MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngacl Structured version   Visualization version   GIF version

Theorem rngacl 20102
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 20098 . 2 (𝑅 ∈ Rng → 𝑅 ∈ Grp)
2 rngacl.b . . 3 𝐵 = (Base‘𝑅)
3 rngacl.p . . 3 + = (+g𝑅)
42, 3grpcl 18898 . 2 ((𝑅 ∈ Grp ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
51, 4syl3an1 1161 1 ((𝑅 ∈ Rng ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1534  wcel 2099  cfv 6548  (class class class)co 7420  Basecbs 17180  +gcplusg 17233  Grpcgrp 18890  Rngcrng 20092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-iota 6500  df-fv 6556  df-ov 7423  df-mgm 18600  df-sgrp 18679  df-mnd 18695  df-grp 18893  df-abl 19738  df-rng 20093
This theorem is referenced by:  imasrng  20117  qusrng  20120  cntzsubrng  20504  rngqiprngghmlem2  21178  rngqiprngghm  21189
  Copyright terms: Public domain W3C validator