Theorem slmd0cl 29102

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 = (0g‘𝐹)

Assertion

Ref	Expression
slmd0cl	⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾)

Proof of Theorem slmd0cl

Step	Hyp	Ref	Expression
1		slmd0cl.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑊)
2	1	slmdsrg 29091	. 2 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3		slmd0cl.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
4		slmd0cl.z	. . 3 ⊢ 0 = (0g‘𝐹)
5	3, 4	srg0cl 18342	. 2 ⊢ (𝐹 ∈ SRing → 0 ∈ 𝐾)
6	2, 5	syl 17	1 ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  Basecbs 15695  Scalarcsca 15771  0_gc0g 15923  SRingcsrg 18328  SLModcslmd 29084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-cmn 18018  df-srg 18329  df-slmd 29085

This theorem is referenced by:  slmd0vs  29108