Theorem subrgbas

Description: Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014)

	Ref	Expression
Hypothesis	subrgbas.b	⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
Assertion	subrgbas	⊢ ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) → 𝐴 = ( Base ‘ 𝑆 ) )

Proof

Step	Hyp	Ref	Expression
1		subrgbas.b	⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
2		subrgsubg	⊢ ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) → 𝐴 ∈ ( SubGrp ‘ 𝑅 ) )
3	1	subgbas	⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝑅 ) → 𝐴 = ( Base ‘ 𝑆 ) )
4	2 3	syl	⊢ ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) → 𝐴 = ( Base ‘ 𝑆 ) )