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 |
|
mirln2.m |
⊢ 𝑀 = ( 𝑆 ‘ 𝐴 ) |
8 |
|
mirln2.d |
⊢ ( 𝜑 → 𝐷 ∈ ran 𝐿 ) |
9 |
|
mirln2.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
10 |
|
mirln2.1 |
⊢ ( 𝜑 → 𝐵 ∈ 𝐷 ) |
11 |
|
mirln2.2 |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ∈ 𝐷 ) |
12 |
1 4 3 6 8 10
|
tglnpt |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
13 |
1 2 3 4 5 6 9 7 12
|
mirinv |
⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝐵 ) = 𝐵 ↔ 𝐴 = 𝐵 ) ) |
14 |
13
|
biimpa |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → 𝐴 = 𝐵 ) |
15 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → 𝐵 ∈ 𝐷 ) |
16 |
14 15
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) = 𝐵 ) → 𝐴 ∈ 𝐷 ) |
17 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → 𝐺 ∈ TarskiG ) |
18 |
1 4 3 6 8 11
|
tglnpt |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐵 ) ∈ 𝑃 ) |
19 |
18
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → ( 𝑀 ‘ 𝐵 ) ∈ 𝑃 ) |
20 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → 𝐵 ∈ 𝑃 ) |
21 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → 𝐴 ∈ 𝑃 ) |
22 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) |
23 |
1 2 3 4 5 17 21 7 20
|
mirbtwn |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → 𝐴 ∈ ( ( 𝑀 ‘ 𝐵 ) 𝐼 𝐵 ) ) |
24 |
1 3 4 17 19 20 21 22 23
|
btwnlng1 |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → 𝐴 ∈ ( ( 𝑀 ‘ 𝐵 ) 𝐿 𝐵 ) ) |
25 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → 𝐷 ∈ ran 𝐿 ) |
26 |
11
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → ( 𝑀 ‘ 𝐵 ) ∈ 𝐷 ) |
27 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → 𝐵 ∈ 𝐷 ) |
28 |
1 3 4 17 19 20 22 22 25 26 27
|
tglinethru |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → 𝐷 = ( ( 𝑀 ‘ 𝐵 ) 𝐿 𝐵 ) ) |
29 |
24 28
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ ( 𝑀 ‘ 𝐵 ) ≠ 𝐵 ) → 𝐴 ∈ 𝐷 ) |
30 |
16 29
|
pm2.61dane |
⊢ ( 𝜑 → 𝐴 ∈ 𝐷 ) |