equiv_iff_eq_of_Z_eq' — Mathlib

English

If P ≈ Q, then P x · Q z = Q x · P z (same as above with X).

Русский

Если P ≈ Q, то P x · Q z = Q x · P z.

LaTeX

$$P(0)·Q(2) = Q(0)·P(2)$$

Lean4

theorem equiv_iff_eq_of_Z_eq' {P Q : Fin 3 → R} (hz : P z = Q z) (hQz : Q z ∈ nonZeroDivisors R) : P ≈ Q ↔ P = Q :=
  by
  refine ⟨?_, Quotient.exact.comp <| congrArg _⟩
  rintro ⟨u, rfl⟩
  simp only [Units.smul_def, (mul_cancel_right_mem_nonZeroDivisors hQz).mp <| one_mul (Q z) ▸ hz]
  rw [one_smul]

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