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Theorem slmd0cl 32946
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 32935 . 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 20147 . 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 6553  Basecbs 17187  Scalarcsca 17243  0gc0g 17428  SRingcsrg 20133  SLModcslmd 32928
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-riota 7382  df-ov 7429  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-cmn 19744  df-srg 20134  df-slmd 32929
This theorem is referenced by:  slmd0vs  32952
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