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Theorem brfs 34550
Description: Binary relation form of the general five segment predicate. (Contributed by Scott Fenton, 5-Oct-2013.)
Assertion
Ref Expression
brfs (((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⟩⟩ FiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))))

Proof of Theorem brfs
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 𝑝 𝑞 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5107 . . 3 (𝑎 = 𝐴 → (𝑎 Colinear ⟨𝑏, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝑏, 𝑐⟩))
2 opeq1 4829 . . . 4 (𝑎 = 𝐴 → ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ = ⟨𝐴, ⟨𝑏, 𝑐⟩⟩)
32breq1d 5114 . . 3 (𝑎 = 𝐴 → (⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
4 opeq1 4829 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑑⟩ = ⟨𝐴, 𝑑⟩)
54breq1d 5114 . . . 4 (𝑎 = 𝐴 → (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩))
65anbi1d 631 . . 3 (𝑎 = 𝐴 → ((⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩)))
71, 3, 63anbi123d 1437 . 2 (𝑎 = 𝐴 → ((𝑎 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩)) ↔ (𝐴 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩))))
8 opeq1 4829 . . . 4 (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
98breq2d 5116 . . 3 (𝑏 = 𝐵 → (𝐴 Colinear ⟨𝑏, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝐵, 𝑐⟩))
108opeq2d 4836 . . . 4 (𝑏 = 𝐵 → ⟨𝐴, ⟨𝑏, 𝑐⟩⟩ = ⟨𝐴, ⟨𝐵, 𝑐⟩⟩)
1110breq1d 5114 . . 3 (𝑏 = 𝐵 → (⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
12 opeq1 4829 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, 𝑑⟩ = ⟨𝐵, 𝑑⟩)
1312breq1d 5114 . . . 4 (𝑏 = 𝐵 → (⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩))
1413anbi2d 630 . . 3 (𝑏 = 𝐵 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩)))
159, 11, 143anbi123d 1437 . 2 (𝑏 = 𝐵 → ((𝐴 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝑐⟩ ∧ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩))))
16 opeq2 4830 . . . 4 (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
1716breq2d 5116 . . 3 (𝑐 = 𝐶 → (𝐴 Colinear ⟨𝐵, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝐵, 𝐶⟩))
1816opeq2d 4836 . . . 4 (𝑐 = 𝐶 → ⟨𝐴, ⟨𝐵, 𝑐⟩⟩ = ⟨𝐴, ⟨𝐵, 𝐶⟩⟩)
1918breq1d 5114 . . 3 (𝑐 = 𝐶 → (⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
2017, 193anbi12d 1438 . 2 (𝑐 = 𝐶 → ((𝐴 Colinear ⟨𝐵, 𝑐⟩ ∧ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩))))
21 opeq2 4830 . . . . 5 (𝑑 = 𝐷 → ⟨𝐴, 𝑑⟩ = ⟨𝐴, 𝐷⟩)
2221breq1d 5114 . . . 4 (𝑑 = 𝐷 → (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩))
23 opeq2 4830 . . . . 5 (𝑑 = 𝐷 → ⟨𝐵, 𝑑⟩ = ⟨𝐵, 𝐷⟩)
2423breq1d 5114 . . . 4 (𝑑 = 𝐷 → (⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩))
2522, 24anbi12d 632 . . 3 (𝑑 = 𝐷 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)))
26253anbi3d 1443 . 2 (𝑑 = 𝐷 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩))))
27 opeq1 4829 . . . 4 (𝑒 = 𝐸 → ⟨𝑒, ⟨𝑓, 𝑔⟩⟩ = ⟨𝐸, ⟨𝑓, 𝑔⟩⟩)
2827breq2d 5116 . . 3 (𝑒 = 𝐸 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩))
29 opeq1 4829 . . . . 5 (𝑒 = 𝐸 → ⟨𝑒, ⟩ = ⟨𝐸, ⟩)
3029breq2d 5116 . . . 4 (𝑒 = 𝐸 → (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩))
3130anbi1d 631 . . 3 (𝑒 = 𝐸 → ((⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)))
3228, 313anbi23d 1440 . 2 (𝑒 = 𝐸 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩))))
33 opeq1 4829 . . . . 5 (𝑓 = 𝐹 → ⟨𝑓, 𝑔⟩ = ⟨𝐹, 𝑔⟩)
3433opeq2d 4836 . . . 4 (𝑓 = 𝐹 → ⟨𝐸, ⟨𝑓, 𝑔⟩⟩ = ⟨𝐸, ⟨𝐹, 𝑔⟩⟩)
3534breq2d 5116 . . 3 (𝑓 = 𝐹 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩))
36 opeq1 4829 . . . . 5 (𝑓 = 𝐹 → ⟨𝑓, ⟩ = ⟨𝐹, ⟩)
3736breq2d 5116 . . . 4 (𝑓 = 𝐹 → (⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩))
3837anbi2d 630 . . 3 (𝑓 = 𝐹 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩)))
3935, 383anbi23d 1440 . 2 (𝑓 = 𝐹 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩))))
40 opeq2 4830 . . . . 5 (𝑔 = 𝐺 → ⟨𝐹, 𝑔⟩ = ⟨𝐹, 𝐺⟩)
4140opeq2d 4836 . . . 4 (𝑔 = 𝐺 → ⟨𝐸, ⟨𝐹, 𝑔⟩⟩ = ⟨𝐸, ⟨𝐹, 𝐺⟩⟩)
4241breq2d 5116 . . 3 (𝑔 = 𝐺 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩))
43423anbi2d 1442 . 2 (𝑔 = 𝐺 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩))))
44 opeq2 4830 . . . . 5 ( = 𝐻 → ⟨𝐸, ⟩ = ⟨𝐸, 𝐻⟩)
4544breq2d 5116 . . . 4 ( = 𝐻 → (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩))
46 opeq2 4830 . . . . 5 ( = 𝐻 → ⟨𝐹, ⟩ = ⟨𝐹, 𝐻⟩)
4746breq2d 5116 . . . 4 ( = 𝐻 → (⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))
4845, 47anbi12d 632 . . 3 ( = 𝐻 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩)))
49483anbi3d 1443 . 2 ( = 𝐻 → ((𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩)) ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩ ∧ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))))
50 fveq2 6838 . 2 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
51 df-fs 34513 . 2 FiveSeg = {⟨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, ⟩⟩ ∧ (𝑎 Colinear ⟨𝑏, 𝑐⟩ ∧ ⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ∧ (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩)))}
527, 15, 20, 26, 32, 39, 43, 49, 50, 51br8 34107 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⟩⟩ FiveSeg ⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ (𝐴 Colinear ⟨𝐵, 𝐶⟩ ∧ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨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 397  w3a 1088   = wceq 1542  wcel 2107  cop 4591   class class class wbr 5104  cfv 6492  cn 12087  𝔼cee 27623  Cgrccgr 27625  Cgr3ccgr3 34507   Colinear ccolin 34508   FiveSeg cfs 34509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-iota 6444  df-fv 6500  df-fs 34513
This theorem is referenced by:  fscgr  34551  linecgr  34552
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