Metamath Proof Explorer

Theorem lsmelvali

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	lsmelvali	$⊢ ((T \in SubGrp (G) \land U \in SubGrp (G)) \land (X \in T \land Y \in U)) \to X \underset{˙}{+} Y \in (T \underset{˙}{\oplus} U)$

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	3	adantr	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G)) \to G \in Grp$
5		eqid	$⊢ {Base}_{G} = {Base}_{G}$
6	5	subgss	$⊢ T \in SubGrp (G) \to T \subseteq {Base}_{G}$
7	6	adantr	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G)) \to T \subseteq {Base}_{G}$
8	5	subgss	$⊢ U \in SubGrp (G) \to U \subseteq {Base}_{G}$
9	8	adantl	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G)) \to U \subseteq {Base}_{G}$
10	4 7 9	3jca	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G)) \to (G \in Grp \land T \subseteq {Base}_{G} \land U \subseteq {Base}_{G})$
11	5 1 2	lsmelvalix	$⊢ ((G \in Grp \land T \subseteq {Base}_{G} \land U \subseteq {Base}_{G}) \land (X \in T \land Y \in U)) \to X \underset{˙}{+} Y \in (T \underset{˙}{\oplus} U)$
12	10 11	sylan	$⊢ ((T \in SubGrp (G) \land U \in SubGrp (G)) \land (X \in T \land Y \in U)) \to X \underset{˙}{+} Y \in (T \underset{˙}{\oplus} U)$