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Metamath Proof Explorer

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 ) )