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

Theorem lineunray 35107
Description: A line is composed of a point and the two rays emerging from it. Theorem 6.15 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
lineunray ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© β†’ (𝑃Line𝑄) = (((𝑃Ray𝑄) βˆͺ {𝑃}) βˆͺ (𝑃Ray𝑅))))

Proof of Theorem lineunray
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simpl1 1191 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
2 simpr 485 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ π‘₯ ∈ (π”Όβ€˜π‘))
3 simpl21 1251 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
4 simpl22 1252 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝑄 ∈ (π”Όβ€˜π‘))
5 brcolinear 35019 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘))) β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© ↔ (π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ© ∨ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩ ∨ 𝑄 Btwn ⟨π‘₯, π‘ƒβŸ©)))
61, 2, 3, 4, 5syl13anc 1372 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© ↔ (π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ© ∨ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩ ∨ 𝑄 Btwn ⟨π‘₯, π‘ƒβŸ©)))
76adantr 481 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© ↔ (π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ© ∨ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩ ∨ 𝑄 Btwn ⟨π‘₯, π‘ƒβŸ©)))
8 olc 866 . . . . . . . . . . . . . 14 (π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ© β†’ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))
98orcd 871 . . . . . . . . . . . . 13 (π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ© β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))
109a1i 11 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ© β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
11 simpl3l 1228 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝑃 β‰  𝑄)
1211necomd 2996 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝑄 β‰  𝑃)
1312adantr 481 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩)) β†’ 𝑄 β‰  𝑃)
14 simprl 769 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩)) β†’ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©)
15 simprr 771 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩)) β†’ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩)
1613, 14, 153jca 1128 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩)) β†’ (𝑄 β‰  𝑃 ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩))
17 simpl23 1253 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝑅 ∈ (π”Όβ€˜π‘))
18 btwnconn2 35062 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ β„• ∧ (𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘)) ∧ (𝑅 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘))) β†’ ((𝑄 β‰  𝑃 ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩) β†’ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))
191, 4, 3, 17, 2, 18syl122anc 1379 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((𝑄 β‰  𝑃 ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩) β†’ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))
2019adantr 481 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩)) β†’ ((𝑄 β‰  𝑃 ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩) β†’ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))
2116, 20mpd 15 . . . . . . . . . . . . . 14 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩)) β†’ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))
2221olcd 872 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩)) β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))
2322expr 457 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (𝑃 Btwn βŸ¨π‘„, π‘₯⟩ β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
24 btwncom 34974 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„• ∧ (𝑄 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ (𝑄 Btwn ⟨π‘₯, π‘ƒβŸ© ↔ 𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩))
251, 4, 2, 3, 24syl13anc 1372 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝑄 Btwn ⟨π‘₯, π‘ƒβŸ© ↔ 𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩))
26 orc 865 . . . . . . . . . . . . . . 15 (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ β†’ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))
2726orcd 871 . . . . . . . . . . . . . 14 (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))
2825, 27syl6bi 252 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝑄 Btwn ⟨π‘₯, π‘ƒβŸ© β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
2928adantr 481 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (𝑄 Btwn ⟨π‘₯, π‘ƒβŸ© β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
3010, 23, 293jaod 1428 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ ((π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ© ∨ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩ ∨ 𝑄 Btwn ⟨π‘₯, π‘ƒβŸ©) β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
317, 30sylbid 239 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
32 olc 866 . . . . . . . . . 10 (((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ (π‘₯ = 𝑃 ∨ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
3331, 32syl6 35 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© β†’ (π‘₯ = 𝑃 ∨ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))))
34 colineartriv1 35027 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ 𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘)) β†’ 𝑃 Colinear βŸ¨π‘ƒ, π‘„βŸ©)
351, 3, 4, 34syl3anc 1371 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝑃 Colinear βŸ¨π‘ƒ, π‘„βŸ©)
36 breq1 5150 . . . . . . . . . . . 12 (π‘₯ = 𝑃 β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© ↔ 𝑃 Colinear βŸ¨π‘ƒ, π‘„βŸ©))
3735, 36syl5ibrcom 246 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (π‘₯ = 𝑃 β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
3837adantr 481 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (π‘₯ = 𝑃 β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
39 btwncolinear3 35031 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘))) β†’ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
401, 3, 2, 4, 39syl13anc 1372 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
41 btwncolinear5 35033 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘))) β†’ (π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ© β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
421, 3, 4, 2, 41syl13anc 1372 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ© β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
4340, 42jaod 857 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
4443adantr 481 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
45 simpl3r 1229 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝑃 β‰  𝑅)
4645adantr 481 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩)) β†’ 𝑃 β‰  𝑅)
47 simprl 769 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩)) β†’ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©)
48 simprr 771 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩)) β†’ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩)
4946, 47, 483jca 1128 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩)) β†’ (𝑃 β‰  𝑅 ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩))
50 btwnouttr 34984 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ β„• ∧ (𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘)) ∧ (𝑅 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘))) β†’ ((𝑃 β‰  𝑅 ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩) β†’ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩))
511, 4, 3, 17, 2, 50syl122anc 1379 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((𝑃 β‰  𝑅 ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩) β†’ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩))
5251adantr 481 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩)) β†’ ((𝑃 β‰  𝑅 ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩) β†’ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩))
5349, 52mpd 15 . . . . . . . . . . . . . 14 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩)) β†’ 𝑃 Btwn βŸ¨π‘„, π‘₯⟩)
54 btwncolinear4 35032 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ β„• ∧ (𝑄 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ (𝑃 Btwn βŸ¨π‘„, π‘₯⟩ β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
