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Theorem rngansg 20117
Description: Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025.)
Assertion
Ref Expression
rngansg (𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))

Proof of Theorem rngansg
StepHypRef Expression
1 rngabl 20102 . 2 (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2 ablnsg 19809 . 2 (𝑅 ∈ Abel → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
31, 2syl 17 1 (𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6553  SubGrpcsubg 19082  NrmSGrpcnsg 19083  Abelcabl 19743  Rngcrng 20099
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-subg 19085  df-nsg 19086  df-cmn 19744  df-abl 19745  df-rng 20100
This theorem is referenced by: (None)
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