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Theorem outsideofeq 31906
Description: Uniqueness law for OutsideOf. Analogue of segconeq 31786. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsideofeq ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → (((𝐴OutsideOf⟨𝑋, 𝑅⟩ ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ (𝐴OutsideOf⟨𝑌, 𝑅⟩ ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) → 𝑋 = 𝑌))

Proof of Theorem outsideofeq
StepHypRef Expression
1 simp1 1059 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
2 simp21 1092 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
3 simp32 1096 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → 𝑋 ∈ (𝔼‘𝑁))
4 simp22 1093 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → 𝑅 ∈ (𝔼‘𝑁))
5 broutsideof2 31898 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝐴OutsideOf⟨𝑋, 𝑅⟩ ↔ (𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩))))
61, 2, 3, 4, 5syl13anc 1325 . . . 4 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → (𝐴OutsideOf⟨𝑋, 𝑅⟩ ↔ (𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩))))
76anbi1d 740 . . 3 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝐴OutsideOf⟨𝑋, 𝑅⟩ ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ↔ ((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩)))
8 simp33 1097 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → 𝑌 ∈ (𝔼‘𝑁))
9 broutsideof2 31898 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁))) → (𝐴OutsideOf⟨𝑌, 𝑅⟩ ↔ (𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩))))
101, 2, 8, 4, 9syl13anc 1325 . . . 4 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → (𝐴OutsideOf⟨𝑌, 𝑅⟩ ↔ (𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩))))
1110anbi1d 740 . . 3 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝐴OutsideOf⟨𝑌, 𝑅⟩ ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩) ↔ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)))
127, 11anbi12d 746 . 2 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → (((𝐴OutsideOf⟨𝑋, 𝑅⟩ ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ (𝐴OutsideOf⟨𝑌, 𝑅⟩ ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) ↔ (((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩))))
13 simpll3 1100 . . . . . . 7 ((((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) → (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩))
14 simprl3 1106 . . . . . . 7 ((((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) → (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩))
1513, 14jca 554 . . . . . 6 ((((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) → ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩) ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)))
1615adantl 482 . . . . 5 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩))) → ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩) ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)))
17 simpll2 1099 . . . . . 6 ((((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) → 𝑅𝐴)
1817adantl 482 . . . . 5 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩))) → 𝑅𝐴)
19 simp23 1094 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
20 simp31 1095 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁))
21 simprlr 802 . . . . . 6 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩))) → ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩)
22 simprrr 804 . . . . . 6 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩))) → ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)
231, 2, 3, 2, 8, 19, 20, 21, 22cgrtr3and 31771 . . . . 5 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩))) → ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)
2416, 18, 23jca32 557 . . . 4 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩))) → (((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩) ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)))
25 simprll 801 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑋 Btwn ⟨𝐴, 𝑅⟩)
26 simprlr 802 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑌 Btwn ⟨𝐴, 𝑅⟩)
27 simprrr 804 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)
28 midofsegid 31880 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩ ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩) → 𝑋 = 𝑌))
291, 2, 4, 3, 8, 28syl122anc 1332 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩ ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩) → 𝑋 = 𝑌))
3029adantr 481 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩ ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩) → 𝑋 = 𝑌))
3125, 26, 27, 30mp3and 1424 . . . . . . 7 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑋 = 𝑌)
3231exp32 630 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) → ((𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩) → 𝑋 = 𝑌)))
33 simprlr 802 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑌 Btwn ⟨𝐴, 𝑅⟩)
34 simprll 801 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑅 Btwn ⟨𝐴, 𝑋⟩)
351, 2, 8, 4, 3, 33, 34btwnexchand 31802 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑌 Btwn ⟨𝐴, 𝑋⟩)
36 simprrr 804 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)
371, 2, 3, 8, 35, 36endofsegidand 31862 . . . . . . 7 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑋 = 𝑌)
3837exp32 630 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑌 Btwn ⟨𝐴, 𝑅⟩) → ((𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩) → 𝑋 = 𝑌)))
39 simprll 801 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑋 Btwn ⟨𝐴, 𝑅⟩)
