Theorem xpun 3312

Description: The cross product of two unions.

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 3310	. 2 |- ((A u. B) X. (C u. D)) = (((A u. B) X. C) u. ((A u. B) X. D))
2		xpundir 3311	. . 3 |- ((A u. B) X. C) = ((A X. C) u. (B X. C))
3		xpundir 3311	. . 3 |- ((A u. B) X. D) = ((A X. D) u. (B X. D))
4	2, 3	uneq12i 2234	. 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 2242	. 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 1542	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 992   u. cun 2097   X. cxp 3249

This theorem is referenced by:  infxpidmlem11 7774

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858  df-un 2102  df-opab 2741  df-xp 3265

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