Theorem xpeq1 4553

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 2203	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	anbi1d 460	. . 3 |- ( A = B -> ( ( x e. A /\ y e. C ) <-> ( x e. B /\ y e. C ) ) )
3	2	opabbidv 3994	. 2 |- ( A = B -> { <. x , y >. | ( x e. A /\ y e. C ) } = { <. x , y >. | ( x e. B /\ y e. C ) } )
4		df-xp 4545	. 2 |- ( A X. C ) = { <. x , y >. | ( x e. A /\ y e. C ) }
5		df-xp 4545	. 2 |- ( B X. C ) = { <. x , y >. | ( x e. B /\ y e. C ) }
6	3, 4, 5	3eqtr4g 2197	1 |- ( A = B -> ( A X. C ) = ( B X. C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   {copab 3988   X. cxp 4537

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 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-opab 3990  df-xp 4545

This theorem is referenced by:  xpeq12  4558  xpeq1i  4559  xpeq1d  4562  opthprc  4590  reseq2  4814  xpeq0r  4961  xpdisj1  4963  xpima1  4985  pmvalg  6553  xpsneng  6716  xpcomeng  6722  xpdom2g  6726  xpfi  6818  exmidomni  7014  exmidfodomrlemim  7057  hashxp  10572  txuni2  12425  txbas  12427  txopn  12434  txrest  12445  txdis  12446  txdis1cn  12447  xmettxlem  12678  xmettx  12679