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 |
|
colperp.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
7 |
|
colperp.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
8 |
|
colperp.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
9 |
|
colperp.1 |
⊢ ( 𝜑 → ( 𝐴 𝐿 𝐵 ) ( ⟂G ‘ 𝐺 ) 𝐷 ) |
10 |
|
colperp.2 |
⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) ) |
11 |
|
colperp.3 |
⊢ ( 𝜑 → 𝐴 ≠ 𝐶 ) |
12 |
4 5 9
|
perpln1 |
⊢ ( 𝜑 → ( 𝐴 𝐿 𝐵 ) ∈ ran 𝐿 ) |
13 |
1 3 4 5 6 7 12
|
tglnne |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
14 |
1 3 4 5 6 7 13
|
tglinerflx1 |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 𝐿 𝐵 ) ) |
15 |
13
|
neneqd |
⊢ ( 𝜑 → ¬ 𝐴 = 𝐵 ) |
16 |
10
|
orcomd |
⊢ ( 𝜑 → ( 𝐴 = 𝐵 ∨ 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ) ) |
17 |
16
|
ord |
⊢ ( 𝜑 → ( ¬ 𝐴 = 𝐵 → 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ) ) |
18 |
15 17
|
mpd |
⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ) |
19 |
1 3 4 5 6 8 11 11 12 14 18
|
tglinethru |
⊢ ( 𝜑 → ( 𝐴 𝐿 𝐵 ) = ( 𝐴 𝐿 𝐶 ) ) |
20 |
19 9
|
eqbrtrrd |
⊢ ( 𝜑 → ( 𝐴 𝐿 𝐶 ) ( ⟂G ‘ 𝐺 ) 𝐷 ) |