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Theorem sub2vec 25472
Description: Definition of the subtraction of two vectors. (Contributed by FL, 12-Sep-2010.)
Hypotheses
Ref Expression
vwit.1  |-  0 w  =  (GId `  + w )
vwit.2  |-  + w  =  ( 1st `  ( 2nd `  R ) )
vwit.3  |-  - w  =  (  /g  `  + w )
vwit.4  |-  W  =  ran  + w
vwit.5  |-  ~ w  =  ( inv `  + w )
Assertion
Ref Expression
sub2vec  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  ( V1 - w V 2
)  =  ( V1 + w ( ~ w `  V 2 ) ) )

Proof of Theorem sub2vec
StepHypRef Expression
1 vwit.2 . . . . 5  |-  + w  =  ( 1st `  ( 2nd `  R ) )
21vecax1 25453 . . . 4  |-  ( R  e.  Vec  ->  + w  e.  AbelOp )
3 ablogrpo 20951 . . . 4  |-  ( + w  e.  AbelOp  ->  + w  e.  GrpOp )
42, 3syl 15 . . 3  |-  ( R  e.  Vec  ->  + w  e.  GrpOp )
54adantr 451 . 2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  + w  e.  GrpOp )
6 simprl 732 . 2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  V1  e.  W )
7 simprr 733 . 2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  V 2  e.  W )
8 vwit.4 . . 3  |-  W  =  ran  + w
9 vwit.5 . . 3  |-  ~ w  =  ( inv `  + w )
10 vwit.3 . . 3  |-  - w  =  (  /g  `  + w )
118, 9, 10grpodivval 20910 . 2  |-  ( ( + w  e.  GrpOp  /\  V1  e.  W  /\  V 2  e.  W )  ->  ( V1 - w V 2 )  =  (
V1 + w ( ~ w `  V 2
) ) )
125, 6, 7, 11syl3anc 1182 1  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  ( V1 - w V 2
)  =  ( V1 + w ( ~ w `  V 2 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854   invcgn 20855    /g cgs 20856   AbelOpcablo 20948    Vec cvec 25449
This theorem is referenced by:  dblsubvec  25474  mvecrtol2  25477  mulinvsca  25480
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-mpt2 5863  df-1st 6122  df-2nd 6123  df-gdiv 20861  df-ablo 20949  df-vec 25450
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