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Theorem slmd0vrid 30778
Description: Right identity law for the zero vector. (ax-hvaddid 28708 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 30761 . 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 17920 . 2 ((𝑊 ∈ Mnd ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)
61, 5sylan 580 1 ((𝑊 ∈ SLMod ∧ 𝑋𝑉) → (𝑋 + 0 ) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Mndcmnd 17899  SLModcslmd 30755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-cmn 18837  df-slmd 30756
This theorem is referenced by: (None)
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