Theorem xpundir 4668

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 4617	. 2 |- ( ( A u. B ) X. C ) = { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) }
2		df-xp 4617	. . . 4 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
3		df-xp 4617	. . . 4 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
4	2, 3	uneq12i 3279	. . 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 3268	. . . . . . 7 |- ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
6	5	anbi1i 455	. . . . . 6 |- ( ( x e. ( A u. B ) /\ y e. C ) <-> ( ( x e. A \/ x e. B ) /\ y e. C ) )
7		andir 814	. . . . . 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 183	. . . . 5 |- ( ( x e. ( A u. B ) /\ y e. C ) <-> ( ( x e. A /\ y e. C ) \/ ( x e. B /\ y e. C ) ) )
9	8	opabbii 4056	. . . 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 4068	. . . 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 2194	. . 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 2194	. 2 |- ( ( A X. C ) u. ( B X. C ) ) = { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) }
13	1, 12	eqtr4i 2194	1 |- ( ( A u. B ) X. C ) = ( ( A X. C ) u. ( B X. C ) )

Syntax hints:   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   u. cun 3119   {copab 4049   X. cxp 4609

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-opab 4051  df-xp 4617

This theorem is referenced by:  xpun  4672  resundi  4904  xpfi  6907  xp2dju  7192  hashxp  10761