Theorem xpun 4835

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 4833	. 2 ⊢ ((A ∪ B) × (C ∪ D)) = (((A ∪ B) × C) ∪ ((A ∪ B) × D))
2		xpundir 4834	. . 3 ⊢ ((A ∪ B) × C) = ((A × C) ∪ (B × C))
3		xpundir 4834	. . 3 ⊢ ((A ∪ B) × D) = ((A × D) ∪ (B × D))
4	2, 3	uneq12i 3417	. 2 ⊢ (((A ∪ B) × C) ∪ ((A ∪ B) × D)) = (((A × C) ∪ (B × C)) ∪ ((A × D) ∪ (B × D)))
5		un4 3424	. 2 ⊢ (((A × C) ∪ (B × C)) ∪ ((A × D) ∪ (B × D))) = (((A × C) ∪ (A × D)) ∪ ((B × C) ∪ (B × D)))
6	1, 4, 5	3eqtri 2377	1 ⊢ ((A ∪ B) × (C ∪ D)) = (((A × C) ∪ (A × D)) ∪ ((B × C) ∪ (B × D)))

Syntax hints:   = wceq 1642   ∪ cun 3208   × cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-opab 4624  df-xp 4785

This theorem is referenced by: (None)