Theorem xpundi 4414

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 4369	. 2 |- ( A X. ( B u. C ) ) = { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) }
2		df-xp 4369	. . . 4 |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
3		df-xp 4369	. . . 4 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
4	2, 3	uneq12i 3124	. . 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 3113	. . . . . . 7 |- ( y e. ( B u. C ) <-> ( y e. B \/ y e. C ) )
6	5	anbi2i 444	. . . . . 6 |- ( ( x e. A /\ y e. ( B u. C ) ) <-> ( x e. A /\ ( y e. B \/ y e. C ) ) )
7		andi 764	. . . . . 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 182	. . . . 5 |- ( ( x e. A /\ y e. ( B u. C ) ) <-> ( ( x e. A /\ y e. B ) \/ ( x e. A /\ y e. C ) ) )
9	8	opabbii 3845	. . . 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 3857	. . . 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 2104	. . 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 2104	. 2 |- ( ( A X. B ) u. ( A X. C ) ) = { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) }
13	1, 12	eqtr4i 2104	1 |- ( A X. ( B u. C ) ) = ( ( A X. B ) u. ( A X. C ) )

Syntax hints:   /\ wa 102   \/ wo 661   = wceq 1284   e. wcel 1433   u. cun 2971   {copab 3838   X. cxp 4361

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-opab 3840  df-xp 4369

This theorem is referenced by:  xpun  4419