Theorem subrg1cl

Description: A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014)

	Ref	Expression
Hypothesis	subrg1cl.a	⊢ 1 = ( 1r ‘ 𝑅 )
Assertion	subrg1cl	⊢ ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) → 1 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		subrg1cl.a	⊢ 1 = ( 1r ‘ 𝑅 )
2		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
3	2 1	issubrg	⊢ ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) ↔ ( ( 𝑅 ∈ Ring ∧ ( 𝑅 ↾s 𝐴 ) ∈ Ring ) ∧ ( 𝐴 ⊆ ( Base ‘ 𝑅 ) ∧ 1 ∈ 𝐴 ) ) )
4	3	simprbi	⊢ ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) → ( 𝐴 ⊆ ( Base ‘ 𝑅 ) ∧ 1 ∈ 𝐴 ) )
5	4	simprd	⊢ ( 𝐴 ∈ ( SubRing ‘ 𝑅 ) → 1 ∈ 𝐴 )