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Theorem sspgval 22077
Description: Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspg.y  |-  Y  =  ( BaseSet `  W )
sspg.g  |-  G  =  ( +v `  U
)
sspg.f  |-  F  =  ( +v `  W
)
sspg.h  |-  H  =  ( SubSp `  U )
Assertion
Ref Expression
sspgval  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A F B )  =  ( A G B ) )

Proof of Theorem sspgval
StepHypRef Expression
1 sspg.y . . . 4  |-  Y  =  ( BaseSet `  W )
2 sspg.g . . . 4  |-  G  =  ( +v `  U
)
3 sspg.f . . . 4  |-  F  =  ( +v `  W
)
4 sspg.h . . . 4  |-  H  =  ( SubSp `  U )
51, 2, 3, 4sspg 22076 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  F  =  ( G  |`  ( Y  X.  Y
) ) )
65oveqd 6038 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( A F B )  =  ( A ( G  |`  ( Y  X.  Y
) ) B ) )
7 ovres 6153 . 2  |-  ( ( A  e.  Y  /\  B  e.  Y )  ->  ( A ( G  |`  ( Y  X.  Y
) ) B )  =  ( A G B ) )
86, 7sylan9eq 2440 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  H )  /\  ( A  e.  Y  /\  B  e.  Y
) )  ->  ( A F B )  =  ( A G B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    X. cxp 4817    |` cres 4821   ` cfv 5395  (class class class)co 6021   NrmCVeccnv 21912   +vcpv 21913   BaseSetcba 21914   SubSpcss 22069
This theorem is referenced by:  sspmval  22081  sspival  22086  sspph  22205  minvecolem2  22226  hhshsslem2  22617
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-1st 6289  df-2nd 6290  df-grpo 21628  df-ablo 21719  df-vc 21874  df-nv 21920  df-va 21923  df-ba 21924  df-sm 21925  df-0v 21926  df-nmcv 21928  df-ssp 22070
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