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Theorem slmd0vrid 31002
 Description: Right identity law for the zero vector. (ax-hvaddid 28886 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 + = (+g𝑊)
slmd0vlid.z 0 = (0g𝑊)
Assertion
Ref Expression
slmd0vrid ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)

Proof of Theorem slmd0vrid
StepHypRef Expression
1 slmdmnd 30985 . 2 (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
2 slmd0vlid.v . . 3 𝑉 = (Base‘𝑊)
3 slmd0vlid.a . . 3 + = (+g𝑊)
4 slmd0vlid.z . . 3 0 = (0g𝑊)
52, 3, 4mndrid 17998 . 2 ((𝑊 ∈ Mnd ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)
61, 5sylan 583 1 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  +gcplusg 16623  0gc0g 16771  Mndcmnd 17977  SLModcslmd 30979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-riota 7108  df-ov 7153  df-0g 16773  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-cmn 18975  df-slmd 30980 This theorem is referenced by: (None)
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