pi_span — Mathlib

English

The principal ideal generated by a single component r in a product (Pi) equals the span of r in the product of rings.

Русский

Линейно-произвольный идеал, порожденный компонентой r, в произведении колец равен порожденному r в произведении.

LaTeX

$$$\\pi(\\operatorname{span}\\{r \\cdot\\}) = \\operatorname{span}\\{r\\}$$$

Lean4

@[simp]
theorem pi_span {r : Π i, R i} : pi (span {r ·}) = span { r } := by ext;
  simp_rw [mem_pi, mem_span_singleton', funext_iff, Classical.skolem, Pi.mul_def]

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