Theorem xpundir 4720

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 4669	. 2 |- ( ( A u. B ) X. C ) = { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) }
2		df-xp 4669	. . . 4 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
3		df-xp 4669	. . . 4 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
4	2, 3	uneq12i 3315	. . 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 3304	. . . . . . 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 820	. . . . . 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 4100	. . . 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 4112	. . . 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 2220	. . 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 2220	. 2 |- ( ( A X. C ) u. ( B X. C ) ) = { <. x , y >. | ( x e. ( A u. B ) /\ y e. C ) }
13	1, 12	eqtr4i 2220	1 |- ( ( A u. B ) X. C ) = ( ( A X. C ) u. ( B X. C ) )

Syntax hints:   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2167   u. cun 3155   {copab 4093   X. cxp 4661

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-opab 4095  df-xp 4669

This theorem is referenced by:  xpun  4724  resundi  4959  xpfi  6993  xp2dju  7282  hashxp  10918