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Theorem addnull2 25464
Description: Addition of the null vector. (Contributed by FL, 13-Sep-2010.)
Hypotheses
Ref Expression
sum2vv.1  |-  + w  =  ( 1st `  ( 2nd `  R ) )
sum2vv.2  |-  W  =  ran  + w
addnull1.1  |-  0 w  =  (GId `  + w )
Assertion
Ref Expression
addnull2  |-  ( ( R  e.  Vec  /\  U  e.  W )  ->  ( 0 w + w U )  =  U )

Proof of Theorem addnull2
StepHypRef Expression
1 sum2vv.1 . . . 4  |-  + w  =  ( 1st `  ( 2nd `  R ) )
21vecax1 25453 . . 3  |-  ( R  e.  Vec  ->  + w  e.  AbelOp )
3 ablogrpo 20951 . . 3  |-  ( + w  e.  AbelOp  ->  + w  e.  GrpOp )
42, 3syl 15 . 2  |-  ( R  e.  Vec  ->  + w  e.  GrpOp )
5 sum2vv.2 . . 3  |-  W  =  ran  + w
6 addnull1.1 . . 3  |-  0 w  =  (GId `  + w )
75, 6grpolid 20886 . 2  |-  ( ( + w  e.  GrpOp  /\  U  e.  W )  ->  ( 0 w + w U )  =  U )
84, 7sylan 457 1  |-  ( ( R  e.  Vec  /\  U  e.  W )  ->  ( 0 w + w U )  =  U )
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   AbelOpcablo 20948    Vec cvec 25449
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-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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-ov 5861  df-1st 6122  df-2nd 6123  df-riota 6304  df-grpo 20858  df-gid 20859  df-ablo 20949  df-vec 25450
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