Theorem rngacl 20088

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 20084	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2		rngacl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		rngacl.p	. . 3 ⊢ + = (+g‘𝑅)
4	2, 3	grpcl 18862	. 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 6489  (class class class)co 7355  Basecbs 17127  +_gcplusg 17168  Grpcgrp 18854  Rngcrng 20078

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 2705  ax-nul 5248

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 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-ov 7358  df-mgm 18556  df-sgrp 18635  df-mnd 18651  df-grp 18857  df-abl 19703  df-rng 20079

This theorem is referenced by:  imasrng  20103  qusrng  20106  cntzsubrng  20491  rngqiprngghmlem2  21234  rngqiprngghm  21245