Theorem rngansg

Description: Every additive subgroup of a non-unital ring is normal. (Contributed by AV, 25-Feb-2025)

	Ref	Expression
Assertion	rngansg	⊢ ( 𝑅 ∈ Rng → ( NrmSGrp ‘ 𝑅 ) = ( SubGrp ‘ 𝑅 ) )

Proof

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