Theorem: ContinuousLinearMap.strongTopology.locallyConvexSpace
File: Mathlib/Analysis/LocallyConvex/StrongTopology.lean

tactic: letI : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖
state_before: R : Type u_2
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_1
F : Type u_3
inst✝¹³ : AddCommGroup E
inst✝¹² : TopologicalSpace E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : TopologicalAddGroup F
inst✝⁸ : OrderedSemiring R
inst✝⁷ : NormedField 𝕜₁
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
σ : 𝕜₁ →+* 𝕜₂
inst✝³ : Module R F
inst✝² : ContinuousConstSMul R F
inst✝¹ : LocallyConvexSpace R F
inst✝ : SMulCommClass 𝕜₂ R F
𝔖 : Set (Set E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
⊢ LocallyConvexSpace R (E →SL[σ] F)

tactic: haveI : TopologicalAddGroup (E →SL[σ] F) := strongTopology.topologicalAddGroup _ _ _
state_before: R : Type u_2
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_1
F : Type u_3
inst✝¹³ : AddCommGroup E
inst✝¹² : TopologicalSpace E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : TopologicalAddGroup F
inst✝⁸ : OrderedSemiring R
inst✝⁷ : NormedField 𝕜₁
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
σ : 𝕜₁ →+* 𝕜₂
inst✝³ : Module R F
inst✝² : ContinuousConstSMul R F
inst✝¹ : LocallyConvexSpace R F
inst✝ : SMulCommClass 𝕜₂ R F
𝔖 : Set (Set E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
this : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖
⊢ LocallyConvexSpace R (E →SL[σ] F)

tactic: apply LocallyConvexSpace.ofBasisZero _ _ _ _
  (strongTopology.hasBasis_nhds_zero_of_basis _ _ _ h𝔖₁ h𝔖₂
    (LocallyConvexSpace.convex_basis_zero R F)) _
state_before: R : Type u_2
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_1
F : Type u_3
inst✝¹³ : AddCommGroup E
inst✝¹² : TopologicalSpace E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : TopologicalAddGroup F
inst✝⁸ : OrderedSemiring R
inst✝⁷ : NormedField 𝕜₁
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
σ : 𝕜₁ →+* 𝕜₂
inst✝³ : Module R F
inst✝² : ContinuousConstSMul R F
inst✝¹ : LocallyConvexSpace R F
inst✝ : SMulCommClass 𝕜₂ R F
𝔖 : Set (Set E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
this✝ : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖
this : TopologicalAddGroup (E →SL[σ] F)
⊢ LocallyConvexSpace R (E →SL[σ] F)

tactic: rintro ⟨S, V⟩ ⟨_, _, hVconvex⟩ f hf g hg a b ha hb hab x hx
state_before: R : Type u_2
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_1
F : Type u_3
inst✝¹³ : AddCommGroup E
inst✝¹² : TopologicalSpace E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : TopologicalAddGroup F
inst✝⁸ : OrderedSemiring R
inst✝⁷ : NormedField 𝕜₁
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
σ : 𝕜₁ →+* 𝕜₂
inst✝³ : Module R F
inst✝² : ContinuousConstSMul R F
inst✝¹ : LocallyConvexSpace R F
inst✝ : SMulCommClass 𝕜₂ R F
𝔖 : Set (Set E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
this✝ : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖
this : TopologicalAddGroup (E →SL[σ] F)
⊢ ∀ (i : Set E × Set F),
    i.fst ∈ 𝔖 ∧ i.snd ∈ 𝓝 0 ∧ Convex R i.snd → Convex R {f | ∀ (x : E), x ∈ i.fst → ↑f x ∈ _root_.id i.snd}

tactic: exact hVconvex (hf x hx) (hg x hx) ha hb hab
state_before: case mk.intro.intro
R : Type u_2
𝕜₁ : Type u_4
𝕜₂ : Type u_5
E : Type u_1
F : Type u_3
inst✝¹³ : AddCommGroup E
inst✝¹² : TopologicalSpace E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : TopologicalSpace F
inst✝⁹ : TopologicalAddGroup F
inst✝⁸ : OrderedSemiring R
inst✝⁷ : NormedField 𝕜₁
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : Module 𝕜₂ F
σ : 𝕜₁ →+* 𝕜₂
inst✝³ : Module R F
inst✝² : ContinuousConstSMul R F
inst✝¹ : LocallyConvexSpace R F
inst✝ : SMulCommClass 𝕜₂ R F
𝔖 : Set (Set E)
h𝔖₁ : Set.Nonempty 𝔖
h𝔖₂ : DirectedOn (fun x x_1 => x ⊆ x_1) 𝔖
this✝ : TopologicalSpace (E →SL[σ] F) := strongTopology σ F 𝔖
this : TopologicalAddGroup (E →SL[σ] F)
S : Set E
V : Set F
left✝¹ : (S, V).fst ∈ 𝔖
left✝ : (S, V).snd ∈ 𝓝 0
hVconvex : Convex R (S, V).snd
f : E →SL[σ] F
hf : f ∈ {f | ∀ (x : E), x ∈ (S, V).fst → ↑f x ∈ _root_.id (S, V).snd}
g : E →SL[σ] F
hg : g ∈ {f | ∀ (x : E), x ∈ (S, V).fst → ↑f x ∈ _root_.id (S, V).snd}
a b : R
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
x : E
hx : x ∈ (S, V).fst
⊢ ↑(a • f + b • g) x ∈ _root_.id (S, V).snd