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Theorem slmd0vcl 33227
Description: The zero vector is a vector. (ax-hv0cl 31022 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 33212 . 2 (𝑊 ∈ SLMod → 𝑊 ∈ Mnd)
2 slmd0vcl.v . . 3 𝑉 = (Base‘𝑊)
3 slmd0vcl.z . . 3 0 = (0g𝑊)
42, 3mndidcl 18762 . 2 (𝑊 ∈ Mnd → 0𝑉)
51, 4syl 17 1 (𝑊 ∈ SLMod → 0𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cfv 6561  Basecbs 17247  0gc0g 17484  Mndcmnd 18747  SLModcslmd 33206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-cmn 19800  df-slmd 33207
This theorem is referenced by:  slmdvs0  33231
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