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Theorem srg0cl 19261
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 19251 . 2 (𝑅 ∈ SRing → 𝑅 ∈ Mnd)
2 srg0cl.b . . 3 𝐵 = (Base‘𝑅)
3 srg0cl.z . . 3 0 = (0g𝑅)
42, 3mndidcl 17918 . 2 (𝑅 ∈ Mnd → 0𝐵)
51, 4syl 17 1 (𝑅 ∈ SRing → 0𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cfv 6348  Basecbs 16475  0gc0g 16705  Mndcmnd 17903  SRingcsrg 19247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7106  df-ov 7151  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-cmn 18900  df-srg 19248
This theorem is referenced by:  srgisid  19270  srgen1zr  19272  srglmhm  19277  srgrmhm  19278  slmd0cl  30839  slmdvs0  30846
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