Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vecax1 Unicode version

Theorem vecax1 25464
Description: 1st "axiom" of a vector space or module. The vector addition is an abelian group. (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by FL, 14-Sep-2010.)
Hypothesis
Ref Expression
vecax1.1  |-  + w  =  ( 1st `  ( 2nd `  R ) )
Assertion
Ref Expression
vecax1  |-  ( R  e.  Vec  ->  + w  e.  AbelOp )

Proof of Theorem vecax1
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2285 . . 3  |-  ran  ( 1st `  ( 1st `  R
) )  =  ran  ( 1st `  ( 1st `  R ) )
2 eqid 2285 . . 3  |-  ( 1st `  ( 1st `  R
) )  =  ( 1st `  ( 1st `  R ) )
3 eqid 2285 . . 3  |-  ( 2nd `  ( 1st `  R
) )  =  ( 2nd `  ( 1st `  R ) )
4 vecax1.1 . . 3  |-  + w  =  ( 1st `  ( 2nd `  R ) )
5 eqid 2285 . . 3  |-  ( 2nd `  ( 2nd `  R
) )  =  ( 2nd `  ( 2nd `  R ) )
6 eqid 2285 . . 3  |-  ran  + w  =  ran  + w
71, 2, 3, 4, 5, 6vecval3b 25463 . 2  |-  ( R  e.  Vec  ->  ( + w  e.  AbelOp  /\  ( 2nd `  ( 2nd `  R
) ) : ( ran  ( 1st `  ( 1st `  R ) )  X.  ran  + w
) --> ran  + w  /\  A. w  e.  ran  + w ( ( (GId
`  ( 2nd `  ( 1st `  R ) ) ) ( 2nd `  ( 2nd `  R ) ) w )  =  w  /\  A. x  e. 
ran  ( 1st `  ( 1st `  R ) ) ( A. z  e. 
ran  + w ( x ( 2nd `  ( 2nd `  R ) ) ( w + w
z ) )  =  ( ( x ( 2nd `  ( 2nd `  R ) ) w ) + w (
x ( 2nd `  ( 2nd `  R ) ) z ) )  /\  A. y  e.  ran  ( 1st `  ( 1st `  R
) ) ( ( ( x ( 1st `  ( 1st `  R
) ) y ) ( 2nd `  ( 2nd `  R ) ) w )  =  ( ( x ( 2nd `  ( 2nd `  R
) ) w ) + w ( y ( 2nd `  ( 2nd `  R ) ) w ) )  /\  ( ( x ( 2nd `  ( 1st `  R ) ) y ) ( 2nd `  ( 2nd `  R ) ) w )  =  ( x ( 2nd `  ( 2nd `  R ) ) ( y ( 2nd `  ( 2nd `  R
) ) w ) ) ) ) ) ) )
87simp1d 967 1  |-  ( R  e.  Vec  ->  + w  e.  AbelOp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   A.wral 2545    X. cxp 4689   ran crn 4692   -->wf 5253   ` cfv 5257  (class class class)co 5860   1stc1st 6122   2ndc2nd 6123  GIdcgi 20856   AbelOpcablo 20950    Vec cvec 25460
This theorem is referenced by:  claddinvvec  25471  vec2inv  25472  sum2vv  25473  addnull1  25474  addnull2  25475  addvecass  25476  addvecom  25477  invaddvec  25478  vecsrcan  25480  vecslcan  25481  vwit  25482  sub2vec  25483  mvecrtol  25484  vecrcan  25486  veclcan  25487  mvecrtol2  25488  mulinvsca  25491  muldisc  25492  svli2  25495  svs2  25498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-ov 5863  df-1st 6124  df-2nd 6125  df-vec 25461
  Copyright terms: Public domain W3C validator