Theorem xpundir 4789

Description: Distributive law for cross product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)

Assertion

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

Proof of Theorem xpundir

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

Step	Hyp	Ref	Expression
1		df-xp 4737	. 2 |- ( ( A u. B ) X. C ) = { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) }
2		df-xp 4737	. . . 4 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
3		df-xp 4737	. . . 4 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
4	2, 3	uneq12i 3361	. . 3 |- ( ( A X. C ) u. ( B X. C ) ) = ( { <. x , y >. | ( x e. A /\ y e. C ) } u. { <. x , y >. | ( x e. B /\ y e. C ) } )
5		elun 3350	. . . . . . 7 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6	5	anbi1i 458	. . . . . 6 |- ( ( x e. ( A u. B ) /\ y e. C ) <-> ( ( x e. A \/ x e. B ) /\ y e. C ) )
7		andir 827	. . . . . 6 |- ( ( ( x e. A \/ x e. B ) /\ y e. C ) <-> ( ( x e. A /\ y e. C ) \/ ( x e. B /\ y e. C ) ) )
8	6, 7	bitri 184	. . . . 5 |- ( ( x e. ( A u. B ) /\ y e. C ) <-> ( ( x e. A /\ y e. C ) \/ ( x e. B /\ y e. C ) ) )
9	8	opabbii 4161	. . . 4 |- { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) } = { <. x , y >. | ( ( x e. A /\ y e. C ) \/ ( x e. B /\ y e. C ) ) }
10		unopab 4173	. . . 4 |- ( { <. 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 ) ) }
11	9, 10	eqtr4i 2255	. . 3 |- { <. 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 ) } )
12	4, 11	eqtr4i 2255	. 2 |- ( ( A X. C ) u. ( B X. C ) ) = { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) }
13	1, 12	eqtr4i 2255	1 |- ( ( A u. B ) X. C ) = ( ( A X. C ) u. ( B X. C ) )

Syntax hints:   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2202   u. cun 3199   {copab 4154   X. cxp 4729

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-opab 4156  df-xp 4737

This theorem is referenced by:  xpun  4793  resundi  5032  xpfi  7167  xp2dju  7490  hashxp  11153