Theorem rngansg 20197

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

Step	Hyp	Ref	Expression
1		rngabl 20182	. 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Abel)
2		ablnsg 19868	. 2 ⊢ (𝑅 ∈ Abel → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ Rng → (NrmSGrp‘𝑅) = (SubGrp‘𝑅))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ‘cfv 6515  SubGrpcsubg 19143  NrmSGrpcnsg 19144  Abelcabl 19802  Rngcrng 20179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6471  df-fun 6517  df-fv 6523  df-ov 7393  df-subg 19146  df-nsg 19147  df-cmn 19803  df-abl 19804  df-rng 20180

This theorem is referenced by: (None)