map_id — Mathlib

English

For f: α → C and g: β → C with HasProduct, and p: β → α, q: ∀ b, f(p b) ⟶ g b, and b ∈ β, we have Pi.map' p q ≫ Pi.π g b = Pi.π f (p b) ≫ q b.

Русский

Для f: α → C и g: β → C с.HasProduct, p: β → α, q: ∀ b, f(p b) ⟶ g b, верно: Pi.map' p q ≫ Pi.π g b = Pi.π f (p b) ≫ q b.

LaTeX

$$$\Pi.map' p q \;\;≫\;\Pi.\pi g b = \Pi.\pi f (p b) \;\;≫\; q b$$$

Lean4

@[simp]
theorem map_id {f : α → C} [HasProduct f] : Pi.map (fun a => 𝟙 (f a)) = 𝟙 (∏ᶜ f) := by ext; simp

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