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 e. CMnd /\ S e. ( SubMnd ` G ) ) -> H e. CMnd )

Proof

Step	Hyp	Ref	Expression
1		subgabl.h	|- H = ( G |`s S )
2	1	submmnd	|- ( S e. ( SubMnd ` G ) -> H e. Mnd )
3	1	subcmn	|- ( ( G e. CMnd /\ H e. Mnd ) -> H e. CMnd )
4	2 3	sylan2	|- ( ( G e. CMnd /\ S e. ( SubMnd ` G ) ) -> H e. CMnd )