Theorem xpeq2 4549

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 2201	. . . 4 |- ( A = B -> ( y e. A <-> y e. B ) )
2	1	anbi2d 459	. . 3 |- ( A = B -> ( ( x e. C /\ y e. A ) <-> ( x e. C /\ y e. B ) ) )
3	2	opabbidv 3989	. 2 |- ( A = B -> { <. x , y >. | ( x e. C /\ y e. A ) } = { <. x , y >. | ( x e. C /\ y e. B ) } )
4		df-xp 4540	. 2 |- ( C X. A ) = { <. x , y >. | ( x e. C /\ y e. A ) }
5		df-xp 4540	. 2 |- ( C X. B ) = { <. x , y >. | ( x e. C /\ y e. B ) }
6	3, 4, 5	3eqtr4g 2195	1 |- ( A = B -> ( C X. A ) = ( C X. B ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   {copab 3983   X. cxp 4532

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-opab 3985  df-xp 4540

This theorem is referenced by:  xpeq12  4553  xpeq2i  4555  xpeq2d  4558  xpeq0r  4956  xpdisj2  4959  pmvalg  6546  xpcomeng  6715  djueq12  6917  txuni2  12414  txbas  12416  txopn  12423  txrest  12434  txdis  12435  txdis1cn  12436  xmettxlem  12667  xmettx  12668