Theorem rngacl 20180

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 20176	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2		rngacl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		rngacl.p	. . 3 ⊢ + = (+g‘𝑅)
4	2, 3	grpcl 18972	. 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
5	1, 4	syl3an1 1162	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  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-ext 2706  ax-nul 5312

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-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  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-iota 6516  df-fv 6571  df-ov 7434  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-abl 19816  df-rng 20171

This theorem is referenced by:  imasrng  20195  qusrng  20198  cntzsubrng  20584  rngqiprngghmlem2  21316  rngqiprngghm  21327