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 ) |
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 ) |