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Theorem slmd0vcl 30336
Description: The zero vector is a vector. (ax-hv0cl 28432 analog.) (Contributed by NM, 10-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmd0vcl.v 𝑉 = (Base‘𝑊)
slmd0vcl.z 0 = (0g𝑊)
Assertion
Ref Expression
slmd0vcl (𝑊 ∈ SLMod → 0𝑉)

Proof of Theorem slmd0vcl
StepHypRef Expression
1 slmdmnd 30321 . 2 (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
2 slmd0vcl.v . . 3 𝑉 = (Base‘𝑊)
3 slmd0vcl.z . . 3 0 = (0g𝑊)
42, 3mndidcl 17694 . 2 (𝑊 ∈ Mnd → 0𝑉)
51, 4syl 17 1 (𝑊 ∈ SLMod → 0𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  cfv 6135  Basecbs 16255  0gc0g 16486  Mndcmnd 17680  SLModcslmd 30315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-riota 6883  df-ov 6925  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-cmn 18581  df-slmd 30316
This theorem is referenced by:  slmdvs0  30340
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