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Theorem slmd0cl 32866
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 32855 . 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 20102 . 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 1533   ∈ wcel 2098  β€˜cfv 6536  Basecbs 17150  Scalarcsca 17206  0gc0g 17391  SRingcsrg 20088  SLModcslmd 32848
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 6488  df-fun 6538  df-fv 6544  df-riota 7360  df-ov 7407  df-0g 17393  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-cmn 19699  df-srg 20089  df-slmd 32849
This theorem is referenced by:  slmd0vs  32872
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