Step |
Hyp |
Ref |
Expression |
1 |
|
hpgid.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
hpgid.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
3 |
|
hpgid.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
4 |
|
hpgid.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
hpgid.d |
⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 ) |
6 |
|
hpgid.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
7 |
|
hpgid.o |
⊢ 𝑂 = { 〈 𝑎 , 𝑏 〉 ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) } |
8 |
|
colopp.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
9 |
|
colopp.p |
⊢ ( 𝜑 → 𝐶 ∈ 𝐷 ) |
10 |
|
colopp.1 |
⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ∨ 𝐴 = 𝐵 ) ) |
11 |
4
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐺 ∈ TarskiG ) |
12 |
6
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐴 ∈ 𝑃 ) |
13 |
8
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐵 ∈ 𝑃 ) |
14 |
|
eqid |
⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 ) |
15 |
5
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐷 ∈ ran 𝐿 ) |
16 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) |
17 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝑦 ∈ 𝐷 ) |
18 |
|
eleq1w |
⊢ ( 𝑡 = 𝑦 → ( 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ↔ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
19 |
18
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) ∧ 𝑡 = 𝑦 ) → ( 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ↔ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
20 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) |
21 |
17 19 20
|
rspcedvd |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) |
22 |
1 14 2 7 6 8
|
islnopp |
⊢ ( 𝜑 → ( 𝐴 𝑂 𝐵 ↔ ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) ) ) |
23 |
22
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → ( 𝐴 𝑂 𝐵 ↔ ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) ) ) |
24 |
16 21 23
|
mpbir2and |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐴 𝑂 𝐵 ) |
25 |
1 14 2 7 3 15 11 12 13 24
|
oppne3 |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐴 ≠ 𝐵 ) |
26 |
1 2 3 11 12 13 25
|
tgelrnln |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → ( 𝐴 𝐿 𝐵 ) ∈ ran 𝐿 ) |
27 |
1 2 3 11 12 13 25
|
tglinerflx1 |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐴 ∈ ( 𝐴 𝐿 𝐵 ) ) |
28 |
16
|
simpld |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → ¬ 𝐴 ∈ 𝐷 ) |
29 |
|
nelne1 |
⊢ ( ( 𝐴 ∈ ( 𝐴 𝐿 𝐵 ) ∧ ¬ 𝐴 ∈ 𝐷 ) → ( 𝐴 𝐿 𝐵 ) ≠ 𝐷 ) |
30 |
27 28 29
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → ( 𝐴 𝐿 𝐵 ) ≠ 𝐷 ) |
31 |
25
|
neneqd |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → ¬ 𝐴 = 𝐵 ) |
32 |
10
|
orcomd |
⊢ ( 𝜑 → ( 𝐴 = 𝐵 ∨ 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ) ) |
33 |
32
|
ord |
⊢ ( 𝜑 → ( ¬ 𝐴 = 𝐵 → 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ) ) |
34 |
33
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → ( ¬ 𝐴 = 𝐵 → 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ) ) |
35 |
31 34
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐶 ∈ ( 𝐴 𝐿 𝐵 ) ) |
36 |
9
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐶 ∈ 𝐷 ) |
37 |
35 36
|
elind |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐶 ∈ ( ( 𝐴 𝐿 𝐵 ) ∩ 𝐷 ) ) |
38 |
1 3 2 11 15 17
|
tglnpt |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝑦 ∈ 𝑃 ) |
39 |
1 2 3 11 12 13 38 25 20
|
btwnlng1 |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝑦 ∈ ( 𝐴 𝐿 𝐵 ) ) |
40 |
39 17
|
elind |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝑦 ∈ ( ( 𝐴 𝐿 𝐵 ) ∩ 𝐷 ) ) |
41 |
1 2 3 11 26 15 30 37 40
|
tglineineq |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐶 = 𝑦 ) |
42 |
41 20
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) |
43 |
42
|
adantllr |
⊢ ( ( ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) ∧ 𝑦 ∈ 𝐷 ) ∧ 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) |
44 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) → ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) |
45 |
18
|
cbvrexvw |
⊢ ( ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐷 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) |
46 |
44 45
|
sylib |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) → ∃ 𝑦 ∈ 𝐷 𝑦 ∈ ( 𝐴 𝐼 𝐵 ) ) |
47 |
43 46
|
r19.29a |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) |
48 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐶 ∈ 𝐷 ) |
49 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ∧ 𝑡 = 𝐶 ) → 𝑡 = 𝐶 ) |
50 |
49
|
eleq1d |
⊢ ( ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ∧ 𝑡 = 𝐶 ) → ( 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ↔ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
51 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) |
52 |
48 50 51
|
rspcedvd |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) → ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) |
53 |
52
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) → ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) |
54 |
47 53
|
impbida |
⊢ ( ( 𝜑 ∧ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) → ( ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ↔ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
55 |
54
|
pm5.32da |
⊢ ( 𝜑 → ( ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) ↔ ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) ) |
56 |
|
3anrot |
⊢ ( ( 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ∧ ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ↔ ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
57 |
|
df-3an |
⊢ ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ↔ ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
58 |
56 57
|
bitri |
⊢ ( ( 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ∧ ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ↔ ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
59 |
58
|
a1i |
⊢ ( 𝜑 → ( ( 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ∧ ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ↔ ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) ) |
60 |
55 22 59
|
3bitr4d |
⊢ ( 𝜑 → ( 𝐴 𝑂 𝐵 ↔ ( 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ∧ ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) ) |