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Theorem rngacl 20065
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 + = (+g𝑅)
Assertion
Ref Expression
rngacl ((𝑅 ∈ Rng ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)

Proof of Theorem rngacl
StepHypRef Expression
1 rnggrp 20061 . 2 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2 rngacl.b . . 3 𝐵 = (Base‘𝑅)
3 rngacl.p . . 3 + = (+g𝑅)
42, 3grpcl 18869 . 2 ((𝑅 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
51, 4syl3an1 1160 1 ((𝑅 ∈ Rng ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  cfv 6536  (class class class)co 7404  Basecbs 17151  +gcplusg 17204  Grpcgrp 18861  Rngcrng 20055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-nul 5299
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6488  df-fv 6544  df-ov 7407  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-abl 19701  df-rng 20056
This theorem is referenced by:  imasrng  20080  qusrng  20083  cntzsubrng  20465  rngqiprngghmlem2  21139  rngqiprngghm  21150
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