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Theorem rng0cl 20181
Description: The zero element of a non-unital ring belongs to its base set. (Contributed by AV, 16-Feb-2025.)
Hypotheses
Ref Expression
rng0cl.b 𝐵 = (Base‘𝑅)
rng0cl.z 0 = (0g𝑅)
Assertion
Ref Expression
rng0cl (𝑅 ∈ Rng → 0𝐵)

Proof of Theorem rng0cl
StepHypRef Expression
1 rnggrp 20176 . 2 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2 rng0cl.b . . 3 𝐵 = (Base‘𝑅)
3 rng0cl.z . . 3 0 = (0g𝑅)
42, 3grpidcl 18996 . 2 (𝑅 ∈ Grp → 0𝐵)
51, 4syl 17 1 (𝑅 ∈ Rng → 0𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cfv 6563  Basecbs 17245  0gc0g 17486  Grpcgrp 18964  Rngcrng 20170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-abl 19816  df-rng 20171
This theorem is referenced by:  rngrz  20184  cntzsubrng  20584  rnglidl0  21257  rngqiprngimf1lem  21322
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