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Theorem brifs 31789
Description: Binary relationship form of the inner five segment predicate. (Contributed by Scott Fenton, 26-Sep-2013.)
Assertion
Ref Expression
brifs (((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⟩⟩ InnerFiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝐺⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩))))

Proof of Theorem brifs
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 4370 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑐⟩ = ⟨𝐴, 𝑐⟩)
21breq2d 4625 . . . 4 (𝑎 = 𝐴 → (𝑏 Btwn ⟨𝑎, 𝑐⟩ ↔ 𝑏 Btwn ⟨𝐴, 𝑐⟩))
32anbi1d 740 . . 3 (𝑎 = 𝐴 → ((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩)))
41breq1d 4623 . . . 4 (𝑎 = 𝐴 → (⟨𝑎, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ↔ ⟨𝐴, 𝑐⟩Cgr⟨𝑒, 𝑔⟩))
54anbi1d 740 . . 3 (𝑎 = 𝐴 → ((⟨𝑎, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩)))
6 opeq1 4370 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑑⟩ = ⟨𝐴, 𝑑⟩)
76breq1d 4623 . . . 4 (𝑎 = 𝐴 → (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩))
87anbi1d 740 . . 3 (𝑎 = 𝐴 → ((⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑔, ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑔, ⟩)))
93, 5, 83anbi123d 1396 . 2 (𝑎 = 𝐴 → (((𝑏 Btwn ⟨𝑎, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝑎, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑔, ⟩)) ↔ ((𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑔, ⟩))))
10 breq1 4616 . . . 4 (𝑏 = 𝐵 → (𝑏 Btwn ⟨𝐴, 𝑐⟩ ↔ 𝐵 Btwn ⟨𝐴, 𝑐⟩))
1110anbi1d 740 . . 3 (𝑏 = 𝐵 → ((𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩)))
12 opeq1 4370 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
1312breq1d 4623 . . . 4 (𝑏 = 𝐵 → (⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩))
1413anbi2d 739 . . 3 (𝑏 = 𝐵 → ((⟨𝐴, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩)))
1511, 143anbi12d 1397 . 2 (𝑏 = 𝐵 → (((𝑏 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑔, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑔, ⟩))))
16 opeq2 4371 . . . . 5 (𝑐 = 𝐶 → ⟨𝐴, 𝑐⟩ = ⟨𝐴, 𝐶⟩)
1716breq2d 4625 . . . 4 (𝑐 = 𝐶 → (𝐵 Btwn ⟨𝐴, 𝑐⟩ ↔ 𝐵 Btwn ⟨𝐴, 𝐶⟩))
1817anbi1d 740 . . 3 (𝑐 = 𝐶 → ((𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩)))
1916breq1d 4623 . . . 4 (𝑐 = 𝐶 → (⟨𝐴, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ↔ ⟨𝐴, 𝐶⟩Cgr⟨𝑒, 𝑔⟩))
20 opeq2 4371 . . . . 5 (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
2120breq1d 4623 . . . 4 (𝑐 = 𝐶 → (⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩))
2219, 21anbi12d 746 . . 3 (𝑐 = 𝐶 → ((⟨𝐴, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝐶⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩)))
23 opeq1 4370 . . . . 5 (𝑐 = 𝐶 → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝑑⟩)
2423breq1d 4623 . . . 4 (𝑐 = 𝐶 → (⟨𝑐, 𝑑⟩Cgr⟨𝑔, ⟩ ↔ ⟨𝐶, 𝑑⟩Cgr⟨𝑔, ⟩))
2524anbi2d 739 . . 3 (𝑐 = 𝐶 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑔, ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝑔, ⟩)))
2618, 22, 253anbi123d 1396 . 2 (𝑐 = 𝐶 → (((𝐵 Btwn ⟨𝐴, 𝑐⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝐵, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑔, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝑔, ⟩))))
27 opeq2 4371 . . . . 5 (𝑑 = 𝐷 → ⟨𝐴, 𝑑⟩ = ⟨𝐴, 𝐷⟩)
2827breq1d 4623 . . . 4 (𝑑 = 𝐷 → (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩))
29 opeq2 4371 . . . . 5 (𝑑 = 𝐷 → ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩)
3029breq1d 4623 . . . 4 (𝑑 = 𝐷 → (⟨𝐶, 𝑑⟩Cgr⟨𝑔, ⟩ ↔ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩))
3128, 30anbi12d 746 . . 3 (𝑑 = 𝐷 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝑔, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩)))
32313anbi3d 1402 . 2 (𝑑 = 𝐷 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐶, 𝑑⟩Cgr⟨𝑔, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩))))
33 opeq1 4370 . . . . 5 (𝑒 = 𝐸 → ⟨𝑒, 𝑔⟩ = ⟨𝐸, 𝑔⟩)
3433breq2d 4625 . . . 4 (𝑒 = 𝐸 → (𝑓 Btwn ⟨𝑒, 𝑔⟩ ↔ 𝑓 Btwn ⟨𝐸, 𝑔⟩))
3534anbi2d 739 . . 3 (𝑒 = 𝐸 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩)))
3633breq2d 4625 . . . 4 (𝑒 = 𝐸 → (⟨𝐴, 𝐶⟩Cgr⟨𝑒, 𝑔⟩ ↔ ⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝑔⟩))
3736anbi1d 740 . . 3 (𝑒 = 𝐸 → ((⟨𝐴, 𝐶⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩)))
