pi_eq_zero — Mathlib

English

Another formulation: for a family of linear maps, pi f = 0 iff all components are zero.

Русский

Другой формулировкой: произведение/плюс pi f = 0 тогда, когда все компоненты равны нулю.

LaTeX

$$$\\pi f = 0 \\iff \\forall i, f_i = 0$$$

Lean4

theorem pi_eq_zero (f : ∀ i, M →L[R] φ i) : pi f = 0 ↔ ∀ i, f i = 0 :=
  by
  simp only [ContinuousLinearMap.ext_iff, pi_apply, funext_iff]
  exact forall_swap

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