Theorem lsmidm

Description: Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014) (Revised by Mario Carneiro, 21-Jun-2014) (Proof shortened by AV, 27-Dec-2023)

	Ref	Expression
Hypothesis	lsmub1.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
Assertion	lsmidm	$$⊢U\in \mathrm{SubGrp}\left(G\right)\to U\underset{\dot{}}{\oplus }U=U$$

Proof

Step	Hyp	Ref	Expression
1		lsmub1.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
2		subgsubm	$$⊢U\in \mathrm{SubGrp}\left(G\right)\to U\in \mathrm{SubMnd}\left(G\right)$$
3	1	smndlsmidm	$$⊢U\in \mathrm{SubMnd}\left(G\right)\to U\underset{\dot{}}{\oplus }U=U$$
4	2 3	syl	$$⊢U\in \mathrm{SubGrp}\left(G\right)\to U\underset{\dot{}}{\oplus }U=U$$