Theorem xpssres 4849

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 4546	. . 3 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
2		inxp 4668	. . 3 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
3		incom 3263	. . . 4 |- ( A i^i C ) = ( C i^i A )
4		inv1 3394	. . . 4 |- ( B i^i _V ) = B
5	3, 4	xpeq12i 4556	. . 3 |- ( ( A i^i C ) X. ( B i^i _V ) ) = ( ( C i^i A ) X. B )
6	1, 2, 5	3eqtri 2162	. 2 |- ( ( A X. B ) |` C ) = ( ( C i^i A ) X. B )
7		df-ss 3079	. . . 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 4557	. 2 |- ( C C_ A -> ( ( C i^i A ) X. B ) = ( C X. B ) )
10	6, 9	syl5eq 2182	1 |- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) )

Syntax hints:   -> wi 4   = wceq 1331   _Vcvv 2681   i^i cin 3065   C_ wss 3066   X. cxp 4532   |` cres 4536

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-opab 3985  df-xp 4540  df-rel 4541  df-res 4546

This theorem is referenced by:  cnconst2  12391