Theorem xpssres 4919

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 4616	. . 3 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
2		inxp 4738	. . 3 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
3		incom 3314	. . . 4 |- ( A i^i C ) = ( C i^i A )
4		inv1 3445	. . . 4 |- ( B i^i _V ) = B
5	3, 4	xpeq12i 4626	. . 3 |- ( ( A i^i C ) X. ( B i^i _V ) ) = ( ( C i^i A ) X. B )
6	1, 2, 5	3eqtri 2190	. 2 |- ( ( A X. B ) |` C ) = ( ( C i^i A ) X. B )
7		df-ss 3129	. . . 4 |- ( C C_ A <-> ( C i^i A ) = C )
8	7	biimpi 119	. . 3 |- ( C C_ A -> ( C i^i A ) = C )
9	8	xpeq1d 4627	. 2 |- ( C C_ A -> ( ( C i^i A ) X. B ) = ( C X. B ) )
10	6, 9	syl5eq 2211	1 |- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) )

Syntax hints:   -> wi 4   = wceq 1343   _Vcvv 2726   i^i cin 3115   C_ wss 3116   X. cxp 4602   |` cres 4606

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-xp 4610  df-rel 4611  df-res 4616

This theorem is referenced by:  cnconst2  12873