Theorem xpssres 5046

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 4735	. . 3 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
2		inxp 4862	. . 3 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
3		incom 3397	. . . 4 |- ( A i^i C ) = ( C i^i A )
4		inv1 3529	. . . 4 |- ( B i^i _V ) = B
5	3, 4	xpeq12i 4745	. . 3 |- ( ( A i^i C ) X. ( B i^i _V ) ) = ( ( C i^i A ) X. B )
6	1, 2, 5	3eqtri 2254	. 2 |- ( ( A X. B ) |` C ) = ( ( C i^i A ) X. B )
7		df-ss 3211	. . . 4 |- ( C C_ A <-> ( C i^i A ) = C )
8	7	biimpi 120	. . 3 |- ( C C_ A -> ( C i^i A ) = C )
9	8	xpeq1d 4746	. 2 |- ( C C_ A -> ( ( C i^i A ) X. B ) = ( C X. B ) )
10	6, 9	eqtrid 2274	1 |- ( C C_ A -> ( ( A X. B ) |` C ) = ( C X. B ) )

Syntax hints:   -> wi 4   = wceq 1395   _Vcvv 2800   i^i cin 3197   C_ wss 3198   X. cxp 4721   |` cres 4725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-opab 4149  df-xp 4729  df-rel 4730  df-res 4735

This theorem is referenced by:  cnconst2  14947