-- Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors
Goal:
σ : Type u_1
R : Type u_2
inst : Semiring R
φ : MvPowerSeries σ R
hφ :
  Membership.mem (α := R) (γ := Submonoid R) (nonZeroDivisorsLeft R)
    ((constantCoeff : (a : MvPowerSeries σ R) → R) φ : R)
intro intro₁ : ∀ (x : MvPowerSeries σ R) (hx : φ * x = 0), x = 0
h :
  ∀ (x : MvPowerSeries σ R) (hx : φ * x = 0) (d : σ →₀ ℕ),
    Eq.{u_2 + 1} (α := R) ((coeff d : (a : MvPowerSeries σ R) → R) x : R)
      ((coeff d : (a : MvPowerSeries σ R) → R) 0 : R)
h₁ h₂ :
  ∀ (x : MvPowerSeries σ R) (hx : φ * x = 0) (d e : σ →₀ ℕ)
    (he :
      ∀ y < e,
        Eq.{u_2 + 1} (α := R) ((coeff y : (a : MvPowerSeries σ R) → R) x : R)
          ((coeff y : (a : MvPowerSeries σ R) → R) 0 : R)),
    Eq.{u_2 + 1} (α := R) ((coeff e : (a : MvPowerSeries σ R) → R) x : R)
      ((coeff e : (a : MvPowerSeries σ R) → R) 0 : R)
h₃ :
  ∀ (x : MvPowerSeries σ R) (hx : φ * x = 0) (d e : σ →₀ ℕ)
    (he :
      ∀ y < e,
        Eq.{u_2 + 1} (α := R) ((coeff y : (a : MvPowerSeries σ R) → R) x : R)
          ((coeff y : (a : MvPowerSeries σ R) → R) 0 : R)),
    HMul.hMul (α := R) (β := R) ((constantCoeff : (a : MvPowerSeries σ R) → R) φ : R)
        ((coeff e : (a : MvPowerSeries σ R) → R) x : R) =
      (coeff e : (a : MvPowerSeries σ R) → R) (φ * x)
h₄ :
  ∀ (x : MvPowerSeries σ R) (hx : φ * x = 0) (d e : σ →₀ ℕ)
    (he :
      ∀ y < e,
        Eq.{u_2 + 1} (α := R) ((coeff y : (a : MvPowerSeries σ R) → R) x : R)
          ((coeff y : (a : MvPowerSeries σ R) → R) 0 : R)),
    HMul.hMul (α := R) (β := R) ((constantCoeff : (a : MvPowerSeries σ R) → R) φ : R)
        ((coeff e : (a : MvPowerSeries σ R) → R) x : R) =
      ∑ p ∈ @antidiagonal _ Finsupp.instAddMonoid _ e,
        HMul.hMul (α := R) (β := R) ((coeff p.1 : (a : MvPowerSeries σ R) → R) φ : R)
          ((coeff p.2 : (a : MvPowerSeries σ R) → R) x : R)
h₅ :
  ∀ (x : MvPowerSeries σ R) (hx : φ * x = 0) (d e : σ →₀ ℕ)
    (he :
      ∀ y < e,
        Eq.{u_2 + 1} (α := R) ((coeff y : (a : MvPowerSeries σ R) → R) x : R)
          ((coeff y : (a : MvPowerSeries σ R) → R) 0 : R)),
    ∀ b ∈ @antidiagonal _ Finsupp.instAddMonoid _ e,
      b ≠ (0, e) →
        HMul.hMul (α := R) (β := R) ((coeff b.1 : (a : MvPowerSeries σ R) → R) φ : R)
            ((coeff b.2 : (a : MvPowerSeries σ R) → R) x : R) =
          0
h₆ h₇ :
  ∀ (x : MvPowerSeries σ R) (hx : φ * x = 0) (d e : σ →₀ ℕ)
    (he :
      ∀ y < e,
        Eq.{u_2 + 1} (α := R) ((coeff y : (a : MvPowerSeries σ R) → R) x : R)
          ((coeff y : (a : MvPowerSeries σ R) → R) 0 : R)),
    (0, e) ∉ @antidiagonal _ Finsupp.instAddMonoid _ e →
      HMul.hMul (α := R) (β := R) ((coeff (0, e).1 : (a : MvPowerSeries σ R) → R) φ : R)
          ((coeff (0, e).2 : (a : MvPowerSeries σ R) → R) x : R) =
        0
⊢ Membership.mem (γ := Submonoid (MvPowerSeries σ R)) (nonZeroDivisorsLeft (MvPowerSeries σ R)) φ

Tactic:
classical
  intro x hx
  ext d
  apply WellFoundedLT.induction d
  intro e he
  rw [map_zero, ← mul_left_mem_nonZeroDivisorsLeft_eq_zero_iff hφ,
    ← map_zero (f := coeff e), ← hx]
  convert! (coeff_mul e φ x).symm
  rw [Finset.sum_eq_single (0, e), coeff_zero_eq_constantCoeff]
  · rintro ⟨_, u⟩ huv _
    suffices u < e by simp only [he u this, mul_zero, map_zero]
    apply lt_of_le_of_ne
    · simp only [← mem_antidiagonal.mp huv, le_add_iff_nonneg_left, zero_le]
    · rintro rfl
      simp_all
  · simp only [mem_antidiagonal, zero_add, not_true_eq_false, coeff_zero_eq_constantCoeff,
      false_implies]

Premises used: MvPowerSeries.coeff_zero_eq_constantCoeff, zero_le, map_zero, not_true_eq_false, zero_add, mul_left_mem_nonZeroDivisorsLeft_eq_zero_iff, false_implies, Finset.HasAntidiagonal.mem_antidiagonal, le_add_iff_nonneg_left, rfl, lt_of_le_of_ne