comap_prod — Mathlib

English

For kernels κ:β→γ and η:β→δ, and a measurable f: α → β, the comap of their product along f equals the product of comaps: (κ ×K η).comap f hf = (κ.comap f hf) ×K (η.comap f hf).

Русский

Если κ:β→γ и η:β→δ — ядра, и f: α → β измеримо, то образování по компаксу вдоль f распадается на произведение образований: (κ ×K η).comap f hf = (κ.comap f hf) ×K (η.comap f hf).

LaTeX

$$$\\displaystyle (\\kappa \\times_K \\eta).comap f hf = (\\kappa.comap f hf) \\times_K (\\eta.comap f hf)$$$

Lean4

theorem comap_prod (κ : Kernel β γ) [IsSFiniteKernel κ] (η : Kernel β δ) [IsSFiniteKernel η] {f : α → β}
    (hf : Measurable f) : (κ ×ₖ η).comap f hf = (κ.comap f hf) ×ₖ (η.comap f hf) :=
  by
  ext1 x
  rw [comap_apply, prod_apply, prod_apply, comap_apply, comap_apply]

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