Step |
Hyp |
Ref |
Expression |
1 |
|
lspindp1.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lspindp1.o |
⊢ 0 = ( 0g ‘ 𝑊 ) |
3 |
|
lspindp1.n |
⊢ 𝑁 = ( LSpan ‘ 𝑊 ) |
4 |
|
lspindp1.w |
⊢ ( 𝜑 → 𝑊 ∈ LVec ) |
5 |
|
lspindp2.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑉 ) |
6 |
|
lspindp2.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑉 ∖ { 0 } ) ) |
7 |
|
lspindp2.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑉 ) |
8 |
|
lspindp2.q |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑋 } ) ≠ ( 𝑁 ‘ { 𝑌 } ) ) |
9 |
|
lspindp2.e |
⊢ ( 𝜑 → ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ) |
10 |
8
|
necomd |
⊢ ( 𝜑 → ( 𝑁 ‘ { 𝑌 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ) |
11 |
|
prcom |
⊢ { 𝑋 , 𝑌 } = { 𝑌 , 𝑋 } |
12 |
11
|
fveq2i |
⊢ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) = ( 𝑁 ‘ { 𝑌 , 𝑋 } ) |
13 |
12
|
eleq2i |
⊢ ( 𝑍 ∈ ( 𝑁 ‘ { 𝑋 , 𝑌 } ) ↔ 𝑍 ∈ ( 𝑁 ‘ { 𝑌 , 𝑋 } ) ) |
14 |
9 13
|
sylnib |
⊢ ( 𝜑 → ¬ 𝑍 ∈ ( 𝑁 ‘ { 𝑌 , 𝑋 } ) ) |
15 |
1 2 3 4 6 5 7 10 14
|
lspindp1 |
⊢ ( 𝜑 → ( ( 𝑁 ‘ { 𝑍 } ) ≠ ( 𝑁 ‘ { 𝑋 } ) ∧ ¬ 𝑌 ∈ ( 𝑁 ‘ { 𝑍 , 𝑋 } ) ) ) |