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Theorem slmd0cl 30468
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 30457 . 2 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 slmd0cl.k . . 3 𝐾 = (Base‘𝐹)
4 slmd0cl.z . . 3 0 = (0g𝐹)
53, 4srg0cl 18982 . 2 (𝐹 ∈ SRing → 0𝐾)
62, 5syl 17 1 (𝑊 ∈ SLMod → 0𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wcel 2048  cfv 6182  Basecbs 16329  Scalarcsca 16414  0gc0g 16559  SRingcsrg 18968  SLModcslmd 30450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-iota 6146  df-fun 6184  df-fv 6190  df-riota 6931  df-ov 6973  df-0g 16561  df-mgm 17700  df-sgrp 17742  df-mnd 17753  df-cmn 18658  df-srg 18969  df-slmd 30451
This theorem is referenced by:  slmd0vs  30474
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