Step |
Hyp |
Ref |
Expression |
1 |
|
tkgeom.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
tkgeom.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
tkgeom.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
tkgeom.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
tgcgrextend.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
6 |
|
tgcgrextend.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
7 |
|
tgcgrextend.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
8 |
|
tgcgrextend.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) |
9 |
|
tgcgrextend.e |
⊢ ( 𝜑 → 𝐸 ∈ 𝑃 ) |
10 |
|
tgcgrextend.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑃 ) |
11 |
|
tgsegconeq.1 |
⊢ ( 𝜑 → 𝐷 ≠ 𝐴 ) |
12 |
|
tgsegconeq.2 |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐷 𝐼 𝐸 ) ) |
13 |
|
tgsegconeq.3 |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐷 𝐼 𝐹 ) ) |
14 |
|
tgsegconeq.4 |
⊢ ( 𝜑 → ( 𝐴 − 𝐸 ) = ( 𝐵 − 𝐶 ) ) |
15 |
|
tgsegconeq.5 |
⊢ ( 𝜑 → ( 𝐴 − 𝐹 ) = ( 𝐵 − 𝐶 ) ) |
16 |
|
eqidd |
⊢ ( 𝜑 → ( 𝐷 − 𝐴 ) = ( 𝐷 − 𝐴 ) ) |
17 |
|
eqidd |
⊢ ( 𝜑 → ( 𝐴 − 𝐸 ) = ( 𝐴 − 𝐸 ) ) |
18 |
14 15
|
eqtr4d |
⊢ ( 𝜑 → ( 𝐴 − 𝐸 ) = ( 𝐴 − 𝐹 ) ) |
19 |
1 2 3 4 8 5 9 8 5 10 12 13 16 18
|
tgcgrextend |
⊢ ( 𝜑 → ( 𝐷 − 𝐸 ) = ( 𝐷 − 𝐹 ) ) |
20 |
1 2 3 4 8 5 9 8 5 9 9 10 11 12 12 16 17 19 18
|
axtg5seg |
⊢ ( 𝜑 → ( 𝐸 − 𝐸 ) = ( 𝐸 − 𝐹 ) ) |
21 |
20
|
eqcomd |
⊢ ( 𝜑 → ( 𝐸 − 𝐹 ) = ( 𝐸 − 𝐸 ) ) |
22 |
1 2 3 4 9 10 9 21
|
axtgcgrid |
⊢ ( 𝜑 → 𝐸 = 𝐹 ) |