Theorem inxp 4855

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 4853	. . 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 3387	. . . . . 6 |- ( x e. ( A i^i C ) <-> ( x e. A /\ x e. C ) )
4		elin 3387	. . . . . 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 4150	. . 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 2250	. 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 4724	. . 3 |- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }
10		df-xp 4724	. . 3 |- ( C X. D ) = { <. x , y >. | ( x e. C /\ y e. D ) }
11	9, 10	ineq12i 3403	. 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 4724	. 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 2260	1 |- ( ( A X. B ) i^i ( C X. D ) ) = ( ( A i^i C ) X. ( B i^i D ) )

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   i^i cin 3196   {copab 4143   X. cxp 4716

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145  df-xp 4724  df-rel 4725

This theorem is referenced by:  xpindi  4856  xpindir  4857  dmxpin  4945  xpssres  5039  xpdisj1  5152  xpdisj2  5153  imainrect  5173  xpima1  5174  xpima2m  5175  hashxp  11043  txbas  14926  txrest  14944  metreslem  15048