Theorem submcl

Description: Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015)

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

Proof

Step	Hyp	Ref	Expression
1		submcl.p	⊢ + = ( +g ‘ 𝑀 )
2		submrcl	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → 𝑀 ∈ Mnd )
3		eqid	⊢ ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
4		eqid	⊢ ( 0g ‘ 𝑀 ) = ( 0g ‘ 𝑀 )
5	3 4 1	issubm	⊢ ( 𝑀 ∈ Mnd → ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ) )
6	2 5	syl	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ↔ ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) ) )
7	6	ibi	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → ( 𝑆 ⊆ ( Base ‘ 𝑀 ) ∧ ( 0g ‘ 𝑀 ) ∈ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) )
8	7	simp3d	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) → ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 )
9		ovrspc2v	⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 + 𝑦 ) ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
10	8 9	sylan2	⊢ ( ( ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ∧ 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
11	10	ancoms	⊢ ( ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ∧ ( 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )
12	11	3impb	⊢ ( ( 𝑆 ∈ ( SubMnd ‘ 𝑀 ) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆 ) → ( 𝑋 + 𝑌 ) ∈ 𝑆 )