Theorem submgmcl

Description: Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020)

	Ref	Expression
Hypothesis	submgmcl.p	⊢ + = ( +g ‘ 𝑀 )
Assertion	submgmcl	⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		submgmcl.p	⊢ + = ( +g ‘ 𝑀 )
2		submgmrcl	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → 𝑀 ∈ Mgm )
3		eqid	⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
4	3 1	issubmgm	⊢ ( 𝑀 ∈ Mgm → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ) )
5	2 4	syl	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ) )
6	5	ibi	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) )
7	6	simprd	⊢ ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 )
8		ovrspc2v	⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
9	7 8	sylan2	⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ∧ 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
10	9	ancoms	⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
11	10	3impb	⊢ ( ( 𝑆 ∈ ( SubMgm ‘ 𝑀 ) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )