# Cartesian product types

Content created by Egbert Rijke, Fredrik Bakke and Jonathan Prieto-Cubides.

Created on 2022-01-26.

module foundation.cartesian-product-types where

open import foundation-core.cartesian-product-types public

Imports
open import foundation.dependent-pair-types
open import foundation.subuniverses
open import foundation.universe-levels

open import foundation-core.identity-types
open import foundation-core.transport-along-identifications


## Properties

### Transport in a family of cartesian products

tr-product :
{l1 l2 : Level} {A : UU l1} {a0 a1 : A}
(B C : A → UU l2) (p : a0 ＝ a1) (u : B a0 × C a0) →
(tr (λ a → B a × C a) p u) ＝ (pair (tr B p (pr1 u)) (tr C p (pr2 u)))
tr-product B C refl u = refl


### Subuniverses closed under cartesian product types

is-closed-under-products-subuniverses :
{l1 l2 l3 l4 l5 : Level} (P : subuniverse l1 l2) (Q : subuniverse l3 l4)
(R : subuniverse (l1 ⊔ l3) l5) → UU (lsuc l1 ⊔ l2 ⊔ lsuc l3 ⊔ l4 ⊔ l5)
is-closed-under-products-subuniverses {l1} {l2} {l3} P Q R =
{X : UU l1} {Y : UU l3} →
is-in-subuniverse P X → is-in-subuniverse Q Y → is-in-subuniverse R (X × Y)

is-closed-under-products-subuniverse :
{l1 l2 : Level} (P : subuniverse l1 l2) → UU (lsuc l1 ⊔ l2)
is-closed-under-products-subuniverse P =
is-closed-under-products-subuniverses P P P