Theorem xpun 5746

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 5741	. 2 ⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 ∪ 𝐵) × 𝐶) ∪ ((𝐴 ∪ 𝐵) × 𝐷))
2		xpundir 5742	. . 3 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
3		xpundir 5742	. . 3 ⊢ ((𝐴 ∪ 𝐵) × 𝐷) = ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))
4	2, 3	uneq12i 4158	. 2 ⊢ (((𝐴 ∪ 𝐵) × 𝐶) ∪ ((𝐴 ∪ 𝐵) × 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷)))
5		un4 4166	. 2 ⊢ (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
6	1, 4, 5	3eqtri 2760	1 ⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))

Syntax hints:   = wceq 1534   ∪ cun 3943   × cxp 5671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3472  df-un 3950  df-opab 5206  df-xp 5679

This theorem is referenced by:  ex-xp  30240