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Theorem sspsval 22080
Description: Scalar multiplication on a subspace in terms of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
ssps.y  |-  Y  =  ( BaseSet `  W )
ssps.s  |-  S  =  ( .s OLD `  U
)
ssps.r  |-  R  =  ( .s OLD `  W
)
ssps.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
sspsval  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  CC  /\  B  e.  Y ) )  ->  ( A R B )  =  ( A S B ) )

Proof of Theorem sspsval
StepHypRef Expression
1 ssps.y . . . 4  |-  Y  =  ( BaseSet `  W )
2 ssps.s . . . 4  |-  S  =  ( .s OLD `  U
)
3 ssps.r . . . 4  |-  R  =  ( .s OLD `  W
)
4 ssps.h . . . 4  |-  H  =  ( SubSp `  U )
51, 2, 3, 4ssps 22079 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  R  =  ( S  |`  ( CC  X.  Y
) ) )
65oveqd 6039 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( A R B )  =  ( A ( S  |`  ( CC  X.  Y
) ) B ) )
7 ovres 6154 . 2  |-  ( ( A  e.  CC  /\  B  e.  Y )  ->  ( A ( S  |`  ( CC  X.  Y
) ) B )  =  ( A S B ) )
86, 7sylan9eq 2441 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  CC  /\  B  e.  Y ) )  ->  ( A R B )  =  ( A S B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    X. cxp 4818    |` cres 4822   ` cfv 5396  (class class class)co 6022   CCcc 8923   NrmCVeccnv 21913   BaseSetcba 21915   .s OLDcns 21916   SubSpcss 22070
This theorem is referenced by:  sspmval  22082  sspival  22087  minvecolem2  22227  hhshsslem2  22618
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-1st 6290  df-2nd 6291  df-vc 21875  df-nv 21921  df-va 21924  df-ba 21925  df-sm 21926  df-0v 21927  df-nmcv 21929  df-ssp 22071
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