Step |
Hyp |
Ref |
Expression |
1 |
|
hpg.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
hpg.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
hpg.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
hpg.o |
⊢ 𝑂 = { 〈 𝑎 , 𝑏 〉 ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) } |
5 |
|
opphl.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
6 |
|
opphl.d |
⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 ) |
7 |
|
opphl.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
8 |
|
opphllem1.s |
⊢ 𝑆 = ( ( pInvG ‘ 𝐺 ) ‘ 𝑀 ) |
9 |
|
opphllem1.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
10 |
|
opphllem1.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
11 |
|
opphllem1.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
12 |
|
opphllem1.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝐷 ) |
13 |
|
opphllem1.o |
⊢ ( 𝜑 → 𝐴 𝑂 𝐶 ) |
14 |
|
opphllem1.m |
⊢ ( 𝜑 → 𝑀 ∈ 𝐷 ) |
15 |
|
opphllem1.n |
⊢ ( 𝜑 → 𝐴 = ( 𝑆 ‘ 𝐶 ) ) |
16 |
|
opphllem1.x |
⊢ ( 𝜑 → 𝐴 ≠ 𝑅 ) |
17 |
|
opphllem1.y |
⊢ ( 𝜑 → 𝐵 ≠ 𝑅 ) |
18 |
|
opphllem2.z |
⊢ ( 𝜑 → ( 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ∨ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) ) |
19 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝐷 ∈ ran 𝐿 ) |
20 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝐺 ∈ TarskiG ) |
21 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝐶 ∈ 𝑃 ) |
22 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝐵 ∈ 𝑃 ) |
23 |
|
eqid |
⊢ ( pInvG ‘ 𝐺 ) = ( pInvG ‘ 𝐺 ) |
24 |
1 5 3 7 6 14
|
tglnpt |
⊢ ( 𝜑 → 𝑀 ∈ 𝑃 ) |
25 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝑀 ∈ 𝑃 ) |
26 |
1 2 3 5 23 20 25 8 22
|
mircl |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ( 𝑆 ‘ 𝐵 ) ∈ 𝑃 ) |
27 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝑀 ∈ 𝐷 ) |
28 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝑅 ∈ 𝐷 ) |
29 |
1 2 3 5 23 20 8 19 27 28
|
mirln |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ( 𝑆 ‘ 𝑅 ) ∈ 𝐷 ) |
30 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 = 𝐵 ) → 𝐴 = 𝐵 ) |
31 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 = 𝐵 ) → 𝐵 ∈ 𝐷 ) |
32 |
30 31
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ 𝐷 ) |
33 |
7
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐺 ∈ TarskiG ) |
34 |
10
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐵 ∈ 𝑃 ) |
35 |
1 5 3 7 6 12
|
tglnpt |
⊢ ( 𝜑 → 𝑅 ∈ 𝑃 ) |
36 |
35
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝑅 ∈ 𝑃 ) |
37 |
9
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐴 ∈ 𝑃 ) |
38 |
17
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐵 ≠ 𝑅 ) |
39 |
38
|
necomd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝑅 ≠ 𝐵 ) |
40 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) |
41 |
1 3 5 33 36 34 37 39 40
|
btwnlng1 |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐴 ∈ ( 𝑅 𝐿 𝐵 ) ) |
42 |
1 3 5 33 34 36 37 38 41
|
lncom |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐴 ∈ ( 𝐵 𝐿 𝑅 ) ) |
43 |
6
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐷 ∈ ran 𝐿 ) |
44 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐵 ∈ 𝐷 ) |
45 |
12
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝑅 ∈ 𝐷 ) |
46 |
1 3 5 33 34 36 38 38 43 44 45
|
tglinethru |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐷 = ( 𝐵 𝐿 𝑅 ) ) |
47 |
42 46
|
eleqtrrd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) ∧ 𝐴 ≠ 𝐵 ) → 𝐴 ∈ 𝐷 ) |
48 |
32 47
|
pm2.61dane |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) → 𝐴 ∈ 𝐷 ) |
49 |
1 2 3 4 5 6 7 9 11 13
|
oppne1 |
⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐷 ) |
50 |
49
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ 𝐵 ∈ 𝐷 ) → ¬ 𝐴 ∈ 𝐷 ) |
51 |
48 50
|
pm2.65da |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ¬ 𝐵 ∈ 𝐷 ) |
52 |
20
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) → 𝐺 ∈ TarskiG ) |
