Theorem xpun 5690

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 5685	. 2 ⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 ∪ 𝐵) × 𝐶) ∪ ((𝐴 ∪ 𝐵) × 𝐷))
2		xpundir 5686	. . 3 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
3		xpundir 5686	. . 3 ⊢ ((𝐴 ∪ 𝐵) × 𝐷) = ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))
4	2, 3	uneq12i 4116	. 2 ⊢ (((𝐴 ∪ 𝐵) × 𝐶) ∪ ((𝐴 ∪ 𝐵) × 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷)))
5		un4 4125	. 2 ⊢ (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
6	1, 4, 5	3eqtri 2758	1 ⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))

Syntax hints:   = wceq 1541   ∪ cun 3900   × cxp 5614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3907  df-opab 5154  df-xp 5622

This theorem is referenced by:  ex-xp  30411