Theorem xpssres 4663

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

Step	Hyp	Ref	Expression
1		df-res 4375	. . 3 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
2		inxp 4488	. . 3 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
3		incom 3158	. . . 4 |- ( A i^i C ) = ( C i^i A )
4		inv1 3280	. . . 4 |- ( B i^i _V ) = B
5	3, 4	xpeq12i 4385	. . 3 |- ( ( A i^i C ) X. ( B i^i _V ) ) = ( ( C i^i A ) X. B )
6	1, 2, 5	3eqtri 2105	. 2 |- ( ( A X. B ) |` C ) = ( ( C i^i A ) X. B )
7		df-ss 2986	. . . 4 |- ( C C_ A <-> ( C i^i A ) = C )
8	7	biimpi 118	. . 3 |- ( C C_ A -> ( C i^i A ) = C )
9	8	xpeq1d 4386	. 2 |- ( C C_ A -> ( ( C i^i A ) X. B ) = ( C X. B ) )
10	6, 9	syl5eq 2125	1 |- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) )

Syntax hints:   -> wi 4   = wceq 1284   _Vcvv 2601   i^i cin 2972   C_ wss 2973   X. cxp 4361   |` cres 4365

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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-opab 3840  df-xp 4369  df-rel 4370  df-res 4375

This theorem is referenced by: (None)