-- Lean 4 proof step from Mathlib.Analysis.Normed.Lp.PiLp (goal)
Goal:
p_1 : ℝ≥0∞
𝕜 : Type u_1
ι_1 : Type u_2
α_1 : ι_1 → Type u_3
β : ι_1 → Type u_4
hp_1 : Fact (1 ≤ p_1)
inst : Fintype ι_1
inst_1 : Semiring 𝕜
inst_2 : (i : ι_1) → SeminormedAddCommGroup (α_1 i)
inst_3 : (i : ι_1) → SeminormedAddCommGroup (β i)
inst_4 : (i : ι_1) → Module.{u_1, u_3} 𝕜 (α_1 i)
inst_5 : (i : ι_1) → Module.{u_1, u_4} 𝕜 (β i)
c : 𝕜
ι : Type u_5
κ : Type u_6
p : ℝ≥0∞
α : ι ⊕ κ → Type u_7
inst_6 : Fintype ι
inst_7 : Fintype κ
inst_8 : Fact (1 ≤ p)
inst_9 : (i : ι ⊕ κ) → SeminormedAddCommGroup (α i)
inst_10 : (i : ι ⊕ κ) → Module.{u_1, u_7} 𝕜 (α i)
x : (i : ι ⊕ κ) → α i
hp : 1 ≤ p.toReal
this : 0 < p.toReal
hpt : p ≠ ∞
⊢ (↑(nnnorm (E := WithLp p (WithLp p ((i : ι) → α (Sum.inl i)) × WithLp p ((i : κ) → α (Sum.inr i))))
          ((WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i) ≪≫ₗ LinearEquiv.sumPiEquivProdPi 𝕜 ι κ α ≪≫ₗ
                    LinearEquiv.prodCongr (LinearEquiv.symm (WithLp.linearEquiv p 𝕜 ((s : ι) → α (Sum.inl s))))
                      (LinearEquiv.symm (WithLp.linearEquiv p 𝕜 ((t : κ) → α (Sum.inr t)))) ≪≫ₗ
                  LinearEquiv.symm
                    (WithLp.linearEquiv p 𝕜 (WithLp p ((s : ι) → α (Sum.inl s)) × WithLp p ((t : κ) → α (Sum.inr t)))) :
                (a : WithLp p ((i : ι ⊕ κ) → α i)) →
                  WithLp p (WithLp p ((i : ι) → α (Sum.inl i)) × WithLp p ((i : κ) → α (Sum.inr i))))
              (((LinearEquiv.symm (WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)) :
                    ((i : ι ⊕ κ) → α i) ≃ₗ[𝕜] WithLp p ((i : ι ⊕ κ) → α i)) :
                  (a : (i : ι ⊕ κ) → α i) → WithLp p ((i : ι ⊕ κ) → α i))
                x) :
            WithLp p (WithLp p ((i : ι) → α (Sum.inl i)) × WithLp p ((i : κ) → α (Sum.inr i))))) :
      ℝ) =
    (↑(nnnorm (E := WithLp p ((i : ι ⊕ κ) → α i))
          (((LinearEquiv.symm (WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)) :
                  ((i : ι ⊕ κ) → α i) ≃ₗ[𝕜] WithLp p ((i : ι ⊕ κ) → α i)) :
                (a : (i : ι ⊕ κ) → α i) → WithLp p ((i : ι ⊕ κ) → α i))
              x :
            WithLp p ((i : ι ⊕ κ) → α i))) :
      ℝ)
have inr : nnnorm (E := WithLp p (WithLp p ((i : ι) → α (Sum.inl i)) × WithLp p ((i : κ) → α (Sum.inr i))))
    ((WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i) ≪≫ₗ LinearEquiv.sumPiEquivProdPi 𝕜 ι κ α ≪≫ₗ
              LinearEquiv.prodCongr (LinearEquiv.symm (WithLp.linearEquiv p 𝕜 ((s : ι) → α (Sum.inl s))))
                (LinearEquiv.symm (WithLp.linearEquiv p 𝕜 ((t : κ) → α (Sum.inr t)))) ≪≫ₗ
            LinearEquiv.symm
              (WithLp.linearEquiv p 𝕜 (WithLp p ((s : ι) → α (Sum.inl s)) × WithLp p ((t : κ) → α (Sum.inr t)))) :
          (a : WithLp p ((i : ι ⊕ κ) → α i)) →
            WithLp p (WithLp p ((i : ι) → α (Sum.inl i)) × WithLp p ((i : κ) → α (Sum.inr i))))
        (((LinearEquiv.symm (WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)) :
              ((i : ι ⊕ κ) → α i) ≃ₗ[𝕜] WithLp p ((i : ι ⊕ κ) → α i)) :
            (a : (i : ι ⊕ κ) → α i) → WithLp p ((i : ι ⊕ κ) → α i))
          x) :
      WithLp p (WithLp p ((i : ι) → α (Sum.inl i)) × WithLp p ((i : κ) → α (Sum.inr i)))) =
  nnnorm (E := WithLp p ((i : ι ⊕ κ) → α i))
    (((LinearEquiv.symm (WithLp.linearEquiv p 𝕜 ((i : ι ⊕ κ) → α i)) :
            ((i : ι ⊕ κ) → α i) ≃ₗ[𝕜] WithLp p ((i : ι ⊕ κ) → α i)) :
          (a : (i : ι ⊕ κ) → α i) → WithLp p ((i : ι ⊕ κ) → α i))
        x :
      WithLp p ((i : ι ⊕ κ) → α i))
Tactic: congr 1