Theorem xpundi 3220

Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52.

Assertion

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

Proof of Theorem xpundi

Step	Hyp	Ref	Expression
1		elun 2169	. . . . . 6 |- (y e. (B u. C) <-> (y e. B \/ y e. C))
2	1	anbi2i 480	. . . . 5 |- ((x e. A /\ y e. (B u. C)) <-> (x e. A /\ (y e. B \/ y e. C)))
3		andi 603	. . . . 5 |- ((x e. A /\ (y e. B \/ y e. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
4	2, 3	bitr 173	. . . 4 |- ((x e. A /\ y e. (B u. C)) <-> ((x e. A /\ y e. B) \/ (x e. A /\ y e. C)))
5	4	opabbii 2666	. . 3 |- {<.x, y>. | (x e. A /\ y e. (B u. C))} = {<.x, y>. | ((x e. A /\ y e. B) \/ (x e. A /\ y e. C))}
6		unopab 2674	. . 3 |- ({<.x, y>. | (x e. A /\ y e. B)} u. {<.x, y>. | (x e. A /\ y e. C)}) = {<.x, y>. | ((x e. A /\ y e. B) \/ (x e. A /\ y e. C))}
7	5, 6	eqtr4 1495	. 2 |- {<.x, y>. | (x e. A /\ y e. (B u. C))} = ({<.x, y>. | (x e. A /\ y e. B)} u. {<.x, y>. | (x e. A /\ y e. C)})
8		df-xp 3179	. 2 |- (A X. (B u. C)) = {<.x, y>. | (x e. A /\ y e. (B u. C))}
9		df-xp 3179	. . 3 |- (A X. B) = {<.x, y>. | (x e. A /\ y e. B)}
10		df-xp 3179	. . 3 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
11	9, 10	uneq12i 2178	. 2 |- ((A X. B) u. (A X. C)) = ({<.x, y>. | (x e. A /\ y e. B)} u. {<.x, y>. | (x e. A /\ y e. C)})
12	7, 8, 11	3eqtr4 1502	1 |- (A X. (B u. C)) = ((A X. B) u. (A X. C))

Syntax hints:   \/ wo 222   /\ wa 223   = wceq 954   e. wcel 956   u. cun 2041  {copab 2661   X. cxp 3163

This theorem is referenced by:  xpun 3222  xp2cda 4908  xpcdaen 4911  alephadd 7532

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-opab 2662  df-xp 3179

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