Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  slmd0vrid Structured version   Unicode version

Theorem slmd0vrid 28385
Description: Right identity law for the zero vector. (ax-hvaddid 26500 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmd0vlid.v  |-  V  =  ( Base `  W
)
slmd0vlid.a  |-  .+  =  ( +g  `  W )
slmd0vlid.z  |-  .0.  =  ( 0g `  W )
Assertion
Ref Expression
slmd0vrid  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  ( X  .+  .0.  )  =  X )

Proof of Theorem slmd0vrid
StepHypRef Expression
1 slmdmnd 28368 . 2  |-  ( W  e. SLMod  ->  W  e.  Mnd )
2 slmd0vlid.v . . 3  |-  V  =  ( Base `  W
)
3 slmd0vlid.a . . 3  |-  .+  =  ( +g  `  W )
4 slmd0vlid.z . . 3  |-  .0.  =  ( 0g `  W )
52, 3, 4mndrid 16513 . 2  |-  ( ( W  e.  Mnd  /\  X  e.  V )  ->  ( X  .+  .0.  )  =  X )
61, 5sylan 473 1  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  ( X  .+  .0.  )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15153   0gc0g 15301   Mndcmnd 16490  SLModcslmd 28362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-riota 6267  df-ov 6308  df-0g 15303  df-mgm 16443  df-sgrp 16482  df-mnd 16492  df-cmn 17371  df-slmd 28363
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator