Theorem xpun 5607

Description: The Cartesian product of two unions. (Contributed by NM, 12-Aug-2004.)

Assertion

Ref	Expression
xpun	⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))

Proof of Theorem xpun

Step	Hyp	Ref	Expression
1		xpundi 5602	. 2 ⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 ∪ 𝐵) × 𝐶) ∪ ((𝐴 ∪ 𝐵) × 𝐷))
2		xpundir 5603	. . 3 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
3		xpundir 5603	. . 3 ⊢ ((𝐴 ∪ 𝐵) × 𝐷) = ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))
4	2, 3	uneq12i 4061	. 2 ⊢ (((𝐴 ∪ 𝐵) × 𝐶) ∪ ((𝐴 ∪ 𝐵) × 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷)))
5		un4 4069	. 2 ⊢ (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
6	1, 4, 5	3eqtri 2763	1 ⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))

Syntax hints:   = wceq 1543   ∪ cun 3851   × cxp 5534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-v 3400  df-un 3858  df-opab 5102  df-xp 5542

This theorem is referenced by:  ex-xp  28473