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Theorem rng0cl 20097
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 20092 . 2 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2 rng0cl.b . . 3 𝐵 = (Base‘𝑅)
3 rng0cl.z . . 3 0 = (0g𝑅)
42, 3grpidcl 18916 . 2 (𝑅 ∈ Grp → 0𝐵)
51, 4syl 17 1 (𝑅 ∈ Rng → 0𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  cfv 6543  Basecbs 17174  0gc0g 17415  Grpcgrp 18884  Rngcrng 20086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-riota 7371  df-ov 7418  df-0g 17417  df-mgm 18594  df-sgrp 18673  df-mnd 18689  df-grp 18887  df-abl 19732  df-rng 20087
This theorem is referenced by:  rngrz  20100  cntzsubrng  20498  rnglidl0  21119  rngqiprngimf1lem  21178
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