Step | Hyp | Ref
| Expression |
1 | | opeq1 4831 |
. . . . 5
⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑐⟩ = ⟨𝐴, 𝑐⟩) |
2 | 1 | breq2d 5118 |
. . . 4
⊢ (𝑎 = 𝐴 → (𝑏 Btwn ⟨𝑎, 𝑐⟩ ↔ 𝑏 Btwn ⟨𝐴, 𝑐⟩)) |
3 | 2 | anbi1d 631 |
. . 3
⊢ (𝑎 = 𝐴 → ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩))) |
4 | | opeq1 4831 |
. . . . 5
⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩) |
5 | 4 | breq1d 5116 |
. . . 4
⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩)) |
6 | 5 | anbi1d 631 |
. . 3
⊢ (𝑎 = 𝐴 → ((⟨𝑎, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩))) |
7 | | opeq1 4831 |
. . . . 5
⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑑⟩ = ⟨𝐴, 𝑑⟩) |
8 | 7 | breq1d 5116 |
. . . 4
⊢ (𝑎 = 𝐴 → (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩)) |
9 | 8 | anbi1d 631 |
. . 3
⊢ (𝑎 = 𝐴 → ((⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩))) |
10 | 3, 6, 9 | 3anbi123d 1437 |
. 2
⊢ (𝑎 = 𝐴 → (((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝑎, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ ((𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)))) |
11 | | breq1 5109 |
. . . 4
⊢ (𝑏 = 𝐵 → (𝑏 Btwn ⟨𝐴, 𝑐⟩ ↔ 𝐵 Btwn ⟨𝐴, 𝑐⟩)) |
12 | 11 | anbi1d 631 |
. . 3
⊢ (𝑏 = 𝐵 → ((𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩))) |
13 | | opeq2 4832 |
. . . . 5
⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩) |
14 | 13 | breq1d 5116 |
. . . 4
⊢ (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩)) |
15 | | opeq1 4831 |
. . . . 5
⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩) |
16 | 15 | breq1d 5116 |
. . . 4
⊢ (𝑏 = 𝐵 → (⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩)) |
17 | 14, 16 | anbi12d 632 |
. . 3
⊢ (𝑏 = 𝐵 → ((⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩))) |
18 | | opeq1 4831 |
. . . . 5
⊢ (𝑏 = 𝐵 → ⟨𝑏, 𝑑⟩ = ⟨𝐵, 𝑑⟩) |
19 | 18 | breq1d 5116 |
. . . 4
⊢ (𝑏 = 𝐵 → (⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) |
20 | 19 | anbi2d 630 |
. . 3
⊢ (𝑏 = 𝐵 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩))) |
21 | 12, 17, 20 | 3anbi123d 1437 |
. 2
⊢ (𝑏 = 𝐵 → (((𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)))) |
22 | | opeq2 4832 |
. . . . 5
⊢ (𝑐 = 𝐶 → ⟨𝐴, 𝑐⟩ = ⟨𝐴, 𝐶⟩) |
23 | 22 | breq2d 5118 |
. . . 4
⊢ (𝑐 = 𝐶 → (𝐵 Btwn ⟨𝐴, 𝑐⟩ ↔ 𝐵 Btwn ⟨𝐴, 𝐶⟩)) |
24 | 23 | anbi1d 631 |
. . 3
⊢ (𝑐 = 𝐶 → ((𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩))) |
25 | | opeq2 4832 |
. . . . 5
⊢ (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩) |
26 | 25 | breq1d 5116 |
. . . 4
⊢ (𝑐 = 𝐶 → (⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩)) |
27 | 26 | anbi2d 630 |
. . 3
⊢ (𝑐 = 𝐶 → ((⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩))) |
28 | 24, 27 | 3anbi12d 1438 |
. 2
⊢ (𝑐 = 𝐶 → (((𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)))) |
29 | | opeq2 4832 |
. . . . 5
⊢ (𝑑 = 𝐷 → ⟨𝐴, 𝑑⟩ = ⟨𝐴, 𝐷⟩) |
30 | 29 | breq1d 5116 |
. . . 4
⊢ (𝑑 = 𝐷 → (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩)) |
31 | | opeq2 4832 |
. . . . 5
⊢ (𝑑 = 𝐷 → ⟨𝐵, 𝑑⟩ = ⟨𝐵, 𝐷⟩) |
32 | 31 | breq1d 5116 |
. . . 4
⊢ (𝑑 = 𝐷 → (⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)) |
33 | 30, 32 | anbi12d 632 |
. . 3
⊢ (𝑑 = 𝐷 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩))) |
34 | 33 | 3anbi3d 1443 |
. 2
⊢ (𝑑 = 𝐷 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ℎ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)))) |
35 | | opeq1 4831 |
. . . . 5
⊢ (𝑒 = 𝐸 → ⟨𝑒, 𝑔⟩ = ⟨𝐸, 𝑔⟩) |
36 | 35 | breq2d 5118 |
. . . 4
⊢ (𝑒 = 𝐸 → (𝑓 Btwn ⟨𝑒, 𝑔⟩ ↔ 𝑓 Btwn ⟨𝐸, 𝑔⟩)) |
37 | 36 | anbi2d 630 |
. . 3
⊢ (𝑒 = 𝐸 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩))) |
38 | | opeq1 4831 |
