Theorem mndlsmidm

Description: Subgroup sum is idempotent for monoids. This corresponds to the observation in Lang p. 6. (Contributed by AV, 27-Dec-2023)

	Ref	Expression
Hypotheses	mndlsmidm.p	|- .(+) = ( LSSum ` G )
	mndlsmidm.b	|- B = ( Base ` G )
Assertion	mndlsmidm	|- ( G e. Mnd -> ( B .(+) B ) = B )

Proof

Step	Hyp	Ref	Expression
1		mndlsmidm.p	|- .(+) = ( LSSum ` G )
2		mndlsmidm.b	|- B = ( Base ` G )
3	2	submid	|- ( G e. Mnd -> B e. ( SubMnd ` G ) )
4	1	smndlsmidm	|- ( B e. ( SubMnd ` G ) -> ( B .(+) B ) = B )
5	3 4	syl	|- ( G e. Mnd -> ( B .(+) B ) = B )