Theorem inxp 4761

Description: The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
inxp	|- ( ( A X. B ) i^i ( C X. D ) ) = ( ( A i^i C ) X. ( B i^i D ) )

Proof of Theorem inxp

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

Step	Hyp	Ref	Expression
1		inopab 4759	. . 3 |- ( { <. x , y >. | ( x e. A /\ y e. B ) } i^i { <. x , y >. | ( x e. C /\ y e. D ) } ) = { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ( x e. C /\ y e. D ) ) }
2		an4 586	. . . . 5 |- ( ( ( x e. A /\ y e. B ) /\ ( x e. C /\ y e. D ) ) <-> ( ( x e. A /\ x e. C ) /\ ( y e. B /\ y e. D ) ) )
3		elin 3318	. . . . . 6 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3318	. . . . . 6 |- ( y e. ( B i^i D ) <-> ( y e. B /\ y e. D ) )
5	3, 4	anbi12i 460	. . . . 5 |- ( ( x e. ( A i^i C ) /\ y e. ( B i^i D ) ) <-> ( ( x e. A /\ x e. C ) /\ ( y e. B /\ y e. D ) ) )
6	2, 5	bitr4i 187	. . . 4 |- ( ( ( x e. A /\ y e. B ) /\ ( x e. C /\ y e. D ) ) <-> ( x e. ( A i^i C ) /\ y e. ( B i^i D ) ) )
7	6	opabbii 4070	. . 3 |- { <. x , y >. | ( ( x e. A /\ y e. B ) /\ ( x e. C /\ y e. D ) ) } = { <. x , y >. | ( x e. ( A i^i C ) /\ y e. ( B i^i D ) ) }
8	1, 7	eqtri 2198	. 2 |- ( { <. x , y >. | ( x e. A /\ y e. B ) } i^i { <. x , y >. | ( x e. C /\ y e. D ) } ) = { <. x , y >. | ( x e. ( A i^i C ) /\ y e. ( B i^i D ) ) }
9		df-xp 4632	. . 3 |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
10		df-xp 4632	. . 3 |- ( C X. D ) = { <. x , y >. | ( x e. C /\ y e. D ) }
11	9, 10	ineq12i 3334	. 2 |- ( ( A X. B ) i^i ( C X. D ) ) = ( { <. x , y >. | ( x e. A /\ y e. B ) } i^i { <. x , y >. | ( x e. C /\ y e. D ) } )
12		df-xp 4632	. 2 |- ( ( A i^i C ) X. ( B i^i D ) ) = { <. x , y >. | ( x e. ( A i^i C ) /\ y e. ( B i^i D ) ) }
13	8, 11, 12	3eqtr4i 2208	1 |- ( ( A X. B ) i^i ( C X. D ) ) = ( ( A i^i C ) X. ( B i^i D ) )

Syntax hints:   /\ wa 104   = wceq 1353   e. wcel 2148   i^i cin 3128   {copab 4063   X. cxp 4624

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-opab 4065  df-xp 4632  df-rel 4633

This theorem is referenced by:  xpindi  4762  xpindir  4763  dmxpin  4849  xpssres  4942  xpdisj1  5053  xpdisj2  5054  imainrect  5074  xpima1  5075  xpima2m  5076  hashxp  10801  txbas  13651  txrest  13669  metreslem  13773