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Theorem slmd0cl 30906
 Description: The ring zero in a semimodule belongs to the ring base set. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
slmd0cl.f 𝐹 = (Scalar‘𝑊)
slmd0cl.k 𝐾 = (Base‘𝐹)
slmd0cl.z 0 = (0g𝐹)
Assertion
Ref Expression
slmd0cl (𝑊 ∈ SLMod → 0𝐾)

Proof of Theorem slmd0cl
StepHypRef Expression
1 slmd0cl.f . . 3 𝐹 = (Scalar‘𝑊)
21slmdsrg 30895 . 2 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 slmd0cl.k . . 3 𝐾 = (Base‘𝐹)
4 slmd0cl.z . . 3 0 = (0g𝐹)
53, 4srg0cl 19266 . 2 (𝐹 ∈ SRing → 0𝐾)
62, 5syl 17 1 (𝑊 ∈ SLMod → 0𝐾)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ‘cfv 6325  Basecbs 16478  Scalarcsca 16563  0gc0g 16708  SRingcsrg 19252  SLModcslmd 30888 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-iota 6284  df-fun 6327  df-fv 6333  df-riota 7094  df-ov 7139  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-cmn 18904  df-srg 19253  df-slmd 30889 This theorem is referenced by:  slmd0vs  30912
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