lsmless1

Description: Subset implies subgroup sum subset. (Contributed by NM, 6-Feb-2014) (Revised by Mario Carneiro, 19-Apr-2016)

		Ref	Expression
	Hypothesis	lsmub1.p	$⊢ \underset{˙}{\oplus} = LSSum (G)$
	Assertion	lsmless1	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G) \land S \subseteq T) \to S \underset{˙}{\oplus} U \subseteq T \underset{˙}{\oplus} U$

Step	Hyp	Ref	Expression
1		lsmub1.p	$⊢ \underset{˙}{\oplus} = LSSum (G)$
2		subgrcl	$⊢ T \in SubGrp (G) \to G \in Grp$
3	2	3ad2ant1	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G) \land S \subseteq T) \to G \in Grp$
4		eqid	$⊢ {Base}_{G} = {Base}_{G}$
5	4	subgss	$⊢ T \in SubGrp (G) \to T \subseteq {Base}_{G}$
6	5	3ad2ant1	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G) \land S \subseteq T) \to T \subseteq {Base}_{G}$
7	4	subgss	$⊢ U \in SubGrp (G) \to U \subseteq {Base}_{G}$
8	7	3ad2ant2	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G) \land S \subseteq T) \to U \subseteq {Base}_{G}$
9		simp3	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G) \land S \subseteq T) \to S \subseteq T$
10	4 1	lsmless1x	$⊢ ((G \in Grp \land T \subseteq {Base}_{G} \land U \subseteq {Base}_{G}) \land S \subseteq T) \to S \underset{˙}{\oplus} U \subseteq T \underset{˙}{\oplus} U$
11	3 6 8 9 10	syl31anc	$⊢ (T \in SubGrp (G) \land U \in SubGrp (G) \land S \subseteq T) \to S \underset{˙}{\oplus} U \subseteq T \underset{˙}{\oplus} U$

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