divNat_inj — Mathlib

English

Equality of divNat representations implies cross-multiplication equality: divNat(n1,d1) = divNat(n2,d2) with d1,d2 ≠ 0 iff n1·d2 = n2·d1.

Русский

Существование равенства divNat(n1,d1) = divNat(n2,d2) при d1,d2 ≠ 0 эквивалентно n1·d2 = n2·d1.

LaTeX

$$$\divNat(n_1,d_1) = \divNat(n_2,d_2) \iff n_1 \cdot d_2 = n_2 \cdot d_1 \quad (d_1 \neq 0, d_2 \neq 0)$$$

Lean4

theorem divNat_inj (h₁ : d₁ ≠ 0) (h₂ : d₂ ≠ 0) : divNat n₁ d₁ = divNat n₂ d₂ ↔ n₁ * d₂ = n₂ * d₁ := by rw [← coe_inj];
  simp [Rat.mkRat_eq_iff, h₁, h₂]; norm_cast

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