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Theorem slmd0cl 29899
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 29888 . 2 (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3 slmd0cl.k . . 3 𝐾 = (Base‘𝐹)
4 slmd0cl.z . . 3 0 = (0g𝐹)
53, 4srg0cl 18565 . 2 (𝐹 ∈ SRing → 0𝐾)
62, 5syl 17 1 (𝑊 ∈ SLMod → 0𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  cfv 5926  Basecbs 15904  Scalarcsca 15991  0gc0g 16147  SRingcsrg 18551  SLModcslmd 29881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-riota 6651  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-cmn 18241  df-srg 18552  df-slmd 29882
This theorem is referenced by:  slmd0vs  29905
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