Theorem xpundi 4590

Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)

Assertion

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

Proof of Theorem xpundi

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-xp 4540	. 2 |- ( A X. ( B u. C ) ) = { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) }
2		df-xp 4540	. . . 4 |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
3		df-xp 4540	. . . 4 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
4	2, 3	uneq12i 3223	. . 3 |- ( ( A X. B ) u. ( A X. C ) ) = ( { <. x , y >. | ( x e. A /\ y e. B ) } u. { <. x , y >. | ( x e. A /\ y e. C ) } )
5		elun 3212	. . . . . . 7 |- ( y e. ( B u. C ) <-> ( y e. B \/ y e. C ) )
6	5	anbi2i 452	. . . . . 6 |- ( ( x e. A /\ y e. ( B u. C ) ) <-> ( x e. A /\ ( y e. B \/ y e. C ) ) )
7		andi 807	. . . . . 6 |- ( ( x e. A /\ ( y e. B \/ y e. C ) ) <-> ( ( x e. A /\ y e. B ) \/ ( x e. A /\ y e. C ) ) )
8	6, 7	bitri 183	. . . . 5 |- ( ( x e. A /\ y e. ( B u. C ) ) <-> ( ( x e. A /\ y e. B ) \/ ( x e. A /\ y e. C ) ) )
9	8	opabbii 3990	. . . 4 |- { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) } = { <. x , y >. | ( ( x e. A /\ y e. B ) \/ ( x e. A /\ y e. C ) ) }
10		unopab 4002	. . . 4 |- ( { <. 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 ) ) }
11	9, 10	eqtr4i 2161	. . 3 |- { <. 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 ) } )
12	4, 11	eqtr4i 2161	. 2 |- ( ( A X. B ) u. ( A X. C ) ) = { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) }
13	1, 12	eqtr4i 2161	1 |- ( A X. ( B u. C ) ) = ( ( A X. B ) u. ( A X. C ) )

Syntax hints:   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   u. cun 3064   {copab 3983   X. cxp 4532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-opab 3985  df-xp 4540

This theorem is referenced by:  xpun  4595  djuassen  7066  xpdjuen  7067