Theorem lsmelval

Description: Subgroup sum membership (for a left module or left vector space). (Contributed by NM, 4-Feb-2014) (Revised by Mario Carneiro, 19-Apr-2016)

	Ref	Expression
Hypotheses	lsmelval.a	$$⊢\underset{\dot{}}{+}={+}_G$$
	lsmelval.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
Assertion	lsmelval	$$⊢\left(T\in \mathrm{SubGrp}\left(G\right)\wedge U\in \mathrm{SubGrp}\left(G\right)\right)\to \left(X\in \left(T\underset{\dot{}}{\oplus }U\right)\leftrightarrow \exists y\in T\exists z\in UX=y\underset{\dot{}}{+}z\right)$$

Proof

Step	Hyp	Ref	Expression
1		lsmelval.a	$$⊢\underset{\dot{}}{+}={+}_G$$
2		lsmelval.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
3		subgrcl	$$⊢T\in \mathrm{SubGrp}\left(G\right)\to G\in \mathrm{Grp}$$
4		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
5	4	subgss	$$⊢T\in \mathrm{SubGrp}\left(G\right)\to T\subseteq {\mathrm{Base}}_G$$
6	4	subgss	$$⊢U\in \mathrm{SubGrp}\left(G\right)\to U\subseteq {\mathrm{Base}}_G$$
7	4 1 2	lsmelvalx	$$⊢\left(G\in \mathrm{Grp}\wedge T\subseteq {\mathrm{Base}}_G\wedge U\subseteq {\mathrm{Base}}_G\right)\to \left(X\in \left(T\underset{\dot{}}{\oplus }U\right)\leftrightarrow \exists y\in T\exists z\in UX=y\underset{\dot{}}{+}z\right)$$
8	3 5 6 7	syl2an3an	$$⊢\left(T\in \mathrm{SubGrp}\left(G\right)\wedge U\in \mathrm{SubGrp}\left(G\right)\right)\to \left(X\in \left(T\underset{\dot{}}{\oplus }U\right)\leftrightarrow \exists y\in T\exists z\in UX=y\underset{\dot{}}{+}z\right)$$