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Theorem brfs 33930
 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 5035 . . 3 (𝑎 = 𝐴 → (𝑎 Colinear ⟨𝑏, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝑏, 𝑐⟩))
2 opeq1 4761 . . . 4 (𝑎 = 𝐴 → ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ = ⟨𝐴, ⟨𝑏, 𝑐⟩⟩)
32breq1d 5042 . . 3 (𝑎 = 𝐴 → (⟨𝑎, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
4 opeq1 4761 . . . . 5 (𝑎 = 𝐴 → ⟨𝑎, 𝑑⟩ = ⟨𝐴, 𝑑⟩)
54breq1d 5042 . . . 4 (𝑎 = 𝐴 → (⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩))
65anbi1d 632 . . 3 (𝑎 = 𝐴 → ((⟨𝑎, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩)))
71, 3, 63anbi123d 1433 . 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 4761 . . . 4 (𝑏 = 𝐵 → ⟨𝑏, 𝑐⟩ = ⟨𝐵, 𝑐⟩)
98breq2d 5044 . . 3 (𝑏 = 𝐵 → (𝐴 Colinear ⟨𝑏, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝐵, 𝑐⟩))
108opeq2d 4770 . . . 4 (𝑏 = 𝐵 → ⟨𝐴, ⟨𝑏, 𝑐⟩⟩ = ⟨𝐴, ⟨𝐵, 𝑐⟩⟩)
1110breq1d 5042 . . 3 (𝑏 = 𝐵 → (⟨𝐴, ⟨𝑏, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
12 opeq1 4761 . . . . 5 (𝑏 = 𝐵 → ⟨𝑏, 𝑑⟩ = ⟨𝐵, 𝑑⟩)
1312breq1d 5042 . . . 4 (𝑏 = 𝐵 → (⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩))
1413anbi2d 631 . . 3 (𝑏 = 𝐵 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝑏, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩)))
159, 11, 143anbi123d 1433 . 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 4763 . . . 4 (𝑐 = 𝐶 → ⟨𝐵, 𝑐⟩ = ⟨𝐵, 𝐶⟩)
1716breq2d 5044 . . 3 (𝑐 = 𝐶 → (𝐴 Colinear ⟨𝐵, 𝑐⟩ ↔ 𝐴 Colinear ⟨𝐵, 𝐶⟩))
1816opeq2d 4770 . . . 4 (𝑐 = 𝐶 → ⟨𝐴, ⟨𝐵, 𝑐⟩⟩ = ⟨𝐴, ⟨𝐵, 𝐶⟩⟩)
1918breq1d 5042 . . 3 (𝑐 = 𝐶 → (⟨𝐴, ⟨𝐵, 𝑐⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩))
2017, 193anbi12d 1434 . 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 4763 . . . . 5 (𝑑 = 𝐷 → ⟨𝐴, 𝑑⟩ = ⟨𝐴, 𝐷⟩)
2221breq1d 5042 . . . 4 (𝑑 = 𝐷 → (⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩))
23 opeq2 4763 . . . . 5 (𝑑 = 𝐷 → ⟨𝐵, 𝑑⟩ = ⟨𝐵, 𝐷⟩)
2423breq1d 5042 . . . 4 (𝑑 = 𝐷 → (⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩))
2522, 24anbi12d 633 . . 3 (𝑑 = 𝐷 → ((⟨𝐴, 𝑑⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝑑⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)))
26253anbi3d 1439 . 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 4761 . . . 4 (𝑒 = 𝐸 → ⟨𝑒, ⟨𝑓, 𝑔⟩⟩ = ⟨𝐸, ⟨𝑓, 𝑔⟩⟩)
2827breq2d 5044 . . 3 (𝑒 = 𝐸 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝑒, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩))
29 opeq1 4761 . . . . 5 (𝑒 = 𝐸 → ⟨𝑒, ⟩ = ⟨𝐸, ⟩)
3029breq2d 5044 . . . 4 (𝑒 = 𝐸 → (⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩))
3130anbi1d 632 . . 3 (𝑒 = 𝐸 → ((⟨𝐴, 𝐷⟩Cgr⟨𝑒, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩)))
3228, 313anbi23d 1436 . 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 4761 . . . . 5 (𝑓 = 𝐹 → ⟨𝑓, 𝑔⟩ = ⟨𝐹, 𝑔⟩)
3433opeq2d 4770 . . . 4 (𝑓 = 𝐹 → ⟨𝐸, ⟨𝑓, 𝑔⟩⟩ = ⟨𝐸, ⟨𝐹, 𝑔⟩⟩)
3534breq2d 5044 . . 3 (𝑓 = 𝐹 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝑓, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩))
36 opeq1 4761 . . . . 5 (𝑓 = 𝐹 → ⟨𝑓, ⟩ = ⟨𝐹, ⟩)
3736breq2d 5044 . . . 4 (𝑓 = 𝐹 → (⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩))
3837anbi2d 631 . . 3 (𝑓 = 𝐹 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝑓, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩)))
3935, 383anbi23d 1436 . 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 4763 . . . . 5 (𝑔 = 𝐺 → ⟨𝐹, 𝑔⟩ = ⟨𝐹, 𝐺⟩)
4140opeq2d 4770 . . . 4 (𝑔 = 𝐺 → ⟨𝐸, ⟨𝐹, 𝑔⟩⟩ = ⟨𝐸, ⟨𝐹, 𝐺⟩⟩)
4241breq2d 5044 . . 3 (𝑔 = 𝐺 → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝑔⟩⟩ ↔ ⟨𝐴, ⟨𝐵, 𝐶⟩⟩Cgr3⟨𝐸, ⟨𝐹, 𝐺⟩⟩))
43423anbi2d 1438 . 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 4763 . . . . 5 ( = 𝐻 → ⟨𝐸, ⟩ = ⟨𝐸, 𝐻⟩)
4544breq2d 5044 . . . 4 ( = 𝐻 → (⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ↔ ⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩))
46 opeq2 4763 . . . . 5 ( = 𝐻 → ⟨𝐹, ⟩ = ⟨𝐹, 𝐻⟩)
4746breq2d 5044 . . . 4 ( = 𝐻 → (⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩ ↔ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩))
4845, 47anbi12d 633 . . 3 ( = 𝐻 → ((⟨𝐴, 𝐷⟩Cgr⟨𝐸, ⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, ⟩) ↔ (⟨𝐴, 𝐷⟩Cgr⟨𝐸, 𝐻⟩ ∧ ⟨𝐵, 𝐷⟩Cgr⟨𝐹, 𝐻⟩)))
49483anbi3d 1439 . 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 6658 . 2 (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
51 df-fs 33893 . 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 33239 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ⟨cop 4528   class class class wbr 5032  ‘cfv 6335  ℕcn 11674  𝔼cee 26781  Cgrccgr 26783  Cgr3ccgr3 33887   Colinear ccolin 33888   FiveSeg cfs 33889 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-iota 6294  df-fv 6343  df-fs 33893 This theorem is referenced by:  fscgr  33931  linecgr  33932
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