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Theorem brofs 35729
Description: Binary relation form of the outer five segment predicate. (Contributed by Scott Fenton, 21-Sep-2013.)
Assertion
Ref Expression
brofs (((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⟩))))

Proof of Theorem brofs
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 𝑝 𝑞 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 4875 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑐⟩ = ⟨𝐴, 𝑐⟩)
21breq2d 5161 . . . 4 (𝑎 = 𝐴 → (𝑏 Btwn ⟨𝑎, 𝑐⟩ ↔ 𝑏 Btwn ⟨𝐴, 𝑐⟩))
32anbi1d 629 . . 3 (𝑎 = 𝐴 → ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩)))
4 opeq1 4875 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
54breq1d 5159 . . . 4 (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩))
65anbi1d 629 . . 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 4875 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑑⟩ = ⟨𝐴, 𝑑⟩)
87breq1d 5159 . . . 4 (𝑎 = 𝐴 → (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩))
98anbi1d 629 . . 3 (𝑎 = 𝐴 → ((⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩)))
103, 6, 93anbi123d 1432 . 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, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩)) ↔ ((𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩))))
11 breq1 5152 . . . 4 (𝑏 = 𝐵 → (𝑏 Btwn ⟨𝐴, 𝑐⟩ ↔ 𝐵 Btwn ⟨𝐴, 𝑐⟩))
1211anbi1d 629 . . 3 (𝑏 = 𝐵 → ((𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩)))
13 opeq2 4876 . . . . 5 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
1413breq1d 5159 . . . 4 (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩))
15 opeq1 4875 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
1615breq1d 5159 . . . 4 (𝑏 = 𝐵 → (⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩))
1714, 16anbi12d 630 . . 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 4875 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, 𝑑⟩ = ⟨𝐵, 𝑑⟩)
1918breq1d 5159 . . . 4 (𝑏 = 𝐵 → (⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩))
2019anbi2d 628 . . 3 (𝑏 = 𝐵 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩)))
2112, 17, 203anbi123d 1432 . 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, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩))))
22 opeq2 4876 . . . . 5 (𝑐 = 𝐶 → ⟨𝐴, 𝑐⟩ = ⟨𝐴, 𝐶⟩)
2322breq2d 5161 . . . 4 (𝑐 = 𝐶 → (𝐵 Btwn ⟨𝐴, 𝑐⟩ ↔ 𝐵 Btwn ⟨𝐴, 𝐶⟩))
2423anbi1d 629 . . 3 (𝑐 = 𝐶 → ((𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩)))
25 opeq2 4876 . . . . 5 (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
2625breq1d 5159 . . . 4 (𝑐 = 𝐶 → (⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩))
2726anbi2d 628 . . 3 (𝑐 = 𝐶 → ((⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩)))
2824, 273anbi12d 1433 . 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, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩))))
29 opeq2 4876 . . . . 5 (𝑑 = 𝐷 → ⟨𝐴, 𝑑⟩ = ⟨𝐴, 𝐷⟩)
3029breq1d 5159 . . . 4 (𝑑 = 𝐷 → (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩))
31 opeq2 4876 . . . . 5 (𝑑 = 𝐷 → ⟨𝐵, 𝑑⟩ = ⟨𝐵, 𝐷⟩)
3231breq1d 5159 . . . 4 (𝑑 = 𝐷 → (⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩))
3330, 32anbi12d 630 . . 3 (𝑑 = 𝐷 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)))
34333anbi3d 1438 . 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, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩))))
35 opeq1 4875 . . . . 5 (𝑒 = 𝐸 → ⟨𝑒, 𝑔⟩ = ⟨𝐸, 𝑔⟩)
3635breq2d 5161 . . . 4 (𝑒 = 𝐸 → (𝑓 Btwn ⟨𝑒, 𝑔⟩ ↔ 𝑓 Btwn ⟨𝐸, 𝑔⟩))
3736anbi2d 628 . . 3 (𝑒 = 𝐸 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩)))
