Theorem inxp 4811

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 4809	. . 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 3355	. . . . . 6 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3355	. . . . . 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 4110	. . 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 2225	. 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 4680	. . 3 |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
10		df-xp 4680	. . 3 |- ( C X. D ) = { <. x , y >. | ( x e. C /\ y e. D ) }
11	9, 10	ineq12i 3371	. 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 4680	. 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 2235	1 |- ( ( A X. B ) i^i ( C X. D ) ) = ( ( A i^i C ) X. ( B i^i D ) )

Syntax hints:   /\ wa 104   = wceq 1372   e. wcel 2175   i^i cin 3164   {copab 4103   X. cxp 4672

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-opab 4105  df-xp 4680  df-rel 4681

This theorem is referenced by:  xpindi  4812  xpindir  4813  dmxpin  4899  xpssres  4993  xpdisj1  5106  xpdisj2  5107  imainrect  5127  xpima1  5128  xpima2m  5129  hashxp  10969  txbas  14701  txrest  14719  metreslem  14823