Theorem lsmcom

Description: Subgroup sum commutes. (Contributed by NM, 6-Feb-2014) (Revised by Mario Carneiro, 21-Jun-2014)

	Ref	Expression
Hypothesis	lsmcom.s	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
Assertion	lsmcom	$$⊢\left(G\in \mathrm{Abel}\wedge T\in \mathrm{SubGrp}\left(G\right)\wedge U\in \mathrm{SubGrp}\left(G\right)\right)\to T\underset{\dot{}}{\oplus }U=U\underset{\dot{}}{\oplus }T$$

Proof

Step	Hyp	Ref	Expression
1		lsmcom.s	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
2		id	$$⊢G\in \mathrm{Abel}\to G\in \mathrm{Abel}$$
3		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
4	3	subgss	$$⊢T\in \mathrm{SubGrp}\left(G\right)\to T\subseteq {\mathrm{Base}}_G$$
5	3	subgss	$$⊢U\in \mathrm{SubGrp}\left(G\right)\to U\subseteq {\mathrm{Base}}_G$$
6	3 1	lsmcomx	$$⊢\left(G\in \mathrm{Abel}\wedge T\subseteq {\mathrm{Base}}_G\wedge U\subseteq {\mathrm{Base}}_G\right)\to T\underset{\dot{}}{\oplus }U=U\underset{\dot{}}{\oplus }T$$
7	2 4 5 6	syl3an	$$⊢\left(G\in \mathrm{Abel}\wedge T\in \mathrm{SubGrp}\left(G\right)\wedge U\in \mathrm{SubGrp}\left(G\right)\right)\to T\underset{\dot{}}{\oplus }U=U\underset{\dot{}}{\oplus }T$$