Theorem xpeq2 4787

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 2302	. . . 4 |- ( A = B -> ( y e. A <-> y e. B ) )
2	1	anbi2d 468	. . 3 |- ( A = B -> ( ( x e. C /\ y e. A ) <-> ( x e. C /\ y e. B ) ) )
3	2	opabbidv 4195	. 2 |- ( A = B -> { <. x , y >. | ( x e. C /\ y e. A ) } = { <. x , y >. | ( x e. C /\ y e. B ) } )
4		df-xp 4778	. 2 |- ( C X. A ) = { <. x , y >. | ( x e. C /\ y e. A ) }
5		df-xp 4778	. 2 |- ( C X. B ) = { <. x , y >. | ( x e. C /\ y e. B ) }
6	3, 4, 5	3eqtr4g 2296	1 |- ( A = B -> ( C X. A ) = ( C X. B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {copab 4189   X. cxp 4770

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-opab 4191  df-xp 4778

This theorem is used by:  xpeq12  4791  xpeq2i  4793  xpeq2d  4796  xpeq0r  5208  xpdisj2  5211  pmvalg  6927  xpcomeng  7120  djueq12  7373  txuni2  15340  txbas  15342  txopn  15349  txrest  15360  txdis  15361  txdis1cn  15362  xmettxlem  15593  xmettx  15594