Theorem sdrgdrng

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

	Ref	Expression
Hypothesis	sdrgdrng.1	⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
Assertion	sdrgdrng	⊢ ( 𝐴 ∈ ( SubDRing ‘ 𝑅 ) → 𝑆 ∈ DivRing )

Proof

Step	Hyp	Ref	Expression
1		sdrgdrng.1	⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
2		issdrg	⊢ ( 𝐴 ∈ ( SubDRing ‘ 𝑅 ) ↔ ( 𝑅 ∈ DivRing ∧ 𝐴 ∈ ( SubRing ‘ 𝑅 ) ∧ ( 𝑅 ↾s 𝐴 ) ∈ DivRing ) )
3	2	simp3bi	⊢ ( 𝐴 ∈ ( SubDRing ‘ 𝑅 ) → ( 𝑅 ↾s 𝐴 ) ∈ DivRing )
4	1 3	eqeltrid	⊢ ( 𝐴 ∈ ( SubDRing ‘ 𝑅 ) → 𝑆 ∈ DivRing )