Theorem xpun 4780

Description: The cross product of two unions. (Contributed by NM, 12-Aug-2004.)

Assertion

Ref	Expression
xpun	|- ( ( A u. B ) X. ( C u. D ) ) = ( ( ( A X. C ) u. ( A X. D ) ) u. ( ( B X. C ) u. ( B X. D ) ) )

Proof of Theorem xpun

Step	Hyp	Ref	Expression
1		xpundi 4775	. 2 |- ( ( A u. B ) X. ( C u. D ) ) = ( ( ( A u. B ) X. C ) u. ( ( A u. B ) X. D ) )
2		xpundir 4776	. . 3 |- ( ( A u. B ) X. C ) = ( ( A X. C ) u. ( B X. C ) )
3		xpundir 4776	. . 3 |- ( ( A u. B ) X. D ) = ( ( A X. D ) u. ( B X. D ) )
4	2, 3	uneq12i 3356	. 2 |- ( ( ( A u. B ) X. C ) u. ( ( A u. B ) X. D ) ) = ( ( ( A X. C ) u. ( B X. C ) ) u. ( ( A X. D ) u. ( B X. D ) ) )
5		un4 3364	. 2 |- ( ( ( A X. C ) u. ( B X. C ) ) u. ( ( A X. D ) u. ( B X. D ) ) ) = ( ( ( A X. C ) u. ( A X. D ) ) u. ( ( B X. C ) u. ( B X. D ) ) )
6	1, 4, 5	3eqtri 2254	1 |- ( ( A u. B ) X. ( C u. D ) ) = ( ( ( A X. C ) u. ( A X. D ) ) u. ( ( B X. C ) u. ( B X. D ) ) )

Syntax hints:   = wceq 1395   u. cun 3195   X. cxp 4717

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-opab 4146  df-xp 4725

This theorem is referenced by: (None)