prod_ext_iff — Mathlib

English

The coercion of pi equals the pi of the coerced maps: coe (pi f) = pi (coe f).

Русский

Сведение к pi согласуется через коэрцию: coe(pi f) = pi(coe f).

LaTeX

$$$\\text{coe}_{\\pi} = \\text{pi}_{*}$$$

Lean4

theorem prod_ext_iff {f g : ContinuousMultilinearMap R M₁ (M₂ × M₃)} :
    f = g ↔
      (ContinuousLinearMap.fst _ _ _).compContinuousMultilinearMap f =
          (ContinuousLinearMap.fst _ _ _).compContinuousMultilinearMap g ∧
        (ContinuousLinearMap.snd _ _ _).compContinuousMultilinearMap f =
          (ContinuousLinearMap.snd _ _ _).compContinuousMultilinearMap g :=
  by rw [← Prod.mk_inj, ← prodEquiv_symm_apply, ← prodEquiv_symm_apply, Equiv.apply_eq_iff_eq]

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