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 ) ) ) |