Step |
Hyp |
Ref |
Expression |
1 |
|
colperpex.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
colperpex.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
colperpex.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
colperpex.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
5 |
|
colperpex.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
6 |
|
hlperpnel.a |
⊢ ( 𝜑 → 𝐴 ∈ ran 𝐿 ) |
7 |
|
hlperpnel.k |
⊢ 𝐾 = ( hlG ‘ 𝐺 ) |
8 |
|
hlperpnel.1 |
⊢ ( 𝜑 → 𝑈 ∈ 𝐴 ) |
9 |
|
hlperpnel.2 |
⊢ ( 𝜑 → 𝑉 ∈ 𝑃 ) |
10 |
|
hlperpnel.3 |
⊢ ( 𝜑 → 𝑊 ∈ 𝑃 ) |
11 |
|
hlperpnel.4 |
⊢ ( 𝜑 → 𝐴 ( ⟂G ‘ 𝐺 ) ( 𝑈 𝐿 𝑉 ) ) |
12 |
|
hlperpnel.5 |
⊢ ( 𝜑 → 𝑉 ( 𝐾 ‘ 𝑈 ) 𝑊 ) |
13 |
1 4 3 5 6 8
|
tglnpt |
⊢ ( 𝜑 → 𝑈 ∈ 𝑃 ) |
14 |
4 5 11
|
perpln2 |
⊢ ( 𝜑 → ( 𝑈 𝐿 𝑉 ) ∈ ran 𝐿 ) |
15 |
1 3 4 5 13 9 14
|
tglnne |
⊢ ( 𝜑 → 𝑈 ≠ 𝑉 ) |
16 |
1 3 7 9 10 13 5 12
|
hlne2 |
⊢ ( 𝜑 → 𝑊 ≠ 𝑈 ) |
17 |
1 3 7 9 10 13 5 4 12
|
hlln |
⊢ ( 𝜑 → 𝑉 ∈ ( 𝑊 𝐿 𝑈 ) ) |
18 |
1 3 4 5 13 9 10 15 17 16
|
lnrot1 |
⊢ ( 𝜑 → 𝑊 ∈ ( 𝑈 𝐿 𝑉 ) ) |
19 |
1 3 4 5 13 9 15 10 16 18
|
tglineelsb2 |
⊢ ( 𝜑 → ( 𝑈 𝐿 𝑉 ) = ( 𝑈 𝐿 𝑊 ) ) |
20 |
1 2 3 4 5 6 14 11
|
perpcom |
⊢ ( 𝜑 → ( 𝑈 𝐿 𝑉 ) ( ⟂G ‘ 𝐺 ) 𝐴 ) |
21 |
19 20
|
eqbrtrrd |
⊢ ( 𝜑 → ( 𝑈 𝐿 𝑊 ) ( ⟂G ‘ 𝐺 ) 𝐴 ) |
22 |
1 2 3 4 5 6 8 10 21
|
footne |
⊢ ( 𝜑 → ¬ 𝑊 ∈ 𝐴 ) |