Theorem xpundir 3311

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

Assertion

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

Proof of Theorem xpundir

Step	Hyp	Ref	Expression
1		elun 2225	. . . . . 6 |- (x e. (A u. B) <-> (x e. A \/ x e. B))
2	1	anbi1i 484	. . . . 5 |- ((x e. (A u. B) /\ y e. C) <-> ((x e. A \/ x e. B) /\ y e. C))
3		andir 608	. . . . 5 |- (((x e. A \/ x e. B) /\ y e. C) <-> ((x e. A /\ y e. C) \/ (x e. B /\ y e. C)))
4	2, 3	bitri 171	. . . 4 |- ((x e. (A u. B) /\ y e. C) <-> ((x e. A /\ y e. C) \/ (x e. B /\ y e. C)))
5	4	opabbii 2745	. . 3 |- {<.x, y>. | (x e. (A u. B) /\ y e. C)} = {<.x, y>. | ((x e. A /\ y e. C) \/ (x e. B /\ y e. C))}
6		unopab 2753	. . 3 |- ({<.x, y>. | (x e. A /\ y e. C)} u. {<.x, y>. | (x e. B /\ y e. C)}) = {<.x, y>. | ((x e. A /\ y e. C) \/ (x e. B /\ y e. C))}
7	5, 6	eqtr4i 1541	. 2 |- {<.x, y>. | (x e. (A u. B) /\ y e. C)} = ({<.x, y>. | (x e. A /\ y e. C)} u. {<.x, y>. | (x e. B /\ y e. C)})
8		df-xp 3265	. 2 |- ((A u. B) X. C) = {<.x, y>. | (x e. (A u. B) /\ y e. C)}
9		df-xp 3265	. . 3 |- (A X. C) = {<.x, y>. | (x e. A /\ y e. C)}
10		df-xp 3265	. . 3 |- (B X. C) = {<.x, y>. | (x e. B /\ y e. C)}
11	9, 10	uneq12i 2234	. 2 |- ((A X. C) u. (B X. C)) = ({<.x, y>. | (x e. A /\ y e. C)} u. {<.x, y>. | (x e. B /\ y e. C)})
12	7, 8, 11	3eqtr4i 1548	1 |- ((A u. B) X. C) = ((A X. C) u. (B X. C))

Syntax hints:   \/ wo 220   /\ wa 221   = wceq 992   e. wcel 994   u. cun 2097  {copab 2740   X. cxp 3249

This theorem is referenced by:  xpun 3312  resundi 3465  cdaassen 5082  fixp 11840

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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