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Theorem brofs 32433
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 4595 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑐⟩ = ⟨𝐴, 𝑐⟩)
21breq2d 4856 . . . 4 (𝑎 = 𝐴 → (𝑏 Btwn ⟨𝑎, 𝑐⟩ ↔ 𝑏 Btwn ⟨𝐴, 𝑐⟩))
32anbi1d 617 . . 3 (𝑎 = 𝐴 → ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩)))
4 opeq1 4595 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩)
54breq1d 4854 . . . 4 (𝑎 = 𝐴 → (⟨𝑎, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩))
65anbi1d 617 . . 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 4595 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑑⟩ = ⟨𝐴, 𝑑⟩)
87breq1d 4854 . . . 4 (𝑎 = 𝐴 → (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩))
98anbi1d 617 . . 3 (𝑎 = 𝐴 → ((⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩)))
103, 6, 93anbi123d 1553 . 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 4847 . . . 4 (𝑏 = 𝐵 → (𝑏 Btwn ⟨𝐴, 𝑐⟩ ↔ 𝐵 Btwn ⟨𝐴, 𝑐⟩))
1211anbi1d 617 . . 3 (𝑏 = 𝐵 → ((𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩)))
13 opeq2 4596 . . . . 5 (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩)
1413breq1d 4854 . . . 4 (𝑏 = 𝐵 → (⟨𝐴, 𝑏⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩))
15 opeq1 4595 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
1615breq1d 4854 . . . 4 (𝑏 = 𝐵 → (⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩))
1714, 16anbi12d 618 . . 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 4595 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, 𝑑⟩ = ⟨𝐵, 𝑑⟩)
1918breq1d 4854 . . . 4 (𝑏 = 𝐵 → (⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩))
2019anbi2d 616 . . 3 (𝑏 = 𝐵 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩)))
2112, 17, 203anbi123d 1553 . 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 4596 . . . . 5 (𝑐 = 𝐶 → ⟨𝐴, 𝑐⟩ = ⟨𝐴, 𝐶⟩)
2322breq2d 4856 . . . 4 (𝑐 = 𝐶 → (𝐵 Btwn ⟨𝐴, 𝑐⟩ ↔ 𝐵 Btwn ⟨𝐴, 𝐶⟩))
2423anbi1d 617 . . 3 (𝑐 = 𝐶 → ((𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩)))
25 opeq2 4596 . . . . 5 (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
2625breq1d 4854 . . . 4 (𝑐 = 𝐶 → (⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩))
2726anbi2d 616 . . 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 1554 . 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 4596 . . . . 5 (𝑑 = 𝐷 → ⟨𝐴, 𝑑⟩ = ⟨𝐴, 𝐷⟩)
3029breq1d 4854 . . . 4 (𝑑 = 𝐷 → (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩))
31 opeq2 4596 . . . . 5 (𝑑 = 𝐷 → ⟨𝐵, 𝑑⟩ = ⟨𝐵, 𝐷⟩)
3231breq1d 4854 . . . 4 (𝑑 = 𝐷 → (⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩))
3330, 32anbi12d 618 . . 3 (𝑑 = 𝐷 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)))
34333anbi3d 1559 . 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 4595 . . . . 5 (𝑒 = 𝐸 → ⟨𝑒, 𝑔⟩ = ⟨𝐸, 𝑔⟩)
3635breq2d 4856 . . . 4 (𝑒 = 𝐸 → (𝑓 Btwn ⟨𝑒, 𝑔⟩ ↔ 𝑓 Btwn ⟨𝐸, 𝑔⟩))
3736anbi2d 616 . . 3 (𝑒 = 𝐸 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩)))
38 opeq1 4595 . . . . 5 (𝑒 = 𝐸 → ⟨𝑒, 𝑓⟩ = ⟨𝐸, 𝑓⟩)
3938breq2d 4856 . . . 4 (𝑒 = 𝐸 → (⟨𝐴, 𝐵⟩Cgr⟨𝑒, 𝑓⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩))
4039anbi1d 617 . . 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 4595 . . . . 5 (𝑒 = 𝐸 → ⟨𝑒, ⟩ = ⟨𝐸, ⟩)
4241breq2d 4856 . . . 4 (𝑒 = 𝐸 → (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩))
4342anbi1d 617 . . 3 (𝑒 = 𝐸 → ((⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)))
4437, 40, 433anbi123d 1553 . 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 4847 . . . 4 (𝑓 = 𝐹 → (𝑓 Btwn ⟨𝐸, 𝑔⟩ ↔ 𝐹 Btwn ⟨𝐸, 𝑔⟩))
4645anbi2d 616 . . 3 (𝑓 = 𝐹 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩)))
47 opeq2 4596 . . . . 5 (𝑓 = 𝐹 → ⟨𝐸, 𝑓⟩ = ⟨𝐸, 𝐹⟩)
4847breq2d 4856 . . . 4 (𝑓 = 𝐹 → (⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝑓⟩ ↔ ⟨𝐴, 𝐵⟩Cgr⟨𝐸, 𝐹⟩))
49 opeq1 4595 . . . . 5 (𝑓 = 𝐹 → ⟨𝑓, 𝑔⟩ = ⟨𝐹, 𝑔⟩)
5049breq2d 4856 . . . 4 (𝑓 = 𝐹 → (⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩))
5148, 50anbi12d 618 . . 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 4595 . . . . 5 (𝑓 = 𝐹 → ⟨𝑓, ⟩ = ⟨𝐹, ⟩)
5352breq2d 4856 . . . 4 (𝑓 = 𝐹 → (⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩))
5453anbi2d 616 . . 3 (𝑓 = 𝐹 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩)))
5546, 51, 543anbi123d 1553 . 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 4596 . . . . 5 (𝑔 = 𝐺 → ⟨𝐸, 𝑔⟩ = ⟨𝐸, 𝐺⟩)
5756breq2d 4856 . . . 4 (𝑔 = 𝐺 → (𝐹 Btwn ⟨𝐸, 𝑔⟩ ↔ 𝐹 Btwn ⟨𝐸, 𝐺⟩))
5857anbi2d 616 . . 3 (𝑔 = 𝐺 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩)))
59 opeq2 4596 . . . . 5 (𝑔 = 𝐺 → ⟨𝐹, 𝑔⟩ = ⟨𝐹, 𝐺⟩)
6059breq2d 4856 . . . 4 (𝑔 = 𝐺 → (⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩))
6160anbi2d 616 . . 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 1554 . 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 4596 . . . . 5 ( = 𝐻 → ⟨𝐸, ⟩ = ⟨𝐸, 𝐻⟩)
6463breq2d 4856 . . . 4 ( = 𝐻 → (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩))
65 opeq2 4596 . . . . 5 ( = 𝐻 → ⟨𝐹, ⟩ = ⟨𝐹, 𝐻⟩)
6665breq2d 4856 . . . 4 ( = 𝐻 → (⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))
6764, 66anbi12d 618 . . 3 ( = 𝐻 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩)))
68673anbi3d 1559 . 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 6408 . 2 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
70 df-ofs 32411 . 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 31968 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 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  cop 4376   class class class wbr 4844  cfv 6101  cn 11305  𝔼cee 25982   Btwn cbtwn 25983  Cgrccgr 25984   OuterFiveSeg cofs 32410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-iota 6064  df-fv 6109  df-ofs 32411
This theorem is referenced by:  5segofs  32434  ofscom  32435  cgrextend  32436  segconeq  32438  ifscgr  32472  brofs2  32505
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