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Theorem slmd0vcl 32872
Description: The zero vector is a vector. (ax-hv0cl 30765 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 32857 . 2 (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
2 slmd0vcl.v . . 3 𝑉 = (Base‘𝑊)
3 slmd0vcl.z . . 3 0 = (0g𝑊)
42, 3mndidcl 18682 . 2 (𝑊 ∈ Mnd → 0𝑉)
51, 4syl 17 1 (𝑊 ∈ SLMod → 0𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6537  Basecbs 17153  0gc0g 17394  Mndcmnd 18667  SLModcslmd 32851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-riota 7361  df-ov 7408  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-cmn 19702  df-slmd 32852
This theorem is referenced by:  slmdvs0  32876
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