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Theorem xpeq1 4405
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.
StepHypRef Expression
1 eleq2 2146 . . . 4  |-  ( A  =  B  ->  (
x  e.  A  <->  x  e.  B ) )
21anbi1d 453 . . 3  |-  ( A  =  B  ->  (
( x  e.  A  /\  y  e.  C
)  <->  ( x  e.  B  /\  y  e.  C ) ) )
32opabbidv 3864 . 2  |-  ( A  =  B  ->  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  C ) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  C ) } )
4 df-xp 4397 . 2  |-  ( A  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  e.  C ) }
5 df-xp 4397 . 2  |-  ( B  X.  C )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  C ) }
63, 4, 53eqtr4g 2140 1  |-  ( A  =  B  ->  ( A  X.  C )  =  ( B  X.  C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   {copab 3858    X. cxp 4389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-opab 3860  df-xp 4397
This theorem is referenced by:  xpeq12  4410  xpeq1i  4411  xpeq1d  4414  opthprc  4437  reseq2  4655  xpeq0r  4796  xpdisj1  4797  xpima1  4817  xpsneng  6387  xpcomeng  6393  xpdom2g  6397  xpfi  6472  hashxp  9902
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