53 |
25
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) → 𝑀 ∈ 𝑃 ) |
54 |
22
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) → 𝐵 ∈ 𝑃 ) |
55 |
1 2 3 5 23 52 53 8 54
|
mirmir |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) → ( 𝑆 ‘ ( 𝑆 ‘ 𝐵 ) ) = 𝐵 ) |
56 |
19
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) → 𝐷 ∈ ran 𝐿 ) |
57 |
27
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) → 𝑀 ∈ 𝐷 ) |
58 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) → ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) |
59 |
1 2 3 5 23 52 8 56 57 58
|
mirln |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) → ( 𝑆 ‘ ( 𝑆 ‘ 𝐵 ) ) ∈ 𝐷 ) |
60 |
55 59
|
eqeltrrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) → 𝐵 ∈ 𝐷 ) |
61 |
51 60
|
mtand |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ¬ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) |
62 |
1 2 3 5 23 20 25 8 22
|
mirbtwn |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝑀 ∈ ( ( 𝑆 ‘ 𝐵 ) 𝐼 𝐵 ) ) |
63 |
1 2 3 4 26 22 27 61 51 62
|
islnoppd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ( 𝑆 ‘ 𝐵 ) 𝑂 𝐵 ) |
64 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ( 𝑆 ‘ 𝐵 ) = ( 𝑆 ‘ 𝐵 ) ) |
65 |
|
nelne2 |
⊢ ( ( ( 𝑆 ‘ 𝑅 ) ∈ 𝐷 ∧ ¬ ( 𝑆 ‘ 𝐵 ) ∈ 𝐷 ) → ( 𝑆 ‘ 𝑅 ) ≠ ( 𝑆 ‘ 𝐵 ) ) |
66 |
29 61 65
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ( 𝑆 ‘ 𝑅 ) ≠ ( 𝑆 ‘ 𝐵 ) ) |
67 |
66
|
necomd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ( 𝑆 ‘ 𝐵 ) ≠ ( 𝑆 ‘ 𝑅 ) ) |
68 |
1 2 3 4 5 6 7 9 11 13
|
oppne2 |
⊢ ( 𝜑 → ¬ 𝐶 ∈ 𝐷 ) |
69 |
68
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ¬ 𝐶 ∈ 𝐷 ) |
70 |
|
nelne2 |
⊢ ( ( ( 𝑆 ‘ 𝑅 ) ∈ 𝐷 ∧ ¬ 𝐶 ∈ 𝐷 ) → ( 𝑆 ‘ 𝑅 ) ≠ 𝐶 ) |
71 |
29 69 70
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ( 𝑆 ‘ 𝑅 ) ≠ 𝐶 ) |
72 |
71
|
necomd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝐶 ≠ ( 𝑆 ‘ 𝑅 ) ) |
73 |
15
|
eqcomd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐶 ) = 𝐴 ) |
74 |
1 2 3 5 23 7 24 8 11 73
|
mircom |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐴 ) = 𝐶 ) |
75 |
74
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ( 𝑆 ‘ 𝐴 ) = 𝐶 ) |
76 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝑅 ∈ 𝑃 ) |
77 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝐴 ∈ 𝑃 ) |
78 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) |
79 |
1 2 3 5 23 20 25 8 76 77 22 78
|
mirbtwni |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → ( 𝑆 ‘ 𝐴 ) ∈ ( ( 𝑆 ‘ 𝑅 ) 𝐼 ( 𝑆 ‘ 𝐵 ) ) ) |
80 |
75 79
|
eqeltrrd |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝐶 ∈ ( ( 𝑆 ‘ 𝑅 ) 𝐼 ( 𝑆 ‘ 𝐵 ) ) ) |
81 |
1 2 3 4 5 19 20 8 26 21 22 29 63 27 64 67 72 80
|
opphllem1 |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝐶 𝑂 𝐵 ) |
82 |
1 2 3 4 5 19 20 21 22 81
|
oppcom |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝑅 𝐼 𝐵 ) ) → 𝐵 𝑂 𝐶 ) |
83 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐷 ∈ ran 𝐿 ) |
84 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐺 ∈ TarskiG ) |
85 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐴 ∈ 𝑃 ) |
86 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐵 ∈ 𝑃 ) |
87 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐶 ∈ 𝑃 ) |
88 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝑅 ∈ 𝐷 ) |
89 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐴 𝑂 𝐶 ) |
90 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝑀 ∈ 𝐷 ) |
91 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐴 = ( 𝑆 ‘ 𝐶 ) ) |
92 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐴 ≠ 𝑅 ) |
93 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐵 ≠ 𝑅 ) |
94 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) |
95 |
1 2 3 4 5 83 84 8 85 86 87 88 89 90 91 92 93 94
|
opphllem1 |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 𝑅 𝐼 𝐴 ) ) → 𝐵 𝑂 𝐶 ) |
96 |
82 95 18
|
mpjaodan |
⊢ ( 𝜑 → 𝐵 𝑂 𝐶 ) |