Theorem: PiTensorProduct.toDualContinuousMultilinearMap_apply_toFun
File: Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean

Proof steps:
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : ⨂[𝕜] (i : ι), E i
x✝ : ContinuousMultilinearMap 𝕜 E F
⊢ ‖((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear) x✝‖ ≤ projectiveSeminorm x * ‖x✝‖
state_after: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : ⨂[𝕜] (i : ι), E i
x✝ : ContinuousMultilinearMap 𝕜 E F
⊢ ‖(lift x✝.toMultilinearMap) x‖ ≤ projectiveSeminorm x * ‖x✝‖
tactic: simp only [LinearMap.coe_comp, Function.comp_apply,
                  ContinuousMultilinearMap.toMultilinearMapLinear_apply, LinearMap.flip_apply,
                  LinearEquiv.coe_coe]
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x : ⨂[𝕜] (i : ι), E i
x✝ : ContinuousMultilinearMap 𝕜 E F
⊢ ‖(lift x✝.toMultilinearMap) x‖ ≤ projectiveSeminorm x * ‖x✝‖
state_after: no goals
tactic: exact norm_eval_le_projectiveSeminorm _ _ _
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x y : ⨂[𝕜] (i : ι), E i
⊢ (fun x => ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯)
      (x + y) =
    (fun x => ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯)
        x +
      (fun x =>
          ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯)
        y
state_after: case h
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x y : ⨂[𝕜] (i : ι), E i
x✝ : ContinuousMultilinearMap 𝕜 E F
⊢ ((fun x => ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯)
        (x + y))
      x✝ =
    ((fun x =>
            ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯)
          x +
        (fun x =>
            ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯)
          y)
      x✝
tactic: ext _
state_before: case h
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
x y : ⨂[𝕜] (i : ι), E i
x✝ : ContinuousMultilinearMap 𝕜 E F
⊢ ((fun x => ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯)
        (x + y))
      x✝ =
    ((fun x =>
            ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯)
          x +
        (fun x =>
            ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯)
          y)
      x✝
state_after: no goals
tactic: simp only [map_add, LinearMap.mkContinuous_apply, LinearMap.coe_comp, Function.comp_apply,
      ContinuousMultilinearMap.toMultilinearMapLinear_apply, LinearMap.add_apply,
      LinearMap.flip_apply, LinearEquiv.coe_coe, ContinuousLinearMap.add_apply]
state_before: ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a : 𝕜
x : ⨂[𝕜] (i : ι), E i
⊢ {
          toFun := fun x =>
            ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯,
          map_add' := ⋯ }.toFun
      (a • x) =
    (RingHom.id 𝕜) a •
      {
            toFun := fun x =>
              ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯,
            map_add' := ⋯ }.toFun
        x
state_after: case h
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a : 𝕜
x : ⨂[𝕜] (i : ι), E i
x✝ : ContinuousMultilinearMap 𝕜 E F
⊢ ({
            toFun := fun x =>
              ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯,
            map_add' := ⋯ }.toFun
        (a • x))
      x✝ =
    ((RingHom.id 𝕜) a •
        {
              toFun := fun x =>
                ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x)
                  ⋯,
              map_add' := ⋯ }.toFun
          x)
      x✝
tactic: ext _
state_before: case h
ι : Type uι
inst✝⁵ : Fintype ι
𝕜 : Type u𝕜
inst✝⁴ : NontriviallyNormedField 𝕜
E : ι → Type uE
inst✝³ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝² : (i : ι) → NormedSpace 𝕜 (E i)
F : Type uF
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a : 𝕜
x : ⨂[𝕜] (i : ι), E i
x✝ : ContinuousMultilinearMap 𝕜 E F
⊢ ({
            toFun := fun x =>
              ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x) ⋯,
            map_add' := ⋯ }.toFun
        (a • x))
      x✝ =
    ((RingHom.id 𝕜) a •
        {
              toFun := fun x =>
                ((↑lift).flip x ∘ₗ ContinuousMultilinearMap.toMultilinearMapLinear).mkContinuous (projectiveSeminorm x)
                  ⋯,
              map_add' := ⋯ }.toFun
          x)
      x✝
state_after: no goals
tactic: simp only [map_smul, LinearMap.mkContinuous_apply, LinearMap.coe_comp, Function.comp_apply,
      ContinuousMultilinearMap.toMultilinearMapLinear_apply, LinearMap.smul_apply,
      LinearMap.flip_apply, LinearEquiv.coe_coe, RingHom.id_apply, ContinuousLinearMap.coe_smul',
      Pi.smul_apply]