appLE_congr — Mathlib

English

If U = U', V = V' and P is a property that depends functorially on the commutative ring homs, then P applied to f.appLE U V e is equivalent to P applied to f.appLE U' V' (e₁ ▸ e₂ ▸ e). In particular, changing opens by equalities does not affect the truth of P.

Русский

При равенствах U = U', V = V' и заданной свойствa P для отображений колебаний колец, значение P от f.appLE U V e равно значению P от f.appLE U' V' (e₁ ▸ e₂ ▸ e).

LaTeX

$$$P (f.appLE U V e) \\iff P (f.appLE U' V' (e_1 \\; ▸ \\; e_2 \\; ▸ \\; e))$$$

Lean4

theorem appLE_congr (e : V ≤ f ⁻¹ᵁ U) (e₁ : U = U') (e₂ : V = V') (P : ∀ {R S : CommRingCat.{u}} (_ : R ⟶ S), Prop) :
    P (f.appLE U V e) ↔ P (f.appLE U' V' (e₁ ▸ e₂ ▸ e)) := by subst e₁; subst e₂; rfl

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