 𝕜 A f).comp (ContinuousMap.compStarAlgHom' 𝕜 A g)

full_name: ContinuousMap.compStarAlgHom
code: /-- Post-composition with a continuous star algebra homomorphism is a star algebra homomorphism
between spaces of continuous maps. -/
@[simps]
def compStarAlgHom (φ : A →⋆ₐ[𝕜] B) (hφ : Continuous φ) :
    C(X, A) →⋆ₐ[𝕜] C(X, B) 
goal: C(X, A) →⋆ₐ[𝕜] C(X, B)

full_name: ContinuousMap.compStarAlgHom_apply
code: /-- Post-composition with a continuous star algebra homomorphism is a star algebra homomorphism
between spaces of continuous maps. -/
@[simps]
def compStarAlgHom (φ : A →⋆ₐ[𝕜] B) (hφ : Continuous φ) :
    C(X, A) →⋆ₐ[𝕜] C(X, B) 
goal: (ContinuousMap.compStarAlgHom X φ hφ) f = { toFun := ⇑φ, continuous_toFun := hφ }.comp f

full_name: ContinuousMap.compStarAlgHom_id
code: /-- `ContinuousMap.compStarAlgHom` sends the identity `StarAlgHom` on `A` to the identity
`StarAlgHom` on `C(X, A)`. -/
lemma compStarAlgHom_id : compStarAlgHom X (.id 𝕜 A) continuous_id = .id 𝕜 C(X, A) 
goal: ContinuousMap.compStarAlgHom X (StarAlgHom.id 𝕜 A) ⋯ = StarAlgHom.id 𝕜 C(X, A)

full_name: ContinuousMap.compStarAlgHom_comp
code: /-- `ContinuousMap.compStarAlgHom` is functorial. -/
lemma compStarAlgHom_comp (φ : A →⋆ₐ[𝕜] B) (ψ : B →⋆ₐ[𝕜] C) (hφ : Continuous φ)
    (hψ : Continuous ψ) : compStarAlgHom X (ψ.comp φ) (hψ.comp hφ) =
      (compStarAlgHom X ψ hψ).comp (compStarAlgHom X φ hφ) 
goal: ContinuousMap.compStarAlgHom X (ψ.comp φ) ⋯ =
  (ContinuousMap.compStarAlgHom X ψ hψ).comp (ContinuousMap.compStarAlgHom X φ hφ)

full_name: ContinuousMap.periodic_tsum_comp_add_zsmul
code: /-- Summing the translates of `f` by `ℤ • p` gives a map which is periodic with period `p`.
(This is true without any convergence conditions, since if the sum doesn't converge it is taken to
be the zero map, which is periodic.) -/
theorem periodic_tsum_comp_add_zsmul [AddCommGroup X] [TopologicalAddGroup X] [AddCommMonoid Y]
    [ContinuousAdd Y] [T2Space Y] (f : C(X, Y)) (p : X) :
    Function.Periodic (⇑(∑' n : ℤ, f.comp (ContinuousMap.addRight (n • p)))) p 
goal: Function.Periodic (⇑(∑' (n : ℤ), f.comp (ContinuousMap.addRight (n • p)))) p

full_name: Homeomorph.compStarAlgEquiv'_symm_apply
code: /-- `ContinuousMap.compStarAlgHom'` as a `StarAlgEquiv` when the continuous map `f` is
actually a homeomorphism. -/
@[simps]
def compStarAlgEquiv' (f : X ≃ₜ Y) : C(Y, A) ≃⋆ₐ[𝕜] C(X, A) 
goal: (Homeomorph.compStarAlgEquiv' 𝕜 A f).symm a = (ContinuousMap.compStarAlgHom' 𝕜 A f.symm.toContinuousMap) a

full_name: Homeomorph.compStarAlgEquiv'_apply
code: /-- `ContinuousMap.compStarAlgHom'` as a `StarAlgEquiv` when the continuous map `f` is
actually a homeomorphism. -/
@[simps]
def compStarAlgEquiv' (f : X ≃ₜ Y) : C(Y, A) ≃⋆ₐ[𝕜] C(X, A) 
goal: (Homeomorph.compStarAlgEquiv' 𝕜 A f) a = (ContinuousMap.compStarAlgHom' 𝕜 A f.toContinuousMap) a

full_name: Homeomorph.compStarAlgEquiv'
code: /-- `ContinuousMap.compStarAlgHom'` as a `StarAlgEquiv` when the continuous map `f` is
actually a homeomorphism. -/
@[simps]
def compStarAlgEquiv' (f : X ≃ₜ Y) : C(Y, A) ≃⋆ₐ[𝕜] C(X, A) 
goal: C(Y, A) ≃⋆ₐ[𝕜] C(X, A)

full_name: ContinuousMap.evalAlgHom
code: /-- Evaluation of continuous maps at a point, bundled as an algebra homomorphism. -/
@[simps]
def ContinuousMap.evalAlgHom (x : X) : C(X, R) →ₐ[S] R 
goal: C(X, R) →ₐ[S] R

full_name: ContinuousMap.evalAlgHom_apply
code: /-- Evaluation of continuous maps at a point, bundled as an algebra homomorphism. -/
@[simps]
def ContinuousMap.evalAlgHom (x : X) : C(X, R) →ₐ[S] R 
goal: (ContinuousMap.evalAlgHom S R x) f = f x

full_name: ContinuousMap.evalStarAlgHom_apply
code: /-- Evaluation of continuous maps at a point, bundled as a star algebra homomorphism. -/
@[simps!]
def ContinuousMap.evalStarAlgHom [StarRing R] [ContinuousStar R] (x : X) :
    C(X, R) →⋆ₐ[S] R 
goal: (ContinuousMap.evalStarAlgHom S R x) f = f x

full_name: ContinuousMap.evalStarAlgHom
code: /-- Evaluation of continuous maps at a point, bundled as a star algebra homomorphism. -/
@[simps!]
def ContinuousMap.evalStarAlgHom [StarRing R] [ContinuousStar R] (x : X) :
    C(X, R) →⋆ₐ[S] R 
goal: C(X, R) →⋆ₐ[S] R