Step |
Hyp |
Ref |
Expression |
1 |
|
cgraid.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
cgraid.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
3 |
|
cgraid.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
4 |
|
cgraid.k |
⊢ 𝐾 = ( hlG ‘ 𝐺 ) |
5 |
|
cgraid.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
6 |
|
cgraid.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
7 |
|
cgraid.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
8 |
|
cgracom.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) |
9 |
|
cgracom.e |
⊢ ( 𝜑 → 𝐸 ∈ 𝑃 ) |
10 |
|
cgracom.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝑃 ) |
11 |
|
cgrcgra.1 |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
12 |
|
cgrcgra.2 |
⊢ ( 𝜑 → 𝐵 ≠ 𝐶 ) |
13 |
|
cgrcgra.3 |
⊢ ( 𝜑 → 〈“ 𝐴 𝐵 𝐶 ”〉 ( cgrG ‘ 𝐺 ) 〈“ 𝐷 𝐸 𝐹 ”〉 ) |
14 |
|
eqid |
⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 ) |
15 |
|
eqid |
⊢ ( cgrG ‘ 𝐺 ) = ( cgrG ‘ 𝐺 ) |
16 |
1 14 2 15 3 5 6 7 8 9 10 13
|
cgr3simp1 |
⊢ ( 𝜑 → ( 𝐴 ( dist ‘ 𝐺 ) 𝐵 ) = ( 𝐷 ( dist ‘ 𝐺 ) 𝐸 ) ) |
17 |
1 14 2 3 5 6 8 9 16 11
|
tgcgrneq |
⊢ ( 𝜑 → 𝐷 ≠ 𝐸 ) |
18 |
1 2 4 8 5 9 3 17
|
hlid |
⊢ ( 𝜑 → 𝐷 ( 𝐾 ‘ 𝐸 ) 𝐷 ) |
19 |
1 14 2 15 3 5 6 7 8 9 10 13
|
cgr3simp2 |
⊢ ( 𝜑 → ( 𝐵 ( dist ‘ 𝐺 ) 𝐶 ) = ( 𝐸 ( dist ‘ 𝐺 ) 𝐹 ) ) |
20 |
1 14 2 3 6 7 9 10 19
|
tgcgrcomlr |
⊢ ( 𝜑 → ( 𝐶 ( dist ‘ 𝐺 ) 𝐵 ) = ( 𝐹 ( dist ‘ 𝐺 ) 𝐸 ) ) |
21 |
12
|
necomd |
⊢ ( 𝜑 → 𝐶 ≠ 𝐵 ) |
22 |
1 14 2 3 7 6 10 9 20 21
|
tgcgrneq |
⊢ ( 𝜑 → 𝐹 ≠ 𝐸 ) |
23 |
1 2 4 10 5 9 3 22
|
hlid |
⊢ ( 𝜑 → 𝐹 ( 𝐾 ‘ 𝐸 ) 𝐹 ) |
24 |
1 2 4 3 5 6 7 8 9 10 8 10 13 18 23
|
iscgrad |
⊢ ( 𝜑 → 〈“ 𝐴 𝐵 𝐶 ”〉 ( cgrA ‘ 𝐺 ) 〈“ 𝐷 𝐸 𝐹 ”〉 ) |