Theorem sgrpcl

Description: Closure of the operation of a semigroup. (Contributed by AV, 15-Feb-2025)

	Ref	Expression
Hypotheses	sgrpass.b	|- B = ( Base ` G )
	sgrpass.o	|- .o. = ( +g ` G )
Assertion	sgrpcl	|- ( ( G e. Smgrp /\ X e. B /\ Y e. B ) -> ( X .o. Y ) e. B )

Proof

Step	Hyp	Ref	Expression
1		sgrpass.b	|- B = ( Base ` G )
2		sgrpass.o	|- .o. = ( +g ` G )
3		sgrpmgm	|- ( G e. Smgrp -> G e. Mgm )
4	1 2	mgmcl	|- ( ( G e. Mgm /\ X e. B /\ Y e. B ) -> ( X .o. Y ) e. B )
5	3 4	syl3an1	|- ( ( G e. Smgrp /\ X e. B /\ Y e. B ) -> ( X .o. Y ) e. B )