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Theorem srg0cl 18690
Description: The zero element of a semiring belongs to its base set. (Contributed by Mario Carneiro, 12-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.)
Hypotheses
Ref Expression
srg0cl.b 𝐵 = (Base‘𝑅)
srg0cl.z 0 = (0g𝑅)
Assertion
Ref Expression
srg0cl (𝑅 ∈ SRing → 0𝐵)

Proof of Theorem srg0cl
StepHypRef Expression
1 srgmnd 18680 . 2 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
2 srg0cl.b . . 3 𝐵 = (Base‘𝑅)
3 srg0cl.z . . 3 0 = (0g𝑅)
42, 3mndidcl 17480 . 2 (𝑅 ∈ Mnd → 0𝐵)
51, 4syl 17 1 (𝑅 ∈ SRing → 0𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1620  wcel 2127  cfv 6037  Basecbs 16030  0gc0g 16273  Mndcmnd 17466  SRingcsrg 18676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rmo 3046  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-iota 6000  df-fun 6039  df-fv 6045  df-riota 6762  df-ov 6804  df-0g 16275  df-mgm 17414  df-sgrp 17456  df-mnd 17467  df-cmn 18366  df-srg 18677
This theorem is referenced by:  srgisid  18699  srgen1zr  18701  srglmhm  18706  srgrmhm  18707  slmd0cl  30051  slmdvs0  30058
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