Theorem lsmelvalix

Description: Subspace sum membership (for a group or vector space). (Contributed by NM, 4-Feb-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	lsmelvalix	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈 ) ) → ( 𝑋 + 𝑌 ) ∈ ( 𝑇 ⊕ 𝑈 ) )

Proof

Step	Hyp	Ref	Expression
1		lsmfval.v	⊢ 𝐵 = ( Base ‘ 𝐺 )
2		lsmfval.a	⊢ + = ( +g ‘ 𝐺 )
3		lsmfval.s	⊢ ⊕ = ( LSSum ‘ 𝐺 )
4		eqid	⊢ ( 𝑋 + 𝑌 ) = ( 𝑋 + 𝑌 )
5		rspceov	⊢ ( ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈 ∧ ( 𝑋 + 𝑌 ) = ( 𝑋 + 𝑌 ) ) → ∃ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 ( 𝑋 + 𝑌 ) = ( 𝑥 + 𝑦 ) )
6	4 5	mp3an3	⊢ ( ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈 ) → ∃ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 ( 𝑋 + 𝑌 ) = ( 𝑥 + 𝑦 ) )
7	1 2 3	lsmelvalx	⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) → ( ( 𝑋 + 𝑌 ) ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 ( 𝑋 + 𝑌 ) = ( 𝑥 + 𝑦 ) ) )
8	7	biimpar	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ ∃ 𝑥 ∈ 𝑇 ∃ 𝑦 ∈ 𝑈 ( 𝑋 + 𝑌 ) = ( 𝑥 + 𝑦 ) ) → ( 𝑋 + 𝑌 ) ∈ ( 𝑇 ⊕ 𝑈 ) )
9	6 8	sylan2	⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ ( 𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑈 ) ) → ( 𝑋 + 𝑌 ) ∈ ( 𝑇 ⊕ 𝑈 ) )