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Theorem invaddvec 25467
Description: Additive inverse of a sum of vectors. (Contributed by FL, 13-Sep-2010.)
Hypotheses
Ref Expression
sum2vv.1  |-  + w  =  ( 1st `  ( 2nd `  R ) )
sum2vv.2  |-  W  =  ran  + w
invaddvec.2  |-  ~ w  =  ( inv `  + w )
Assertion
Ref Expression
invaddvec  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  ( ~ w `  ( V1 + w V 2 )
)  =  ( ( ~ w `  V1 ) + w ( ~ w `  V 2 ) ) )

Proof of Theorem invaddvec
StepHypRef Expression
1 sum2vv.1 . . . . 5  |-  + w  =  ( 1st `  ( 2nd `  R ) )
21vecax1 25453 . . . 4  |-  ( R  e.  Vec  ->  + w  e.  AbelOp )
3 ablogrpo 20951 . . . 4  |-  ( + w  e.  AbelOp  ->  + w  e.  GrpOp )
4 sum2vv.2 . . . . . 6  |-  W  =  ran  + w
5 invaddvec.2 . . . . . 6  |-  ~ w  =  ( inv `  + w )
64, 5grpoinvop 20908 . . . . 5  |-  ( ( + w  e.  GrpOp  /\  V1  e.  W  /\  V 2  e.  W )  ->  ( ~ w `  ( V1 + w V 2 ) )  =  ( ( ~ w `  V 2 ) + w ( ~ w `  V1 ) ) )
763expib 1154 . . . 4  |-  ( + w  e.  GrpOp  ->  (
( V1  e.  W  /\  V 2  e.  W
)  ->  ( ~ w `  ( V1 + w V 2 ) )  =  ( ( ~ w `  V 2
) + w ( ~ w `  V1 )
) ) )
82, 3, 73syl 18 . . 3  |-  ( R  e.  Vec  ->  (
( V1  e.  W  /\  V 2  e.  W
)  ->  ( ~ w `  ( V1 + w V 2 ) )  =  ( ( ~ w `  V 2
) + w ( ~ w `  V1 )
) ) )
98imp 418 . 2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  ( ~ w `  ( V1 + w V 2 )
)  =  ( ( ~ w `  V 2 ) + w
( ~ w `  V1 ) ) )
101rneqi 4905 . . . . . . 7  |-  ran  + w  =  ran  ( 1st `  ( 2nd `  R
) )
114, 10eqtri 2303 . . . . . 6  |-  W  =  ran  ( 1st `  ( 2nd `  R ) )
121fveq2i 5528 . . . . . . 7  |-  ( inv `  + w )  =  ( inv `  ( 1st `  ( 2nd `  R
) ) )
135, 12eqtri 2303 . . . . . 6  |-  ~ w  =  ( inv `  ( 1st `  ( 2nd `  R
) ) )
1411, 13claddinvvec 25460 . . . . 5  |-  ( ( R  e.  Vec  /\  V1  e.  W )  -> 
( ~ w `  V1 )  e.  W )
1511, 13claddinvvec 25460 . . . . 5  |-  ( ( R  e.  Vec  /\  V 2  e.  W
)  ->  ( ~ w `  V 2 )  e.  W )
1614, 15anim12dan 810 . . . 4  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  (
( ~ w `  V1 )  e.  W  /\  ( ~ w `  V 2 )  e.  W
) )
1716ancomd 438 . . 3  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  (
( ~ w `  V 2 )  e.  W  /\  ( ~ w `  V1 )  e.  W ) )
181, 4addvecom 25466 . . 3  |-  ( ( R  e.  Vec  /\  ( ( ~ w `  V 2 )  e.  W  /\  ( ~ w `  V1 )  e.  W ) )  -> 
( ( ~ w `  V 2 ) + w ( ~ w `  V1 ) )  =  ( ( ~ w `  V1 ) + w
( ~ w `  V 2 ) ) )
1917, 18syldan 456 . 2  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  (
( ~ w `  V 2 ) + w
( ~ w `  V1 ) )  =  ( ( ~ w `  V1 ) + w ( ~ w `  V 2
) ) )
209, 19eqtrd 2315 1  |-  ( ( R  e.  Vec  /\  ( V1  e.  W  /\  V 2  e.  W
) )  ->  ( ~ w `  ( V1 + w V 2 )
)  =  ( ( ~ w `  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   invcgn 20855   AbelOpcablo 20948    Vec cvec 25449
This theorem is referenced by:  dblsubvec  25474
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-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-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-vec 25450
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