Theorem xpeq1 4690

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 2269	. . . 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 4111	. 2 |- ( A = B -> { <. x , y >. | ( x e. A /\ y e. C ) } = { <. x , y >. | ( x e. B /\ y e. C ) } )
4		df-xp 4682	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
5		df-xp 4682	. 2 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
6	3, 4, 5	3eqtr4g 2263	1 |- ( A = B -> ( A X. C ) = ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   {copab 4105   X. cxp 4674

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-opab 4107  df-xp 4682

This theorem is referenced by:  xpeq12  4695  xpeq1i  4696  xpeq1d  4699  opthprc  4727  reseq2  4955  xpeq0r  5106  xpdisj1  5108  xpima1  5130  pmvalg  6748  xpsneng  6919  xpcomeng  6925  xpdom2g  6929  xpfi  7031  exmidomni  7246  exmidfodomrlemim  7311  hashxp  10973  txuni2  14761  txbas  14763  txopn  14770  txrest  14781  txdis  14782  txdis1cn  14783  xmettxlem  15014  xmettx  15015  dvmptid  15221