Theorem xpun 3227

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 3225	. 2 |- ((A u. B) X. (C u. D)) = (((A u. B) X. C) u. ((A u. B) X. D))
2		xpundir 3226	. . 3 |- ((A u. B) X. C) = ((A X. C) u. (B X. C))
3		xpundir 3226	. . 3 |- ((A u. B) X. D) = ((A X. D) u. (B X. D))
4	2, 3	uneq12i 2182	. 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 2190	. 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	3eqtr 1499	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 956   u. cun 2045   X. cxp 3168

This theorem is referenced by:  infxpidmlem11 7562

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-opab 2667  df-xp 3184

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