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Theorem sspgval 21305
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 21304 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  F  =  ( G  |`  ( Y  X.  Y
) ) )
65oveqd 5875 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  H )  ->  ( A F B )  =  ( A ( G  |`  ( Y  X.  Y
) ) B ) )
7 ovres 5987 . 2  |-  ( ( A  e.  Y  /\  B  e.  Y )  ->  ( A ( G  |`  ( Y  X.  Y
) ) B )  =  ( A G B ) )
86, 7sylan9eq 2335 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 358    = wceq 1623    e. wcel 1684    X. cxp 4687    |` cres 4691   ` cfv 5255  (class class class)co 5858   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   SubSpcss 21297
This theorem is referenced by:  sspmval  21309  sspival  21314  sspph  21433  minvecolem2  21454  hhshsslem2  21845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-1st 6122  df-2nd 6123  df-grpo 20858  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-ssp 21298
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