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 |
|
oppcom.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
9 |
|
oppcom.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
10 |
|
oppcom.o |
⊢ ( 𝜑 → 𝐴 𝑂 𝐵 ) |
11 |
1 2 3 4 8 9
|
islnopp |
⊢ ( 𝜑 → ( 𝐴 𝑂 𝐵 ↔ ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) ) ) |
12 |
10 11
|
mpbid |
⊢ ( 𝜑 → ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
13 |
12
|
simpld |
⊢ ( 𝜑 → ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ) |
14 |
13
|
simprd |
⊢ ( 𝜑 → ¬ 𝐵 ∈ 𝐷 ) |
15 |
13
|
simpld |
⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐷 ) |
16 |
12
|
simprd |
⊢ ( 𝜑 → ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) |
17 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐺 ∈ TarskiG ) |
18 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐴 ∈ 𝑃 ) |
19 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) → 𝐺 ∈ TarskiG ) |
20 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) → 𝐷 ∈ ran 𝐿 ) |
21 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) → 𝑡 ∈ 𝐷 ) |
22 |
1 5 3 19 20 21
|
tglnpt |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) → 𝑡 ∈ 𝑃 ) |
23 |
22
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝑡 ∈ 𝑃 ) |
24 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝐵 ∈ 𝑃 ) |
25 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) |
26 |
1 2 3 17 18 23 24 25
|
tgbtwncom |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) → 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) |
27 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) → 𝐺 ∈ TarskiG ) |
28 |
9
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) → 𝐵 ∈ 𝑃 ) |
29 |
22
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) → 𝑡 ∈ 𝑃 ) |
30 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) → 𝐴 ∈ 𝑃 ) |
31 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) → 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) |
32 |
1 2 3 27 28 29 30 31
|
tgbtwncom |
⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) ∧ 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) → 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) |
33 |
26 32
|
impbida |
⊢ ( ( 𝜑 ∧ 𝑡 ∈ 𝐷 ) → ( 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ↔ 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) ) |
34 |
33
|
rexbidva |
⊢ ( 𝜑 → ( ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ↔ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) ) |
35 |
16 34
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) |
36 |
14 15 35
|
jca31 |
⊢ ( 𝜑 → ( ( ¬ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) ) |
37 |
1 2 3 4 9 8
|
islnopp |
⊢ ( 𝜑 → ( 𝐵 𝑂 𝐴 ↔ ( ( ¬ 𝐵 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐵 𝐼 𝐴 ) ) ) ) |
38 |
36 37
|
mpbird |
⊢ ( 𝜑 → 𝐵 𝑂 𝐴 ) |