Theorem rngacl 20235

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 20231	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2		rngacl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		rngacl.p	. . 3 ⊢ + = (+g‘𝑅)
4	2, 3	grpcl 19003	. 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
5	1, 4	syl3an1 1181	1 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  +_gcplusg 17305  Grpcgrp 18995  Rngcrng 20225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-grp 18998  df-abl 19848  df-rng 20226

This theorem is referenced by:  imasrng  20250  qusrng  20253  cntzsubrng  20666  rngqiprngghmlem2  21428  rngqiprngghm  21439