Theorem rng0cl 20076

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 20071	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2		rng0cl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		rng0cl.z	. . 3 ⊢ 0 = (0g‘𝑅)
4	2, 3	grpidcl 18873	. 2 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵)
5	1, 4	syl 17	1 ⊢ (𝑅 ∈ Rng → 0 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ‘cfv 6476  Basecbs 17115  0_gc0g 17338  Grpcgrp 18841  Rngcrng 20065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fv 6484  df-riota 7298  df-ov 7344  df-0g 17340  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-abl 19690  df-rng 20066

This theorem is referenced by:  rngrz  20079  cntzsubrng  20477  rnglidl0  21161  rngqiprngimf1lem  21226