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Theorem lmod0vid 15982
Description: Identity equivalent to the value of the zero vector. Provides a convenient way to compute the value. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
Hypotheses
Ref Expression
0vlid.v  |-  V  =  ( Base `  W
)
0vlid.a  |-  .+  =  ( +g  `  W )
0vlid.z  |-  .0.  =  ( 0g `  W )
Assertion
Ref Expression
lmod0vid  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( X  .+  X
)  =  X  <->  .0.  =  X ) )

Proof of Theorem lmod0vid
StepHypRef Expression
1 lmodgrp 15957 . 2  |-  ( W  e.  LMod  ->  W  e. 
Grp )
2 0vlid.v . . 3  |-  V  =  ( Base `  W
)
3 0vlid.a . . 3  |-  .+  =  ( +g  `  W )
4 0vlid.z . . 3  |-  .0.  =  ( 0g `  W )
52, 3, 4grpid 14840 . 2  |-  ( ( W  e.  Grp  /\  X  e.  V )  ->  ( ( X  .+  X )  =  X  <-> 
.0.  =  X ) )
61, 5sylan 458 1  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( X  .+  X
)  =  X  <->  .0.  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   LModclmod 15950
This theorem is referenced by:  lmod0vs  15983  dva0g  31825  dvh0g  31909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-riota 6549  df-0g 13727  df-mnd 14690  df-grp 14812  df-lmod 15952
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