Step |
Hyp |
Ref |
Expression |
1 |
|
ishlg.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
ishlg.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
3 |
|
ishlg.k |
⊢ 𝐾 = ( hlG ‘ 𝐺 ) |
4 |
|
ishlg.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
5 |
|
ishlg.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
6 |
|
ishlg.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
7 |
|
hlln.1 |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
8 |
|
hltr.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) |
9 |
|
hlcgrex.m |
⊢ − = ( dist ‘ 𝐺 ) |
10 |
|
hlcgrex.1 |
⊢ ( 𝜑 → 𝐷 ≠ 𝐴 ) |
11 |
|
hlcgrex.2 |
⊢ ( 𝜑 → 𝐵 ≠ 𝐶 ) |
12 |
|
hlcgreulem.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑃 ) |
13 |
|
hlcgreulem.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑃 ) |
14 |
|
hlcgreulem.1 |
⊢ ( 𝜑 → 𝑋 ( 𝐾 ‘ 𝐴 ) 𝐷 ) |
15 |
|
hlcgreulem.2 |
⊢ ( 𝜑 → 𝑌 ( 𝐾 ‘ 𝐴 ) 𝐷 ) |
16 |
|
hlcgreulem.3 |
⊢ ( 𝜑 → ( 𝐴 − 𝑋 ) = ( 𝐵 − 𝐶 ) ) |
17 |
|
hlcgreulem.4 |
⊢ ( 𝜑 → ( 𝐴 − 𝑌 ) = ( 𝐵 − 𝐶 ) ) |
18 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐺 ∈ TarskiG ) |
19 |
4
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐴 ∈ 𝑃 ) |
20 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐵 ∈ 𝑃 ) |
21 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐶 ∈ 𝑃 ) |
22 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝑦 ∈ 𝑃 ) |
23 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝑋 ∈ 𝑃 ) |
24 |
13
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝑌 ∈ 𝑃 ) |
25 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐴 ≠ 𝑦 ) |
26 |
25
|
necomd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝑦 ≠ 𝐴 ) |
27 |
8
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐷 ∈ 𝑃 ) |
28 |
1 2 3 12 8 4 7 14
|
hlcomd |
⊢ ( 𝜑 → 𝐷 ( 𝐾 ‘ 𝐴 ) 𝑋 ) |
29 |
28
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐷 ( 𝐾 ‘ 𝐴 ) 𝑋 ) |
30 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ) |
31 |
1 2 3 27 23 22 18 19 29 30
|
btwnhl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐴 ∈ ( 𝑋 𝐼 𝑦 ) ) |
32 |
1 9 2 18 23 19 22 31
|
tgbtwncom |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐴 ∈ ( 𝑦 𝐼 𝑋 ) ) |
33 |
1 2 3 13 8 4 7 15
|
hlcomd |
⊢ ( 𝜑 → 𝐷 ( 𝐾 ‘ 𝐴 ) 𝑌 ) |
34 |
33
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐷 ( 𝐾 ‘ 𝐴 ) 𝑌 ) |
35 |
1 2 3 27 24 22 18 19 34 30
|
btwnhl |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐴 ∈ ( 𝑌 𝐼 𝑦 ) ) |
36 |
1 9 2 18 24 19 22 35
|
tgbtwncom |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝐴 ∈ ( 𝑦 𝐼 𝑌 ) ) |
37 |
16
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → ( 𝐴 − 𝑋 ) = ( 𝐵 − 𝐶 ) ) |
38 |
17
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → ( 𝐴 − 𝑌 ) = ( 𝐵 − 𝐶 ) ) |
39 |
1 9 2 18 19 20 21 22 23 24 26 32 36 37 38
|
tgsegconeq |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) → 𝑋 = 𝑌 ) |
40 |
1
|
fvexi |
⊢ 𝑃 ∈ V |
41 |
40
|
a1i |
⊢ ( 𝜑 → 𝑃 ∈ V ) |
42 |
41 5 6 11
|
nehash2 |
⊢ ( 𝜑 → 2 ≤ ( ♯ ‘ 𝑃 ) ) |
43 |
1 9 2 7 8 4 42
|
tgbtwndiff |
⊢ ( 𝜑 → ∃ 𝑦 ∈ 𝑃 ( 𝐴 ∈ ( 𝐷 𝐼 𝑦 ) ∧ 𝐴 ≠ 𝑦 ) ) |
44 |
39 43
|
r19.29a |
⊢ ( 𝜑 → 𝑋 = 𝑌 ) |