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	⊢ + = ( +g ‘ 𝐺 )
	lsmelval.p	⊢ ⊕ = ( LSSum ‘ 𝐺 )
Assertion	lsmelval	⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 + 𝑧 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lsmelval.a	⊢ + = ( +g ‘ 𝐺 )
2		lsmelval.p	⊢ ⊕ = ( LSSum ‘ 𝐺 )
3		subgrcl	⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp )
4		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
5	4	subgss	⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 𝑇 ⊆ ( Base ‘ 𝐺 ) )
6	4	subgss	⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 𝑈 ⊆ ( Base ‘ 𝐺 ) )
7	4 1 2	lsmelvalx	⊢ ( ( 𝐺 ∈ Grp ∧ 𝑇 ⊆ ( Base ‘ 𝐺 ) ∧ 𝑈 ⊆ ( Base ‘ 𝐺 ) ) → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 + 𝑧 ) ) )
8	3 5 6 7	syl2an3an	⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 + 𝑧 ) ) )