. . . . 5
⊢ (𝑒 = 𝐸 → ⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝑓⟩) |
39 | 38 | breq2d 5118 |
. . . 4
⊢ (𝑒 = 𝐸 → (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩)) |
40 | 39 | anbi1d 631 |
. . 3
⊢ (𝑒 = 𝐸 → ((⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩))) |
41 | | opeq1 4831 |
. . . . 5
⊢ (𝑒 = 𝐸 → ⟨𝑒, ℎ⟩ = ⟨𝐸, ℎ⟩) |
42 | 41 | breq2d 5118 |
. . . 4
⊢ (𝑒 = 𝐸 → (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩)) |
43 | 42 | anbi1d 631 |
. . 3
⊢ (𝑒 = 𝐸 → ((⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩))) |
44 | 37, 40, 43 | 3anbi123d 1437 |
. 2
⊢ (𝑒 = 𝐸 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)))) |
45 | | breq1 5109 |
. . . 4
⊢ (𝑓 = 𝐹 → (𝑓 Btwn ⟨𝐸, 𝑔⟩ ↔ 𝐹 Btwn ⟨𝐸, 𝑔⟩)) |
46 | 45 | anbi2d 630 |
. . 3
⊢ (𝑓 = 𝐹 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩))) |
47 | | opeq2 4832 |
. . . . 5
⊢ (𝑓 = 𝐹 → ⟨𝐸, 𝑓⟩ = ⟨𝐸, 𝐹⟩) |
48 | 47 | breq2d 5118 |
. . . 4
⊢ (𝑓 = 𝐹 → (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩)) |
49 | | opeq1 4831 |
. . . . 5
⊢ (𝑓 = 𝐹 → ⟨𝑓, 𝑔⟩ = ⟨𝐹, 𝑔⟩) |
50 | 49 | breq2d 5118 |
. . . 4
⊢ (𝑓 = 𝐹 → (⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩)) |
51 | 48, 50 | anbi12d 632 |
. . 3
⊢ (𝑓 = 𝐹 → ((⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩))) |
52 | | opeq1 4831 |
. . . . 5
⊢ (𝑓 = 𝐹 → ⟨𝑓, ℎ⟩ = ⟨𝐹, ℎ⟩) |
53 | 52 | breq2d 5118 |
. . . 4
⊢ (𝑓 = 𝐹 → (⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)) |
54 | 53 | anbi2d 630 |
. . 3
⊢ (𝑓 = 𝐹 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩))) |
55 | 46, 51, 54 | 3anbi123d 1437 |
. 2
⊢ (𝑓 = 𝐹 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ℎ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)))) |
56 | | opeq2 4832 |
. . . . 5
⊢ (𝑔 = 𝐺 → ⟨𝐸, 𝑔⟩ = ⟨𝐸, 𝐺⟩) |
57 | 56 | breq2d 5118 |
. . . 4
⊢ (𝑔 = 𝐺 → (𝐹 Btwn ⟨𝐸, 𝑔⟩ ↔ 𝐹 Btwn ⟨𝐸, 𝐺⟩)) |
58 | 57 | anbi2d 630 |
. . 3
⊢ (𝑔 = 𝐺 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩))) |
59 | | opeq2 4832 |
. . . . 5
⊢ (𝑔 = 𝐺 → ⟨𝐹, 𝑔⟩ = ⟨𝐹, 𝐺⟩) |
60 | 59 | breq2d 5118 |
. . . 4
⊢ (𝑔 = 𝐺 → (⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩)) |
61 | 60 | anbi2d 630 |
. . 3
⊢ (𝑔 = 𝐺 → ((⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩))) |
62 | 58, 61 | 3anbi12d 1438 |
. 2
⊢ (𝑔 = 𝐺 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)))) |
63 | | opeq2 4832 |
. . . . 5
⊢ (ℎ = 𝐻 → ⟨𝐸, ℎ⟩ = ⟨𝐸, 𝐻⟩) |
64 | 63 | breq2d 5118 |
. . . 4
⊢ (ℎ = 𝐻 → (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩)) |
65 | | opeq2 4832 |
. . . . 5
⊢ (ℎ = 𝐻 → ⟨𝐹, ℎ⟩ = ⟨𝐹, 𝐻⟩) |
66 | 65 | breq2d 5118 |
. . . 4
⊢ (ℎ = 𝐻 → (⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩)) |
67 | 64, 66 | anbi12d 632 |
. . 3
⊢ (ℎ = 𝐻 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))) |
68 | 67 | 3anbi3d 1443 |
. 2
⊢ (ℎ = 𝐻 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ℎ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ℎ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩)))) |
69 | | fveq2 6843 |
. 2
⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁)) |
70 | | df-ofs 34614 |
. 2
⊢
OuterFiveSeg = {⟨𝑝,
𝑞⟩ ∣
∃𝑛 ∈ ℕ
∃𝑎 ∈
(𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)∃𝑒 ∈ (𝔼‘𝑛)∃𝑓 ∈ (𝔼‘𝑛)∃𝑔 ∈ (𝔼‘𝑛)∃ℎ ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝑎, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ℎ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ℎ⟩)))} |
71 | 10, 21, 28, 34, 44, 55, 62, 68, 69, 70 | br8 34385 |
1
⊢ (((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁)) ∧ (𝐹 ∈ (𝔼‘𝑁) ∧ 𝐺 ∈ (𝔼‘𝑁) ∧ 𝐻 ∈ (𝔼‘𝑁))) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩ OuterFiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩)))) |