Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: case inl
p : ℝ≥0∞
𝕜 : Type u_1
ι✝ : Type u_2
α✝ : ι✝ → Type u_3
β : ι✝ → Type u_4
hp : Fact (1 ≤ p)
inst✝¹⁰ : Fintype ι✝
inst✝⁹ : Semiring 𝕜
inst✝⁸ : (i : ι✝) → SeminormedAddCommGroup (α✝ i)
inst✝⁷ : (i : ι✝) → SeminormedAddCommGroup (β i)
inst✝⁶ : (i : ι✝) → Module 𝕜 (α✝ i)
inst✝⁵ : (i : ι✝) → Module 𝕜 (β i)
c : 𝕜
ι : Type u_5
κ : Type u_6
α : ι ⊕ κ → Type u_7
inst✝⁴ : Fintype ι
inst✝³ : Fintype κ
inst✝² : (i : ι ⊕ κ) → SeminormedAddCommGroup (α i)
inst✝¹ : (i : ι ⊕ κ) → Module 𝕜 (α i)
x : (i : ι ⊕ κ) → α i
inst✝ : Fact (1 ≤ ∞)
__LinearEquiv✝ : WithLp ∞ ((i : ι ⊕ κ) → α i) ≃ₗ[𝕜]
  WithLp ∞ (WithLp ∞ ((i : ι) → α (Sum.inl i)) × WithLp ∞ ((i : κ) → α (Sum.inr i))) :=
  WithLp.linearEquiv ∞ 𝕜 ((i : ι ⊕ κ) → α i) ≪≫ₗ LinearEquiv.sumPiEquivProdPi 𝕜 ι κ α ≪≫ₗ
      (WithLp.linearEquiv ∞ 𝕜 ((s : ι) → α (Sum.inl s))).symm.prodCongr
        (WithLp.linearEquiv ∞ 𝕜 ((t : κ) → α (Sum.inr t))).symm ≪≫ₗ
    (WithLp.linearEquiv ∞ 𝕜 (WithLp ∞ ((s : ι) → α (Sum.inl s)) × WithLp ∞ ((t : κ) → α (Sum.inr t)))).symm
⊢ ‖(WithLp.linearEquiv ∞ 𝕜 ((i : ι ⊕ κ) → α i) ≪≫ₗ LinearEquiv.sumPiEquivProdPi 𝕜 ι κ α ≪≫ₗ
            (WithLp.linearEquiv ∞ 𝕜 ((s : ι) → α (Sum.inl s))).symm.prodCongr
              (WithLp.linearEquiv ∞ 𝕜 ((t : κ) → α (Sum.inr t))).symm ≪≫ₗ
          (WithLp.linearEquiv ∞ 𝕜 (WithLp ∞ ((s : ι) → α (Sum.inl s)) × WithLp ∞ ((t : κ) → α (Sum.inr t)))).symm)
        ((WithLp.linearEquiv ∞ 𝕜 ((i : ι ⊕ κ) → α i)).symm x)‖ =
    ‖(WithLp.linearEquiv ∞ 𝕜 ((i : ι ⊕ κ) → α i)).symm x‖

pp: case inr
p✝ : ℝ≥0∞
𝕜 : Type u_1
ι✝ : Type u_2
α✝ : ι✝ → Type u_3
β : ι✝ → Type u_4
hp✝ : Fact (1 ≤ p✝)
inst✝¹⁰ : Fintype ι✝
inst✝⁹ : Semiring 𝕜
inst✝⁸ : (i : ι✝) → SeminormedAddCommGroup (α✝ i)
inst✝⁷ : (i : ι✝) → SeminormedAddCommGroup (β i)
inst✝⁶ : (i : ι✝) → Module 𝕜 (α✝ i)
inst✝⁵ : (i : ι✝) → Module 𝕜 (β i)
c : 𝕜
ι : Type u_5
κ : Type u_6
p : ℝ≥0∞
α : ι ⊕ κ → Type u_7
inst✝⁴ : Fintype ι
inst✝³ : Fintype κ
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι ⊕ κ) → SeminormedAddCommGroup (α i)
inst✝ : (i : ι ⊕ κ) → Module 𝕜 (α i)
x : (i : ι ⊕ κ) → α i
hp : 1 ≤ p.toReal
⊢ ‖(WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i) ≪≫ₗ LinearEquiv.sumPiEquivProdPi 𝕜 ι κ α ≪≫ₗ
            (WithLp.linearEquiv p 𝕜 ((s : ι) → α (Sum.inl s))).symm.prodCongr
              (WithLp.linearEquiv p 𝕜 ((t : κ) → α (Sum.inr t))).symm ≪≫ₗ
          (WithLp.linearEquiv p 𝕜 (WithLp p ((s : ι) → α (Sum.inl s)) × WithLp p ((t : κ) → α (Sum.inr t)))).symm)
        ((WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)).symm x)‖ =
    ‖(WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)).symm x‖

Tactic:
· simp [← Finset.univ_disjSum_univ, Finset.sup_disjSum, Pi.norm_def]

Goal state after:
case inr
p✝ : ℝ≥0∞
𝕜 : Type u_1
ι✝ : Type u_2
α✝ : ι✝ → Type u_3
β : ι✝ → Type u_4
hp✝ : Fact (1 ≤ p✝)
inst✝¹⁰ : Fintype ι✝
inst✝⁹ : Semiring 𝕜
inst✝⁸ : (i : ι✝) → SeminormedAddCommGroup (α✝ i)
inst✝⁷ : (i : ι✝) → SeminormedAddCommGroup (β i)
inst✝⁶ : (i : ι✝) → Module 𝕜 (α✝ i)
inst✝⁵ : (i : ι✝) → Module 𝕜 (β i)
c : 𝕜
ι : Type u_5
κ : Type u_6
p : ℝ≥0∞
α : ι ⊕ κ → Type u_7
inst✝⁴ : Fintype ι
inst✝³ : Fintype κ
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι ⊕ κ) → SeminormedAddCommGroup (α i)
inst✝ : (i : ι ⊕ κ) → Module 𝕜 (α i)
x : (i : ι ⊕ κ) → α i
hp : 1 ≤ p.toReal
⊢ ‖(WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i) ≪≫ₗ LinearEquiv.sumPiEquivProdPi 𝕜 ι κ α ≪≫ₗ
            (WithLp.linearEquiv p 𝕜 ((s : ι) → α (Sum.inl s))).symm.prodCongr
              (WithLp.linearEquiv p 𝕜 ((t : κ) → α (Sum.inr t))).symm ≪≫ₗ
          (WithLp.linearEquiv p 𝕜 (WithLp p ((s : ι) → α (Sum.inl s)) × WithLp p ((t : κ) → α (Sum.inr t)))).symm)
        ((WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)).symm x)‖ =
    ‖(WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)).symm x‖