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 |
|
perpdrag.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝐷 ) |
7 |
|
perpdrag.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐷 ) |
8 |
|
perpdrag.3 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
9 |
|
perpdrag.4 |
⊢ ( 𝜑 → 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝐵 𝐿 𝐶 ) ) |
10 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝐺 ∈ TarskiG ) |
11 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝐷 ( ⟂G ‘ 𝐺 ) ( 𝐵 𝐿 𝐶 ) ) |
12 |
4 10 11
|
perpln1 |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝐷 ∈ ran 𝐿 ) |
13 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝐴 ∈ 𝐷 ) |
14 |
1 4 3 10 12 13
|
tglnpt |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝐴 ∈ 𝑃 ) |
15 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝑥 ∈ 𝐷 ) |
16 |
1 4 3 10 12 15
|
tglnpt |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝑥 ∈ 𝑃 ) |
17 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝐵 ∈ 𝐷 ) |
18 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝐴 ≠ 𝑥 ) |
19 |
1 3 4 10 14 16 18 18 12 13 15
|
tglinethru |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝐷 = ( 𝐴 𝐿 𝑥 ) ) |
20 |
17 19
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝐵 ∈ ( 𝐴 𝐿 𝑥 ) ) |
21 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 𝐶 ∈ 𝑃 ) |
22 |
19 11
|
eqbrtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → ( 𝐴 𝐿 𝑥 ) ( ⟂G ‘ 𝐺 ) ( 𝐵 𝐿 𝐶 ) ) |
23 |
1 2 3 4 10 14 16 20 21 22
|
perprag |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝑥 ) → 〈“ 𝐴 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |
24 |
4 5 9
|
perpln1 |
⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 ) |
25 |
1 3 4 5 24 6
|
tglnpt2 |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐷 𝐴 ≠ 𝑥 ) |
26 |
23 25
|
r19.29a |
⊢ ( 𝜑 → 〈“ 𝐴 𝐵 𝐶 ”〉 ∈ ( ∟G ‘ 𝐺 ) ) |