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Theorem slmd0vrid 30117
Description: Right identity law for the zero vector. (ax-hvaddid 28202 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 30100 . 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 17526 . 2 ((𝑊 ∈ Mnd ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)
61, 5sylan 490 1 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1629  wcel 2143  cfv 6030  (class class class)co 6791  Basecbs 16070  +gcplusg 16155  0gc0g 16314  Mndcmnd 17508  SLModcslmd 30094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-rmo 3067  df-rab 3068  df-v 3350  df-sbc 3585  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-iota 5993  df-fun 6032  df-fv 6038  df-riota 6752  df-ov 6794  df-0g 16316  df-mgm 17456  df-sgrp 17498  df-mnd 17509  df-cmn 18408  df-slmd 30095
This theorem is referenced by: (None)
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