Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sub2vec Unicode version

Theorem sub2vec 25575
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 25556 . . . 4  |-  ( R  e.  Vec  ->  + w  e.  AbelOp )
3 ablogrpo 20967 . . . 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 20926 . 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 1632    e. wcel 1696   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   GrpOpcgr 20869  GIdcgi 20870   invcgn 20871    /g cgs 20872   AbelOpcablo 20964    Vec cvec 25552
This theorem is referenced by:  dblsubvec  25577  mvecrtol2  25580  mulinvsca  25583
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-gdiv 20877  df-ablo 20965  df-vec 25553
  Copyright terms: Public domain W3C validator