Theorem xpeq12 4793

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 4788	. 2 |- ( A = B -> ( A X. C ) = ( B X. C ) )
2		xpeq2 4789	. 2 |- ( C = D -> ( B X. C ) = ( B X. D ) )
3	1, 2	sylan9eq 2291	1 |- ( ( A = B /\ C = D ) -> ( A X. C ) = ( B X. D ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   X. cxp 4772

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-opab 4193  df-xp 4780

This theorem is used by:  xpeq12i  4796  xpeq12d  4799  xpid11  5005  xp11m  5226  tapeq2  7619  pwsval  14204  txtopon  15363  txbasval  15368  ismet  15445  isxmet  15446