Theorem inxp 4909

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 4907	. . 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 592	. . . . 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 3412	. . . . . 6 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3412	. . . . . 6 |- ( y e. ( B i^i D ) <-> ( y e. B /\ y e. D ) )
5	3, 4	anbi12i 464	. . . . 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 4193	. . 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 2259	. 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 4775	. . 3 |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
10		df-xp 4775	. . 3 |- ( C X. D ) = { <. x , y >. | ( x e. C /\ y e. D ) }
11	9, 10	ineq12i 3430	. 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 4775	. 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 2269	1 |- ( ( A X. B ) i^i ( C X. D ) ) = ( ( A i^i C ) X. ( B i^i D ) )

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   i^i cin 3219   {copab 4186   X. cxp 4767

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-opab 4188  df-xp 4775  df-rel 4776

This theorem is referenced by:  xpindi  4910  xpindir  4911  dmxpin  4999  xpssres  5093  xpdisj1  5207  xpdisj2  5208  imainrect  5228  xpima1  5229  xpima2m  5230  hashxp  11245  txbas  15282  txrest  15300  metreslem  15404