Theorem xpeq2 4562

Description: Equality theorem for cross product. (Contributed by NM, 5-Jul-1994.)

Assertion

Ref	Expression
xpeq2	|- ( A = B -> ( C X. A ) = ( C X. B ) )

Proof of Theorem xpeq2

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

Step	Hyp	Ref	Expression
1		eleq2 2204	. . . 4 |- ( A = B -> ( y e. A <-> y e. B ) )
2	1	anbi2d 460	. . 3 |- ( A = B -> ( ( x e. C /\ y e. A ) <-> ( x e. C /\ y e. B ) ) )
3	2	opabbidv 4002	. 2 |- ( A = B -> { <. x , y >. | ( x e. C /\ y e. A ) } = { <. x , y >. | ( x e. C /\ y e. B ) } )
4		df-xp 4553	. 2 |- ( C X. A ) = { <. x , y >. | ( x e. C /\ y e. A ) }
5		df-xp 4553	. 2 |- ( C X. B ) = { <. x , y >. | ( x e. C /\ y e. B ) }
6	3, 4, 5	3eqtr4g 2198	1 |- ( A = B -> ( C X. A ) = ( C X. B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   {copab 3996   X. cxp 4545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-opab 3998  df-xp 4553

This theorem is referenced by:  xpeq12  4566  xpeq2i  4568  xpeq2d  4571  xpeq0r  4969  xpdisj2  4972  pmvalg  6561  xpcomeng  6730  djueq12  6932  txuni2  12464  txbas  12466  txopn  12473  txrest  12484  txdis  12485  txdis1cn  12486  xmettxlem  12717  xmettx  12718