Theorem subrngrng

Description: A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025)

	Ref	Expression
Hypothesis	subrngrng.1	⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
Assertion	subrngrng	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝑆 ∈ Rng )

Proof

Step	Hyp	Ref	Expression
1		subrngrng.1	⊢ 𝑆 = ( 𝑅 ↾s 𝐴 )
2		simp2	⊢ ( ( 𝑅 ∈ Rng ∧ ( 𝑅 ↾s 𝐴 ) ∈ Rng ∧ 𝐴 ⊆ ( Base ‘ 𝑅 ) ) → ( 𝑅 ↾s 𝐴 ) ∈ Rng )
3		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
4	3	issubrng	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) ↔ ( 𝑅 ∈ Rng ∧ ( 𝑅 ↾s 𝐴 ) ∈ Rng ∧ 𝐴 ⊆ ( Base ‘ 𝑅 ) ) )
5	1	eleq1i	⊢ ( 𝑆 ∈ Rng ↔ ( 𝑅 ↾s 𝐴 ) ∈ Rng )
6	2 4 5	3imtr4i	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝑆 ∈ Rng )