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Theorem slmd0vcl 30881
 Description: The zero vector is a vector. (ax-hv0cl 28789 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 30866 . 2 (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
2 slmd0vcl.v . . 3 𝑉 = (Base‘𝑊)
3 slmd0vcl.z . . 3 0 = (0g𝑊)
42, 3mndidcl 17926 . 2 (𝑊 ∈ Mnd → 0𝑉)
51, 4syl 17 1 (𝑊 ∈ SLMod → 0𝑉)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  ‘cfv 6343  Basecbs 16483  0gc0g 16713  Mndcmnd 17911  SLModcslmd 30860 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-riota 7107  df-ov 7152  df-0g 16715  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-cmn 18908  df-slmd 30861 This theorem is referenced by:  slmdvs0  30885
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