Theorem xpeq1 4763

Description: Equality theorem for cross product. (Contributed by NM, 4-Jul-1994.)

Assertion

Ref	Expression
xpeq1	|- ( A = B -> ( A X. C ) = ( B X. C ) )

Proof of Theorem xpeq1

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

Step	Hyp	Ref	Expression
1		eleq2 2296	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	anbi1d 465	. . 3 |- ( A = B -> ( ( x e. A /\ y e. C ) <-> ( x e. B /\ y e. C ) ) )
3	2	opabbidv 4176	. 2 |- ( A = B -> { <. x , y >. | ( x e. A /\ y e. C ) } = { <. x , y >. | ( x e. B /\ y e. C ) } )
4		df-xp 4755	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
5		df-xp 4755	. 2 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
6	3, 4, 5	3eqtr4g 2290	1 |- ( A = B -> ( A X. C ) = ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   {copab 4170   X. cxp 4747

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-opab 4172  df-xp 4755

This theorem is referenced by:  xpeq12  4768  xpeq1i  4769  xpeq1d  4772  opthprc  4801  reseq2  5033  xpeq0r  5185  xpdisj1  5187  xpima1  5209  pmvalg  6893  xpsneng  7073  xpcomeng  7079  xpdom2g  7083  xpfi  7192  exmidomni  7433  exmidfodomrlemim  7504  hashxp  11191  txuni2  15121  txbas  15123  txopn  15130  txrest  15141  txdis  15142  txdis1cn  15143  xmettxlem  15374  xmettx  15375  dvmptid  15581  incistruhgr  16085