Theorem xpeq12 4738

Description: Equality theorem for cross product. (Contributed by FL, 31-Aug-2009.)

Assertion

Ref	Expression
xpeq12	|- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )

Proof of Theorem xpeq12

Step	Hyp	Ref	Expression
1		xpeq1 4733	. 2 |- ( A = B -> ( A X. C ) = ( B X. C ) )
2		xpeq2 4734	. 2 |- ( C = D -> ( B X. C ) = ( B X. D ) )
3	1, 2	sylan9eq 2282	1 |- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   X. cxp 4717

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-opab 4146  df-xp 4725

This theorem is referenced by:  xpeq12i  4741  xpeq12d  4744  xpid11  4947  xp11m  5167  tapeq2  7439  pwsval  13324  txtopon  14936  txbasval  14941  ismet  15018  isxmet  15019