Theorem rng0cl 46648

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 = (0g‘𝑅)

Assertion

Ref	Expression
rng0cl	⊢ (𝑅 ∈ Rng → 0 ∈ 𝐵)

Proof of Theorem rng0cl

Step	Hyp	Ref	Expression
1		rnggrp 46640	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2		rng0cl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		rng0cl.z	. . 3 ⊢ 0 = (0g‘𝑅)
4	2, 3	grpidcl 18846	. 2 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵)
5	1, 4	syl 17	1 ⊢ (𝑅 ∈ Rng → 0 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ‘cfv 6540  Basecbs 17140  0_gc0g 17381  Grpcgrp 18815  Rngcrng 46634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-riota 7361  df-ov 7408  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-abl 19645  df-rng 46635

This theorem is referenced by:  rngrz  46651  cntzsubrng  46730  rnglidl0  46734  rngqiprngimf1lem  46759