Theorem submcmn

Description: A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016)

	Ref	Expression
Hypothesis	subgabl.h	⊢ 𝐻 = ( 𝐺 ↾s 𝑆 )
Assertion	submcmn	⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) → 𝐻 ∈ CMnd )

Proof

Step	Hyp	Ref	Expression
1		subgabl.h	⊢ 𝐻 = ( 𝐺 ↾s 𝑆 )
2	1	submmnd	⊢ ( 𝑆 ∈ ( SubMnd ‘ 𝐺 ) → 𝐻 ∈ Mnd )
3	1	subcmn	⊢ ( ( 𝐺 ∈ CMnd ∧ 𝐻 ∈ Mnd ) → 𝐻 ∈ CMnd )
4	2 3	sylan2	⊢ ( ( 𝐺 ∈ CMnd ∧ 𝑆 ∈ ( SubMnd ‘ 𝐺 ) ) → 𝐻 ∈ CMnd )