Step |
Hyp |
Ref |
Expression |
1 |
|
mirval.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
mirval.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
mirval.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
mirval.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
5 |
|
mirval.s |
⊢ 𝑆 = ( pInvG ‘ 𝐺 ) |
6 |
|
mirval.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
7 |
|
mirhl.m |
⊢ 𝑀 = ( 𝑆 ‘ 𝐴 ) |
8 |
|
mirhl.k |
⊢ 𝐾 = ( hlG ‘ 𝐺 ) |
9 |
|
mirhl.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
10 |
|
mirhl.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑃 ) |
11 |
|
mirhl.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑃 ) |
12 |
|
mirhl.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑃 ) |
13 |
|
mirhl.1 |
⊢ ( 𝜑 → 𝑋 ( 𝐾 ‘ 𝑍 ) 𝑌 ) |
14 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝐺 ∈ TarskiG ) |
15 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝐴 ∈ 𝑃 ) |
16 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝑋 ∈ 𝑃 ) |
17 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝑍 ∈ 𝑃 ) |
18 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑍 ) ) → ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑍 ) ) |
19 |
1 2 3 4 5 14 15 7 16 17 18
|
mireq |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝑋 = 𝑍 ) |
20 |
1 3 8 10 11 12 6
|
ishlg |
⊢ ( 𝜑 → ( 𝑋 ( 𝐾 ‘ 𝑍 ) 𝑌 ↔ ( 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍 ∧ ( 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) ) ) ) |
21 |
13 20
|
mpbid |
⊢ ( 𝜑 → ( 𝑋 ≠ 𝑍 ∧ 𝑌 ≠ 𝑍 ∧ ( 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) ) ) |
22 |
21
|
simp1d |
⊢ ( 𝜑 → 𝑋 ≠ 𝑍 ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝑋 ≠ 𝑍 ) |
24 |
23
|
neneqd |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑍 ) ) → ¬ 𝑋 = 𝑍 ) |
25 |
19 24
|
pm2.65da |
⊢ ( 𝜑 → ¬ ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑍 ) ) |
26 |
25
|
neqned |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ≠ ( 𝑀 ‘ 𝑍 ) ) |
27 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝐺 ∈ TarskiG ) |
28 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝐴 ∈ 𝑃 ) |
29 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝑌 ∈ 𝑃 ) |
30 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝑍 ∈ 𝑃 ) |
31 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝑍 ) ) → ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝑍 ) ) |
32 |
1 2 3 4 5 27 28 7 29 30 31
|
mireq |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝑌 = 𝑍 ) |
33 |
21
|
simp2d |
⊢ ( 𝜑 → 𝑌 ≠ 𝑍 ) |
34 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝑍 ) ) → 𝑌 ≠ 𝑍 ) |
35 |
34
|
neneqd |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝑍 ) ) → ¬ 𝑌 = 𝑍 ) |
36 |
32 35
|
pm2.65da |
⊢ ( 𝜑 → ¬ ( 𝑀 ‘ 𝑌 ) = ( 𝑀 ‘ 𝑍 ) ) |
37 |
36
|
neqned |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝑌 ) ≠ ( 𝑀 ‘ 𝑍 ) ) |
38 |
21
|
simp3d |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) ) |
39 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) → 𝐺 ∈ TarskiG ) |
40 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) → 𝐴 ∈ 𝑃 ) |
41 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) → 𝑍 ∈ 𝑃 ) |
42 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) → 𝑋 ∈ 𝑃 ) |
43 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) → 𝑌 ∈ 𝑃 ) |
44 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) → 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) |
45 |
1 2 3 4 5 39 40 7 41 42 43 44
|
mirbtwni |
⊢ ( ( 𝜑 ∧ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) → ( 𝑀 ‘ 𝑋 ) ∈ ( ( 𝑀 ‘ 𝑍 ) 𝐼 ( 𝑀 ‘ 𝑌 ) ) ) |
46 |
45
|
ex |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) → ( 𝑀 ‘ 𝑋 ) ∈ ( ( 𝑀 ‘ 𝑍 ) 𝐼 ( 𝑀 ‘ 𝑌 ) ) ) ) |
47 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) → 𝐺 ∈ TarskiG ) |
48 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) → 𝐴 ∈ 𝑃 ) |
49 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) → 𝑍 ∈ 𝑃 ) |
50 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) → 𝑌 ∈ 𝑃 ) |
51 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) → 𝑋 ∈ 𝑃 ) |
52 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) → 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) |
53 |
1 2 3 4 5 47 48 7 49 50 51 52
|
mirbtwni |
⊢ ( ( 𝜑 ∧ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) → ( 𝑀 ‘ 𝑌 ) ∈ ( ( 𝑀 ‘ 𝑍 ) 𝐼 ( 𝑀 ‘ 𝑋 ) ) ) |
54 |
53
|
ex |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) → ( 𝑀 ‘ 𝑌 ) ∈ ( ( 𝑀 ‘ 𝑍 ) 𝐼 ( 𝑀 ‘ 𝑋 ) ) ) ) |
55 |
46 54
|
orim12d |
⊢ ( 𝜑 → ( ( 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) → ( ( 𝑀 ‘ 𝑋 ) ∈ ( ( 𝑀 ‘ 𝑍 ) 𝐼 ( 𝑀 ‘ 𝑌 ) ) ∨ ( 𝑀 ‘ 𝑌 ) ∈ ( ( 𝑀 ‘ 𝑍 ) 𝐼 ( 𝑀 ‘ 𝑋 ) ) ) ) ) |
56 |
38 55
|
mpd |
⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ∈ ( ( 𝑀 ‘ 𝑍 ) 𝐼 ( 𝑀 ‘ 𝑌 ) ) ∨ ( 𝑀 ‘ 𝑌 ) ∈ ( ( 𝑀 ‘ 𝑍 ) 𝐼 ( 𝑀 ‘ 𝑋 ) ) ) ) |
57 |
1 2 3 4 5 6 9 7 10
|
mircl |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ∈ 𝑃 ) |
58 |
1 2 3 4 5 6 9 7 11
|
mircl |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝑌 ) ∈ 𝑃 ) |
59 |
1 2 3 4 5 6 9 7 12
|
mircl |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝑍 ) ∈ 𝑃 ) |
60 |
1 3 8 57 58 59 6
|
ishlg |
⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ( 𝐾 ‘ ( 𝑀 ‘ 𝑍 ) ) ( 𝑀 ‘ 𝑌 ) ↔ ( ( 𝑀 ‘ 𝑋 ) ≠ ( 𝑀 ‘ 𝑍 ) ∧ ( 𝑀 ‘ 𝑌 ) ≠ ( 𝑀 ‘ 𝑍 ) ∧ ( ( 𝑀 ‘ 𝑋 ) ∈ ( ( 𝑀 ‘ 𝑍 ) 𝐼 ( 𝑀 ‘ 𝑌 ) ) ∨ ( 𝑀 ‘ 𝑌 ) ∈ ( ( 𝑀 ‘ 𝑍 ) 𝐼 ( 𝑀 ‘ 𝑋 ) ) ) ) ) ) |
61 |
26 37 56 60
|
mpbir3and |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝑋 ) ( 𝐾 ‘ ( 𝑀 ‘ 𝑍 ) ) ( 𝑀 ‘ 𝑌 ) ) |