Theorem rngacl 20097

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 20093	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2		rngacl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		rngacl.p	. . 3 ⊢ + = (+g‘𝑅)
4	2, 3	grpcl 18871	. 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 2113  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  +_gcplusg 17177  Grpcgrp 18863  Rngcrng 20087

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 2115  ax-9 2123  ax-ext 2708  ax-nul 5251

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 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18866  df-abl 19712  df-rng 20088

This theorem is referenced by:  imasrng  20112  qusrng  20115  cntzsubrng  20500  rngqiprngghmlem2  21243  rngqiprngghm  21254