Theorem subrngsubg

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

	Ref	Expression
Assertion	subrngsubg	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝐴 ∈ ( SubGrp ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		subrngrcl	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝑅 ∈ Rng )
2		rnggrp	⊢ ( 𝑅 ∈ Rng → 𝑅 ∈ Grp )
3	1 2	syl	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝑅 ∈ Grp )
4		eqid	⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
5	4	subrngss	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝐴 ⊆ ( Base ‘ 𝑅 ) )
6		eqid	⊢ ( 𝑅 ↾s 𝐴 ) = ( 𝑅 ↾s 𝐴 )
7	6	subrngrng	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → ( 𝑅 ↾s 𝐴 ) ∈ Rng )
8		rnggrp	⊢ ( ( 𝑅 ↾s 𝐴 ) ∈ Rng → ( 𝑅 ↾s 𝐴 ) ∈ Grp )
9	7 8	syl	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → ( 𝑅 ↾s 𝐴 ) ∈ Grp )
10	4	issubg	⊢ ( 𝐴 ∈ ( SubGrp ‘ 𝑅 ) ↔ ( 𝑅 ∈ Grp ∧ 𝐴 ⊆ ( Base ‘ 𝑅 ) ∧ ( 𝑅 ↾s 𝐴 ) ∈ Grp ) )
11	3 5 9 10	syl3anbrc	⊢ ( 𝐴 ∈ ( SubRng ‘ 𝑅 ) → 𝐴 ∈ ( SubGrp ‘ 𝑅 ) )