smul_equiv_smul — Mathlib

English

For P, Q ∈ Fin 3 → R and units u, v ∈ R, the equivalence u • P ≈ v • Q is equivalent to P ≈ Q, i.e., unit-scaling preserves equivalence classes.

Русский

Для P, Q ∈ Fin 3 → R и единиц u, v ∈ R равносильность u • P ≈ v • Q равна P ≈ Q.

LaTeX

$$u • P ≈ v • Q ↔ P ≈ Q$$

Lean4

theorem smul_equiv_smul (P Q : Fin 3 → R) {u v : R} (hu : IsUnit u) (hv : IsUnit v) : u • P ≈ v • Q ↔ P ≈ Q := by
  rw [← Quotient.eq_iff_equiv, ← Quotient.eq_iff_equiv, smul_eq P hu, smul_eq Q hv]

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