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Metamath Proof Explorer

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	$⊢ \underset{˙}{+} = +_{G}$
		lsmelval.p	$⊢ \underset{˙}{\oplus} = LSSum (G)$
	Assertion	lsmelval	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G)) \to (X \in (T \underset{˙}{\oplus} U) \leftrightarrow \exists y \in T \exists z \in U X = y \underset{˙}{+} z)$

Proof

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