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Theorem xpssres 4673
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 4385 . . 3  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
2 inxp 4498 . . 3  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
3 incom 3157 . . . 4  |-  ( A  i^i  C )  =  ( C  i^i  A
)
4 inv1 3281 . . . 4  |-  ( B  i^i  _V )  =  B
53, 4xpeq12i 4395 . . 3  |-  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  ( ( C  i^i  A
)  X.  B )
61, 2, 53eqtri 2080 . 2  |-  ( ( A  X.  B )  |`  C )  =  ( ( C  i^i  A
)  X.  B )
7 df-ss 2959 . . . 4  |-  ( C 
C_  A  <->  ( C  i^i  A )  =  C )
87biimpi 117 . . 3  |-  ( C 
C_  A  ->  ( C  i^i  A )  =  C )
98xpeq1d 4396 . 2  |-  ( C 
C_  A  ->  (
( C  i^i  A
)  X.  B )  =  ( C  X.  B ) )
106, 9syl5eq 2100 1  |-  ( C 
C_  A  ->  (
( A  X.  B
)  |`  C )  =  ( C  X.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1259   _Vcvv 2574    i^i cin 2944    C_ wss 2945    X. cxp 4371    |` cres 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379  df-rel 4380  df-res 4385
This theorem is referenced by: (None)
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