Theorem xpeq2 4662

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 2253	. . . 4 |- ( A = B -> ( y e. A <-> y e. B ) )
2	1	anbi2d 464	. . 3 |- ( A = B -> ( ( x e. C /\ y e. A ) <-> ( x e. C /\ y e. B ) ) )
3	2	opabbidv 4087	. 2 |- ( A = B -> { <. x , y >. | ( x e. C /\ y e. A ) } = { <. x , y >. | ( x e. C /\ y e. B ) } )
4		df-xp 4653	. 2 |- ( C X. A ) = { <. x , y >. | ( x e. C /\ y e. A ) }
5		df-xp 4653	. 2 |- ( C X. B ) = { <. x , y >. | ( x e. C /\ y e. B ) }
6	3, 4, 5	3eqtr4g 2247	1 |- ( A = B -> ( C X. A ) = ( C X. B ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   {copab 4081   X. cxp 4645

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-opab 4083  df-xp 4653

This theorem is referenced by:  xpeq12  4666  xpeq2i  4668  xpeq2d  4671  xpeq0r  5072  xpdisj2  5075  pmvalg  6689  xpcomeng  6858  djueq12  7072  txuni2  14241  txbas  14243  txopn  14250  txrest  14261  txdis  14262  txdis1cn  14263  xmettxlem  14494  xmettx  14495