Theorem xpeq12 4694

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 4689	. 2 |- ( A = B -> ( A X. C ) = ( B X. C ) )
2		xpeq2 4690	. 2 |- ( C = D -> ( B X. C ) = ( B X. D ) )
3	1, 2	sylan9eq 2258	1 |- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   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:  xpeq12i  4697  xpeq12d  4700  xpid11  4901  xp11m  5121  tapeq2  7365  pwsval  13123  txtopon  14734  txbasval  14739  ismet  14816  isxmet  14817