Metamath Proof Explorer


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 + = ( +g𝐺 )
lsmfval.s = ( LSSum ‘ 𝐺 )
Assertion lsmelvalx ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) → ( 𝑋 ∈ ( 𝑇 𝑈 ) ↔ ∃ 𝑦𝑇𝑧𝑈 𝑋 = ( 𝑦 + 𝑧 ) ) )

Proof

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