38 opeq1 4370 . . . . 5 (𝑒 = 𝐸 → ⟨𝑒, ⟩ = ⟨𝐸, ⟩)
3938breq2d 4625 . . . 4 (𝑒 = 𝐸 → (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩))
4039anbi1d 740 . . 3 (𝑒 = 𝐸 → ((⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩)))
4135, 37, 403anbi123d 1396 . 2 (𝑒 = 𝐸 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝑒, 𝑔⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩))))
42 breq1 4616 . . . 4 (𝑓 = 𝐹 → (𝑓 Btwn ⟨𝐸, 𝑔⟩ ↔ 𝐹 Btwn ⟨𝐸, 𝑔⟩))
4342anbi2d 739 . . 3 (𝑓 = 𝐹 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩)))
44 opeq1 4370 . . . . 5 (𝑓 = 𝐹 → ⟨𝑓, 𝑔⟩ = ⟨𝐹, 𝑔⟩)
4544breq2d 4625 . . . 4 (𝑓 = 𝐹 → (⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩))
4645anbi2d 739 . . 3 (𝑓 = 𝐹 → ((⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ↔ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩)))
4743, 463anbi12d 1397 . 2 (𝑓 = 𝐹 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝑓 Btwn ⟨𝐸, 𝑔⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩))))
48 opeq2 4371 . . . . 5 (𝑔 = 𝐺 → ⟨𝐸, 𝑔⟩ = ⟨𝐸, 𝐺⟩)
4948breq2d 4625 . . . 4 (𝑔 = 𝐺 → (𝐹 Btwn ⟨𝐸, 𝑔⟩ ↔ 𝐹 Btwn ⟨𝐸, 𝐺⟩))
5049anbi2d 739 . . 3 (𝑔 = 𝐺 → ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩) ↔ (𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩)))
5148breq2d 4625 . . . 4 (𝑔 = 𝐺 → (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝑔⟩ ↔ ⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝐺⟩))
52 opeq2 4371 . . . . 5 (𝑔 = 𝐺 → ⟨𝐹, 𝑔⟩ = ⟨𝐹, 𝐺⟩)
5352breq2d 4625 . . . 4 (𝑔 = 𝐺 → (⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩ ↔ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩))
5451, 53anbi12d 746 . . 3 (𝑔 = 𝐺 → ((⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩) ↔ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝐺⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩)))
55 opeq1 4370 . . . . 5 (𝑔 = 𝐺 → ⟨𝑔, ⟩ = ⟨𝐺, ⟩)
5655breq2d 4625 . . . 4 (𝑔 = 𝐺 → (⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩ ↔ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, ⟩))
5756anbi2d 739 . . 3 (𝑔 = 𝐺 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, ⟩)))
5850, 54, 573anbi123d 1396 . 2 (𝑔 = 𝐺 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝑔⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝑔⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝑔⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝑔, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝐺⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, ⟩))))
59 opeq2 4371 . . . . 5 ( = 𝐻 → ⟨𝐸, ⟩ = ⟨𝐸, 𝐻⟩)
6059breq2d 4625 . . . 4 ( = 𝐻 → (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩))
61 opeq2 4371 . . . . 5 ( = 𝐻 → ⟨𝐺, ⟩ = ⟨𝐺, 𝐻⟩)
6261breq2d 4625 . . . 4 ( = 𝐻 → (⟨𝐶, 𝐷⟩Cgr⟨𝐺, ⟩ ↔ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩))
6360, 62anbi12d 746 . . 3 ( = 𝐻 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩)))
64633anbi3d 1402 . 2 ( = 𝐻 → (((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝐺⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, ⟩)) ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝐺⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩))))
65 fveq2 6148 . 2 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
66 df-ifs 31786 . 2 InnerFiveSeg = {⟨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, 𝑐⟩Cgr⟨𝑒, 𝑔⟩ ∧ ⟨𝑏, 𝑐⟩Cgr⟨𝑓, 𝑔⟩) ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑐, 𝑑⟩Cgr⟨𝑔, ⟩)))}
679, 15, 26, 32, 41, 47, 58, 64, 65, 66br8 31351 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⟩⟩ InnerFiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ ((𝐵 Btwn ⟨𝐴, 𝐶⟩ ∧ 𝐹 Btwn ⟨𝐸, 𝐺⟩) ∧ (⟨𝐴, 𝐶⟩Cgr⟨𝐸, 𝐺⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐹, 𝐺⟩) ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐶, 𝐷⟩Cgr⟨𝐺, 𝐻⟩))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cop 4154   class class class wbr 4613  cfv 5847  cn 10964  𝔼cee 25668   Btwn cbtwn 25669  Cgrccgr 25670   InnerFiveSeg cifs 31781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-iota 5810  df-fv 5855  df-ifs 31786
This theorem is referenced by:  ifscgr  31790  cgrsub  31791  btwnxfr  31802  brifs2  31824  btwnconn1lem6  31838
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