Theorem rngacl 20075

Description: Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.)

Hypotheses

Ref	Expression
rngacl.b	⊢ 𝐵 = (Base‘𝑅)
rngacl.p	⊢ + = (+g‘𝑅)

Assertion

Ref	Expression
rngacl	⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Proof of Theorem rngacl

Step	Hyp	Ref	Expression
1		rnggrp 20071	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2		rngacl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		rngacl.p	. . 3 ⊢ + = (+g‘𝑅)
4	2, 3	grpcl 18849	. 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
5	1, 4	syl3an1 1163	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ‘cfv 6476  (class class class)co 7341  Basecbs 17115  +_gcplusg 17156  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-ext 2703  ax-nul 5239

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-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  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-iota 6432  df-fv 6484  df-ov 7344  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-abl 19690  df-rng 20066

This theorem is referenced by:  imasrng  20090  qusrng  20093  cntzsubrng  20477  rngqiprngghmlem2  21220  rngqiprngghm  21231