Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  outsideofeq Structured version   Visualization version   GIF version

Theorem outsideofeq 32362
Description: Uniqueness law for OutsideOf. Analogue of segconeq 32242. (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 1081 . . . . 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 1114 . . . . 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 1118 . . . . 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 1115 . . . . 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 32354 . . . . 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 1368 . . . 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 741 . . 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 1119 . . . . 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 32354 . . . . 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 1368 . . . 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 741 . . 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 747 . 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 1122 . . . . . . 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 1128 . . . . . . 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 553 . . . . . 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 481 . . . . 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 1121 . . . . . 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 481 . . . . 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 1116 . . . . . 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 1117 . . . . . 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 820 . . . . . 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 822 . . . . . 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 32227 . . . . 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 819 . . . . . . . 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 820 . . . . . . . 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 822 . . . . . . . 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 32336 . . . . . . . . . 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 1375 . . . . . . . . 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 480 . . . . . . . 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 1467 . . . . . . 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 820 . . . . . . . . 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 819 . . . . . . . . 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 32258 . . . . . . . 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 822 . . . . . . . 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 32318 . . . . . . 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 819 . . . . . . . . . 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 820 . . . . . . . . . 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 32258 . . . . . . . . 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 822 . . . . . . . . . 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 32223 . . . . . . . . 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 32318 . . . . . . . 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 2657 . . . . . . 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 811 . . . . . . . . . . 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 818 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 32223 . . . . . . . . . . 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 32318 . . . . . . . . . 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 2657 . . . . . . . . 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 811 . . . . . . . . . 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 818 . . . . . . . . . . 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 481 . . . . . . . . . 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 32318 . . . . . . . . 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 821 . . . . . . . . . 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 2878 . . . . . . . . 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 819 . . . . . . . . 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 820 . . . . . . . . 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 32333 . . . . . . . . . . 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 1375 . . . . . . . . . 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 480 . . . . . . . . 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 1467 . . . . . . . 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 395 . . . . . . 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 1007 . . . . 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 448 . . . 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 486 . . 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 449 . 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 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  cop 4216   class class class wbr 4685  cfv 5926  cn 11058  𝔼cee 25813   Btwn cbtwn 25814  Cgrccgr 25815  OutsideOfcoutsideof 32351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-ee 25816  df-btwn 25817  df-cgr 25818  df-ofs 32215  df-colinear 32271  df-ifs 32272  df-cgr3 32273  df-fs 32274  df-outsideof 32352
This theorem is referenced by:  outsideofeu  32363  outsidele  32364
  Copyright terms: Public domain W3C validator