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

Theorem slmd0vrid 27456
Description: Right identity law for the zero vector. (ax-hvaddid 25625 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 27439 . 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 15759 . 2  |-  ( ( W  e.  Mnd  /\  X  e.  V )  ->  ( X  .+  .0.  )  =  X )
61, 5sylan 471 1  |-  ( ( W  e. SLMod  /\  X  e.  V )  ->  ( X  .+  .0.  )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6284   Basecbs 14490   +g cplusg 14555   0gc0g 14695   Mndcmnd 15726  SLModcslmd 27433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-riota 6245  df-ov 6287  df-0g 14697  df-mnd 15732  df-cmn 16606  df-slmd 27434
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator