Theorem xpun 4787

Description: The cross 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 4782	. 2 ⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 ∪ 𝐵) × 𝐶) ∪ ((𝐴 ∪ 𝐵) × 𝐷))
2		xpundir 4783	. . 3 ⊢ ((𝐴 ∪ 𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
3		xpundir 4783	. . 3 ⊢ ((𝐴 ∪ 𝐵) × 𝐷) = ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))
4	2, 3	uneq12i 3359	. 2 ⊢ (((𝐴 ∪ 𝐵) × 𝐶) ∪ ((𝐴 ∪ 𝐵) × 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷)))
5		un4 3367	. 2 ⊢ (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
6	1, 4, 5	3eqtri 2256	1 ⊢ ((𝐴 ∪ 𝐵) × (𝐶 ∪ 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))

Syntax hints:   = wceq 1397   ∪ cun 3198   × cxp 4723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-opab 4151  df-xp 4731

This theorem is referenced by: (None)