Theorem xpssres 4926

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 4623	. . 3 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
2		inxp 4745	. . 3 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
3		incom 3319	. . . 4 |- ( A i^i C ) = ( C i^i A )
4		inv1 3451	. . . 4 |- ( B i^i _V ) = B
5	3, 4	xpeq12i 4633	. . 3 |- ( ( A i^i C ) X. ( B i^i _V ) ) = ( ( C i^i A ) X. B )
6	1, 2, 5	3eqtri 2195	. 2 |- ( ( A X. B ) |` C ) = ( ( C i^i A ) X. B )
7		df-ss 3134	. . . 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 4634	. 2 |- ( C C_ A -> ( ( C i^i A ) X. B ) = ( C X. B ) )
10	6, 9	eqtrid 2215	1 |- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) )

Syntax hints:   -> wi 4   = wceq 1348   _Vcvv 2730   i^i cin 3120   C_ wss 3121   X. cxp 4609   |` cres 4613

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617  df-rel 4618  df-res 4623

This theorem is referenced by:  cnconst2  13027