40 simprlr 802 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑅 Btwn ⟨𝐴, 𝑌⟩)
411, 2, 3, 4, 8, 39, 40btwnexchand 31802 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑋 Btwn ⟨𝐴, 𝑌⟩)
42 simprrr 804 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)
431, 2, 3, 2, 8, 42cgrcomand 31767 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → ⟨𝐴, 𝑌⟩Cgr⟨𝐴, 𝑋⟩)
441, 2, 8, 3, 41, 43endofsegidand 31862 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑌 = 𝑋)
4544eqcomd 2627 . . . . . . 7 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑋 = 𝑌)
4645exp32 630 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) → ((𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩) → 𝑋 = 𝑌)))
47 simprr 795 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)) ∧ 𝑋 Btwn ⟨𝐴, 𝑌⟩)) → 𝑋 Btwn ⟨𝐴, 𝑌⟩)
48 simplrr 800 . . . . . . . . . . . . 13 ((((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)) ∧ 𝑋 Btwn ⟨𝐴, 𝑌⟩) → ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)
4948adantl 482 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)) ∧ 𝑋 Btwn ⟨𝐴, 𝑌⟩)) → ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)
501, 2, 3, 2, 8, 49cgrcomand 31767 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)) ∧ 𝑋 Btwn ⟨𝐴, 𝑌⟩)) → ⟨𝐴, 𝑌⟩Cgr⟨𝐴, 𝑋⟩)
511, 2, 8, 3, 47, 50endofsegidand 31862 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)) ∧ 𝑋 Btwn ⟨𝐴, 𝑌⟩)) → 𝑌 = 𝑋)
5251eqcomd 2627 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)) ∧ 𝑋 Btwn ⟨𝐴, 𝑌⟩)) → 𝑋 = 𝑌)
5352expr 642 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → (𝑋 Btwn ⟨𝐴, 𝑌⟩ → 𝑋 = 𝑌))
54 simprr 795 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)) ∧ 𝑌 Btwn ⟨𝐴, 𝑋⟩)) → 𝑌 Btwn ⟨𝐴, 𝑋⟩)
55 simplrr 800 . . . . . . . . . . 11 ((((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)) ∧ 𝑌 Btwn ⟨𝐴, 𝑋⟩) → ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)
5655adantl 482 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)) ∧ 𝑌 Btwn ⟨𝐴, 𝑋⟩)) → ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)
571, 2, 3, 8, 54, 56endofsegidand 31862 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩)) ∧ 𝑌 Btwn ⟨𝐴, 𝑋⟩)) → 𝑋 = 𝑌)
5857expr 642 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → (𝑌 Btwn ⟨𝐴, 𝑋⟩ → 𝑋 = 𝑌))
59 simprrl 803 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑅𝐴)
6059necomd 2845 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝐴𝑅)
61 simprll 801 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑅 Btwn ⟨𝐴, 𝑋⟩)
62 simprlr 802 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑅 Btwn ⟨𝐴, 𝑌⟩)
63 btwnconn1 31877 . . . . . . . . . . 11 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝐴𝑅𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) → (𝑋 Btwn ⟨𝐴, 𝑌⟩ ∨ 𝑌 Btwn ⟨𝐴, 𝑋⟩)))
641, 2, 4, 3, 8, 63syl122anc 1332 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝐴𝑅𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) → (𝑋 Btwn ⟨𝐴, 𝑌⟩ ∨ 𝑌 Btwn ⟨𝐴, 𝑋⟩)))
6564adantr 481 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → ((𝐴𝑅𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) → (𝑋 Btwn ⟨𝐴, 𝑌⟩ ∨ 𝑌 Btwn ⟨𝐴, 𝑋⟩)))
6660, 61, 62, 65mp3and 1424 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → (𝑋 Btwn ⟨𝐴, 𝑌⟩ ∨ 𝑌 Btwn ⟨𝐴, 𝑋⟩))
6753, 58, 66mpjaod 396 . . . . . . 7 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑋 = 𝑌)
6867exp32 630 . . . . . 6 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((𝑅 Btwn ⟨𝐴, 𝑋⟩ ∧ 𝑅 Btwn ⟨𝐴, 𝑌⟩) → ((𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩) → 𝑋 = 𝑌)))
6932, 38, 46, 68ccased 987 . . . . 5 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → (((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩) ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) → ((𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩) → 𝑋 = 𝑌)))
7069imp32 449 . . . 4 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩) ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ (𝑅𝐴 ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐴, 𝑌⟩))) → 𝑋 = 𝑌)
7124, 70syldan 487 . . 3 (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) ∧ (((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩))) → 𝑋 = 𝑌)
7271ex 450 . 2 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → ((((𝑋𝐴𝑅𝐴 ∧ (𝑋 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑋⟩)) ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ ((𝑌𝐴𝑅𝐴 ∧ (𝑌 Btwn ⟨𝐴, 𝑅⟩ ∨ 𝑅 Btwn ⟨𝐴, 𝑌⟩)) ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) → 𝑋 = 𝑌))
7312, 72sylbid 230 1 ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝑋 ∈ (𝔼‘𝑁) ∧ 𝑌 ∈ (𝔼‘𝑁))) → (((𝐴OutsideOf⟨𝑋, 𝑅⟩ ∧ ⟨𝐴, 𝑋⟩Cgr⟨𝐵, 𝐶⟩) ∧ (𝐴OutsideOf⟨𝑌, 𝑅⟩ ∧ ⟨𝐴, 𝑌⟩Cgr⟨𝐵, 𝐶⟩)) → 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  cop 4159   class class class wbr 4618  cfv 5852  cn 10971  𝔼cee 25681   Btwn cbtwn 25682  Cgrccgr 25683  OutsideOfcoutsideof 31895
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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  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-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-oi 8366  df-card 8716  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-z 11329  df-uz 11639  df-rp 11784  df-ico 12130  df-icc 12131  df-fz 12276  df-fzo 12414  df-seq 12749  df-exp 12808  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-sum 14358  df-ee 25684  df-btwn 25685  df-cgr 25686  df-ofs 31759  df-colinear 31815  df-ifs 31816  df-cgr3 31817  df-fs 31818  df-outsideof 31896
This theorem is referenced by:  outsideofeu  31907  outsidele  31908
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