Theorem sdrgsubrg

Description: A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025)

	Ref	Expression
Assertion	sdrgsubrg	⊢ ( 𝐴 ∈ ( SubDRing ‘ 𝑅 ) → 𝐴 ∈ ( SubRing ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		issdrg	⊢ ( 𝐴 ∈ ( SubDRing ‘ 𝑅 ) ↔ ( 𝑅 ∈ DivRing ∧ 𝐴 ∈ ( SubRing ‘ 𝑅 ) ∧ ( 𝑅 ↾s 𝐴 ) ∈ DivRing ) )
2	1	simp2bi	⊢ ( 𝐴 ∈ ( SubDRing ‘ 𝑅 ) → 𝐴 ∈ ( SubRing ‘ 𝑅 ) )