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	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
	mndlsmidm.b	$$⊢B={\mathrm{Base}}_G$$
Assertion	mndlsmidm	$$⊢G\in \mathrm{Mnd}\to B\underset{\dot{}}{\oplus }B=B$$

Proof

Step	Hyp	Ref	Expression
1		mndlsmidm.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
2		mndlsmidm.b	$$⊢B={\mathrm{Base}}_G$$
3	2	submid	$$⊢G\in \mathrm{Mnd}\to B\in \mathrm{SubMnd}\left(G\right)$$
4	1	smndlsmidm	$$⊢B\in \mathrm{SubMnd}\left(G\right)\to B\underset{\dot{}}{\oplus }B=B$$
5	3 4	syl	$$⊢G\in \mathrm{Mnd}\to B\underset{\dot{}}{\oplus }B=B$$