Theorem mndcl

Description: Closure of the operation of a monoid. (Contributed by NM, 14-Aug-2011) (Revised by Mario Carneiro, 6-Jan-2015) (Proof shortened by AV, 8-Feb-2020)

	Ref	Expression
Hypotheses	mndcl.b	|- B = ( Base ` G )
	mndcl.p	|- .+ = ( +g ` G )
Assertion	mndcl	|- ( ( G e. Mnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )

Proof

Step	Hyp	Ref	Expression
1		mndcl.b	|- B = ( Base ` G )
2		mndcl.p	|- .+ = ( +g ` G )
3		mndmgm	|- ( G e. Mnd -> G e. Mgm )
4	1 2	mgmcl	|- ( ( G e. Mgm /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )
5	3 4	syl3an1	|- ( ( G e. Mnd /\ X e. B /\ Y e. B ) -> ( X .+ Y ) e. B )