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 = (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 29888	. 2 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing)
3		slmd0cl.k	. . 3 ⊢ 𝐾 = (Base‘𝐹)
4		slmd0cl.z	. . 3 ⊢ 0 = (0g‘𝐹)
5	3, 4	srg0cl 18565	. 2 ⊢ (𝐹 ∈ SRing → 0 ∈ 𝐾)
6	2, 5	syl 17	1 ⊢ (𝑊 ∈ SLMod → 0 ∈ 𝐾)

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  ‘cfv 5926  Basecbs 15904  Scalarcsca 15991  0_gc0g 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