38 opeq1 4875 . . . . 5 (𝑒 = 𝐸 → ⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝑓⟩)
3938breq2d 5161 . . . 4 (𝑒 = 𝐸 → (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩))
4039anbi1d 629 . . 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 4875 . . . . 5 (𝑒 = 𝐸 → ⟨𝑒, ⟩ = ⟨𝐸, ⟩)
4241breq2d 5161 . . . 4 (𝑒 = 𝐸 → (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩))
4342anbi1d 629 . . 3 (𝑒 = 𝐸 → ((⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)))
4437, 40, 433anbi123d 1432 . 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, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩))))
45 breq1 5152 . . . 4 (𝑓 = 𝐹 → (𝑓 Btwn ⟨𝐸, 𝑔⟩ ↔ 𝐹 Btwn ⟨𝐸, 𝑔⟩))
4645anbi2d 628 . . 3 (𝑓 = 𝐹 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩)))
47 opeq2 4876 . . . . 5 (𝑓 = 𝐹 → ⟨𝐸, 𝑓⟩ = ⟨𝐸, 𝐹⟩)
4847breq2d 5161 . . . 4 (𝑓 = 𝐹 → (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩))
49 opeq1 4875 . . . . 5 (𝑓 = 𝐹 → ⟨𝑓, 𝑔⟩ = ⟨𝐹, 𝑔⟩)
5049breq2d 5161 . . . 4 (𝑓 = 𝐹 → (⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩))
5148, 50anbi12d 630 . . 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 4875 . . . . 5 (𝑓 = 𝐹 → ⟨𝑓, ⟩ = ⟨𝐹, ⟩)
5352breq2d 5161 . . . 4 (𝑓 = 𝐹 → (⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩))
5453anbi2d 628 . . 3 (𝑓 = 𝐹 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩)))
5546, 51, 543anbi123d 1432 . 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, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩))))
56 opeq2 4876 . . . . 5 (𝑔 = 𝐺 → ⟨𝐸, 𝑔⟩ = ⟨𝐸, 𝐺⟩)
5756breq2d 5161 . . . 4 (𝑔 = 𝐺 → (𝐹 Btwn ⟨𝐸, 𝑔⟩ ↔ 𝐹 Btwn ⟨𝐸, 𝐺⟩))
5857anbi2d 628 . . 3 (𝑔 = 𝐺 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩)))
59 opeq2 4876 . . . . 5 (𝑔 = 𝐺 → ⟨𝐹, 𝑔⟩ = ⟨𝐹, 𝐺⟩)
6059breq2d 5161 . . . 4 (𝑔 = 𝐺 → (⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩))
6160anbi2d 628 . . 3 (𝑔 = 𝐺 → ((⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩) ↔ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩)))
6258, 613anbi12d 1433 . 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, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩))))
63 opeq2 4876 . . . . 5 ( = 𝐻 → ⟨𝐸, ⟩ = ⟨𝐸, 𝐻⟩)
6463breq2d 5161 . . . 4 ( = 𝐻 → (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩))
65 opeq2 4876 . . . . 5 ( = 𝐻 → ⟨𝐹, ⟩ = ⟨𝐹, 𝐻⟩)
6665breq2d 5161 . . . 4 ( = 𝐻 → (⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))
6764, 66anbi12d 630 . . 3 ( = 𝐻 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩)))
68673anbi3d 1438 . 2 ( = 𝐻 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨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⟩))))
69 fveq2 6896 . 2 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
70 df-ofs 35707 . 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)∃ ∈ (𝔼‘𝑛)(𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ⟩⟩ ∧ ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝑎, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩)))}
7110, 21, 28, 34, 44, 55, 62, 68, 69, 70br8 35478 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⟩))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  cop 4636   class class class wbr 5149  cfv 6549  cn 12245  𝔼cee 28771   Btwn cbtwn 28772  Cgrccgr 28773   OuterFiveSeg cofs 35706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-iota 6501  df-fv 6557  df-ofs 35707
This theorem is referenced by:  5segofs  35730  ofscom  35731  cgrextend  35732  segconeq  35734  ifscgr  35768  brofs2  35801
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