Theorem xpundi 4731

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 4681	. 2 |- ( A X. ( B u. C ) ) = { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) }
2		df-xp 4681	. . . 4 |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
3		df-xp 4681	. . . 4 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
4	2, 3	uneq12i 3325	. . 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 3314	. . . . . . 7 |- ( y e. ( B u. C ) <-> ( y e. B \/ y e. C ) )
6	5	anbi2i 457	. . . . . 6 |- ( ( x e. A /\ y e. ( B u. C ) ) <-> ( x e. A /\ ( y e. B \/ y e. C ) ) )
7		andi 820	. . . . . 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 184	. . . . 5 |- ( ( x e. A /\ y e. ( B u. C ) ) <-> ( ( x e. A /\ y e. B ) \/ ( x e. A /\ y e. C ) ) )
9	8	opabbii 4111	. . . 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 4123	. . . 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 2229	. . 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 2229	. 2 |- ( ( A X. B ) u. ( A X. C ) ) = { <. x , y >. | ( x e. A /\ y e. ( B u. C ) ) }
13	1, 12	eqtr4i 2229	1 |- ( A X. ( B u. C ) ) = ( ( A X. B ) u. ( A X. C ) )

Syntax hints:   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2176   u. cun 3164   {copab 4104   X. cxp 4673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-opab 4106  df-xp 4681

This theorem is referenced by:  xpun  4736  djuassen  7329  xpdjuen  7330