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Theorem xpun 4834
 Description: The cross product of two unions. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
xpun ((AB) × (CD)) = (((A × C) ∪ (A × D)) ∪ ((B × C) ∪ (B × D)))

Proof of Theorem xpun
StepHypRef Expression
1 xpundi 4832 . 2 ((AB) × (CD)) = (((AB) × C) ∪ ((AB) × D))
2 xpundir 4833 . . 3 ((AB) × C) = ((A × C) ∪ (B × C))
3 xpundir 4833 . . 3 ((AB) × D) = ((A × D) ∪ (B × D))
42, 3uneq12i 3416 . 2 (((AB) × C) ∪ ((AB) × D)) = (((A × C) ∪ (B × C)) ∪ ((A × D) ∪ (B × D)))
5 un4 3423 . 2 (((A × C) ∪ (B × C)) ∪ ((A × D) ∪ (B × D))) = (((A × C) ∪ (A × D)) ∪ ((B × C) ∪ (B × D)))
61, 4, 53eqtri 2377 1 ((AB) × (CD)) = (((A × C) ∪ (A × D)) ∪ ((B × C) ∪ (B × D)))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ∪ cun 3207   × cxp 4770 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-opab 4623  df-xp 4784 This theorem is referenced by: (None)
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