Step |
Hyp |
Ref |
Expression |
1 |
|
tglngval.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
tglngval.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
3 |
|
tglngval.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
tglngval.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
tglngval.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑃 ) |
6 |
|
tglngval.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑃 ) |
7 |
|
tgcolg.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑃 ) |
8 |
|
lnxfr.r |
⊢ ∼ = ( cgrG ‘ 𝐺 ) |
9 |
|
lnxfr.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
10 |
|
lnxfr.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
11 |
|
lnxfr.d |
⊢ − = ( dist ‘ 𝐺 ) |
12 |
|
tgidinside.1 |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ) |
13 |
|
tgidinside.2 |
⊢ ( 𝜑 → ( 𝑋 − 𝑍 ) = ( 𝑋 − 𝐴 ) ) |
14 |
|
tgidinside.3 |
⊢ ( 𝜑 → ( 𝑌 − 𝑍 ) = ( 𝑌 − 𝐴 ) ) |
15 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝐺 ∈ TarskiG ) |
16 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑋 ∈ 𝑃 ) |
17 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑍 ∈ 𝑃 ) |
18 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ) |
19 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑋 = 𝑌 ) |
20 |
19
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → ( 𝑋 𝐼 𝑋 ) = ( 𝑋 𝐼 𝑌 ) ) |
21 |
18 20
|
eleqtrrd |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑍 ∈ ( 𝑋 𝐼 𝑋 ) ) |
22 |
1 11 3 15 16 17 21
|
axtgbtwnid |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑋 = 𝑍 ) |
23 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝐴 ∈ 𝑃 ) |
24 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → ( 𝑋 − 𝑍 ) = ( 𝑋 − 𝐴 ) ) |
25 |
1 11 3 15 16 17 16 23 24 22
|
tgcgreq |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑋 = 𝐴 ) |
26 |
22 25
|
eqtr3d |
⊢ ( ( 𝜑 ∧ 𝑋 = 𝑌 ) → 𝑍 = 𝐴 ) |
27 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝐺 ∈ TarskiG ) |
28 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑋 ∈ 𝑃 ) |
29 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑌 ∈ 𝑃 ) |
30 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑍 ∈ 𝑃 ) |
31 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝐴 ∈ 𝑃 ) |
32 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝐵 ∈ 𝑃 ) |
33 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑋 ≠ 𝑌 ) |
34 |
1 2 3 4 5 7 6 12
|
btwncolg3 |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑋 𝐿 𝑍 ) ∨ 𝑋 = 𝑍 ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑌 ∈ ( 𝑋 𝐿 𝑍 ) ∨ 𝑋 = 𝑍 ) ) |
36 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑋 − 𝑍 ) = ( 𝑋 − 𝐴 ) ) |
37 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → ( 𝑌 − 𝑍 ) = ( 𝑌 − 𝐴 ) ) |
38 |
1 2 3 27 28 29 30 8 31 32 11 33 35 36 37
|
lnid |
⊢ ( ( 𝜑 ∧ 𝑋 ≠ 𝑌 ) → 𝑍 = 𝐴 ) |
39 |
26 38
|
pm2.61dane |
⊢ ( 𝜑 → 𝑍 = 𝐴 ) |