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Theorem xpssres 4824
Description: Restriction of a constant function (or other cross product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
xpssres  |-  ( C 
C_  A  ->  (
( A  X.  B
)  |`  C )  =  ( C  X.  B
) )

Proof of Theorem xpssres
StepHypRef Expression
1 df-res 4521 . . 3  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
2 inxp 4643 . . 3  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
3 incom 3238 . . . 4  |-  ( A  i^i  C )  =  ( C  i^i  A
)
4 inv1 3369 . . . 4  |-  ( B  i^i  _V )  =  B
53, 4xpeq12i 4531 . . 3  |-  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  ( ( C  i^i  A
)  X.  B )
61, 2, 53eqtri 2142 . 2  |-  ( ( A  X.  B )  |`  C )  =  ( ( C  i^i  A
)  X.  B )
7 df-ss 3054 . . . 4  |-  ( C 
C_  A  <->  ( C  i^i  A )  =  C )
87biimpi 119 . . 3  |-  ( C 
C_  A  ->  ( C  i^i  A )  =  C )
98xpeq1d 4532 . 2  |-  ( C 
C_  A  ->  (
( C  i^i  A
)  X.  B )  =  ( C  X.  B ) )
106, 9syl5eq 2162 1  |-  ( C 
C_  A  ->  (
( A  X.  B
)  |`  C )  =  ( C  X.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   _Vcvv 2660    i^i cin 3040    C_ wss 3041    X. cxp 4507    |` cres 4511
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-opab 3960  df-xp 4515  df-rel 4516  df-res 4521
This theorem is referenced by:  cnconst2  12329
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