Theorem lsmelvalx

Description: Subspace sum membership (for a group or vector space). Extended domain version of lsmelval . (Contributed by NM, 28-Jan-2014) (Revised by Mario Carneiro, 19-Apr-2016)

	Ref	Expression
Hypotheses	lsmfval.v	⊢ 𝐵 = ( Base ‘ 𝐺 )
	lsmfval.a	⊢ + = ( +g ‘ 𝐺 )
	lsmfval.s	⊢ ⊕ = ( LSSum ‘ 𝐺 )
Assertion	lsmelvalx	⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 + 𝑧 ) ) )

Proof

Step	Hyp	Ref	Expression
1		lsmfval.v	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		lsmfval.a	⊢ + = ( +g ‘ 𝐺 )
3		lsmfval.s	⊢ ⊕ = ( LSSum ‘ 𝐺 )
4	1 2 3	lsmvalx	⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) → ( 𝑇 ⊕ 𝑈 ) = ran ( 𝑦 ∈ 𝑇 , 𝑧 ∈ 𝑈 ↦ ( 𝑦 + 𝑧 ) ) )
5	4	eleq2d	⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ 𝑋 ∈ ran ( 𝑦 ∈ 𝑇 , 𝑧 ∈ 𝑈 ↦ ( 𝑦 + 𝑧 ) ) ) )
6		eqid	⊢ ( 𝑦 ∈ 𝑇 , 𝑧 ∈ 𝑈 ↦ ( 𝑦 + 𝑧 ) ) = ( 𝑦 ∈ 𝑇 , 𝑧 ∈ 𝑈 ↦ ( 𝑦 + 𝑧 ) )
7		ovex	⊢ ( 𝑦 + 𝑧 ) ∈ V
8	6 7	elrnmpo	⊢ ( 𝑋 ∈ ran ( 𝑦 ∈ 𝑇 , 𝑧 ∈ 𝑈 ↦ ( 𝑦 + 𝑧 ) ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 + 𝑧 ) )
9	5 8	bitrdi	⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) → ( 𝑋 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑋 = ( 𝑦 + 𝑧 ) ) )