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Theorem slmd0cl 32350
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 = (Scalarβ€˜π‘Š)
slmd0cl.k 𝐾 = (Baseβ€˜πΉ)
slmd0cl.z 0 = (0gβ€˜πΉ)
Assertion
Ref Expression
slmd0cl (π‘Š ∈ SLMod β†’ 0 ∈ 𝐾)

Proof of Theorem slmd0cl
StepHypRef Expression
1 slmd0cl.f . . 3 𝐹 = (Scalarβ€˜π‘Š)
21slmdsrg 32339 . 2 (π‘Š ∈ SLMod β†’ 𝐹 ∈ SRing)
3 slmd0cl.k . . 3 𝐾 = (Baseβ€˜πΉ)
4 slmd0cl.z . . 3 0 = (0gβ€˜πΉ)
53, 4srg0cl 20016 . 2 (𝐹 ∈ SRing β†’ 0 ∈ 𝐾)
62, 5syl 17 1 (π‘Š ∈ SLMod β†’ 0 ∈ 𝐾)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  β€˜cfv 6540  Basecbs 17140  Scalarcsca 17196  0gc0g 17381  SRingcsrg 20002  SLModcslmd 32332
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-riota 7361  df-ov 7408  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-cmn 19644  df-srg 20003  df-slmd 32333
This theorem is referenced by:  slmd0vs  32356
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