Theorem xpeq1 4689

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 4110	. 2 |- ( A = B -> { <. x , y >. | ( x e. A /\ y e. C ) } = { <. x , y >. | ( x e. B /\ y e. C ) } )
4		df-xp 4681	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
5		df-xp 4681	. 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 4104   X. cxp 4673

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 4106  df-xp 4681

This theorem is referenced by:  xpeq12  4694  xpeq1i  4695  xpeq1d  4698  opthprc  4726  reseq2  4954  xpeq0r  5105  xpdisj1  5107  xpima1  5129  pmvalg  6746  xpsneng  6917  xpcomeng  6923  xpdom2g  6927  xpfi  7029  exmidomni  7244  exmidfodomrlemim  7309  hashxp  10971  txuni2  14728  txbas  14730  txopn  14737  txrest  14748  txdis  14749  txdis1cn  14750  xmettxlem  14981  xmettx  14982  dvmptid  15188