551, 4, 2, 3, 54syl13anc 1372 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝑃 Btwn βŸ¨π‘„, π‘₯⟩ β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
5655adantr 481 . . . . . . . . . . . . . 14 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩)) β†’ (𝑃 Btwn βŸ¨π‘„, π‘₯⟩ β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
5753, 56mpd 15 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ 𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩)) β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©)
5857expr 457 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
59 simprr 771 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)
601, 2, 3, 17, 59btwncomand 34975 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ π‘₯ Btwn βŸ¨π‘…, π‘ƒβŸ©)
61 simprl 769 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©)
621, 3, 4, 17, 61btwncomand 34975 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ 𝑃 Btwn βŸ¨π‘…, π‘„βŸ©)
631, 17, 2, 3, 4, 60, 62btwnexch3and 34981 . . . . . . . . . . . . . 14 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ 𝑃 Btwn ⟨π‘₯, π‘„βŸ©)
64 btwncolinear2 35030 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ β„• ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ (𝑃 Btwn ⟨π‘₯, π‘„βŸ© β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
651, 2, 4, 3, 64syl13anc 1372 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝑃 Btwn ⟨π‘₯, π‘„βŸ© β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
6665adantr 481 . . . . . . . . . . . . . 14 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ (𝑃 Btwn ⟨π‘₯, π‘„βŸ© β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
6763, 66mpd 15 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© ∧ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©)
6867expr 457 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ© β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
6958, 68jaod 857 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ ((𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©) β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
7044, 69jaod 857 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
7138, 70jaod 857 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ ((π‘₯ = 𝑃 ∨ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))) β†’ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©))
7233, 71impbid 211 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© ↔ (π‘₯ = 𝑃 ∨ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))))
73 pm5.63 1018 . . . . . . . . 9 ((π‘₯ = 𝑃 ∨ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))) ↔ (π‘₯ = 𝑃 ∨ (Β¬ π‘₯ = 𝑃 ∧ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))))
74 df-ne 2941 . . . . . . . . . . . 12 (π‘₯ β‰  𝑃 ↔ Β¬ π‘₯ = 𝑃)
7574anbi1i 624 . . . . . . . . . . 11 ((π‘₯ β‰  𝑃 ∧ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))) ↔ (Β¬ π‘₯ = 𝑃 ∧ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
76 andi 1006 . . . . . . . . . . 11 ((π‘₯ β‰  𝑃 ∧ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))) ↔ ((π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) ∨ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
7775, 76bitr3i 276 . . . . . . . . . 10 ((Β¬ π‘₯ = 𝑃 ∧ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))) ↔ ((π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) ∨ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
7877orbi2i 911 . . . . . . . . 9 ((π‘₯ = 𝑃 ∨ (Β¬ π‘₯ = 𝑃 ∧ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))) ↔ (π‘₯ = 𝑃 ∨ ((π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) ∨ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))))
7973, 78bitri 274 . . . . . . . 8 ((π‘₯ = 𝑃 ∨ ((𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©) ∨ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))) ↔ (π‘₯ = 𝑃 ∨ ((π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) ∨ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))))
8072, 79bitrdi 286 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© ↔ (π‘₯ = 𝑃 ∨ ((π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) ∨ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))))
81 broutsideof2 35082 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ↔ (𝑄 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))))
821, 3, 4, 2, 81syl13anc 1372 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ↔ (𝑄 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))))
83 3simpc 1150 . . . . . . . . . . . 12 ((𝑄 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) β†’ (π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)))
84 simpl3l 1228 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ (π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)))) β†’ 𝑃 β‰  𝑄)
8584necomd 2996 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ (π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)))) β†’ 𝑄 β‰  𝑃)
86 simprrl 779 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ (π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)))) β†’ π‘₯ β‰  𝑃)
87 simprrr 780 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ (π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)))) β†’ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))
8885, 86, 873jca 1128 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ (π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)))) β†’ (𝑄 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)))
8988expr 457 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) β†’ (𝑄 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))))
9083, 89impbid2 225 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((𝑄 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) ↔ (π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))))
9182, 90bitrd 278 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ↔ (π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))))
92 broutsideof2 35082 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOfβŸ¨π‘…, π‘₯⟩ ↔ (𝑅 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
931, 3, 17, 2, 92syl13anc 1372 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝑃OutsideOfβŸ¨π‘…, π‘₯⟩ ↔ (𝑅 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
94 3simpc 1150 . . . . . . . . . . . 12 ((𝑅 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))
95 simpl3r 1229 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))) β†’ 𝑃 β‰  𝑅)
9695necomd 2996 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))) β†’ 𝑅 β‰  𝑃)
97 simprrl 779 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))) β†’ π‘₯ β‰  𝑃)
98 simprrr 780 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))) β†’ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))
9996, 97, 983jca 1128 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))) β†’ (𝑅 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))
10099expr 457 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) β†’ (𝑅 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
10194, 100impbid2 225 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((𝑅 β‰  𝑃 ∧ π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)) ↔ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
10293, 101bitrd 278 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝑃OutsideOfβŸ¨π‘…, π‘₯⟩ ↔ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))
10391, 102orbi12d 917 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩) ↔ ((π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) ∨ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))))
104103adantr 481 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩) ↔ ((π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) ∨ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©)))))
105104orbi2d 914 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ ((π‘₯ = 𝑃 ∨ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)) ↔ (π‘₯ = 𝑃 ∨ ((π‘₯ β‰  𝑃 ∧ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©)) ∨ (π‘₯ β‰  𝑃 ∧ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))))))
10680, 105bitr4d 281 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© ↔ (π‘₯ = 𝑃 ∨ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩))))
107 orcom 868 . . . . . . 7 ((π‘₯ = 𝑃 ∨ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)) ↔ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩) ∨ π‘₯ = 𝑃))
108 or32 924 . . . . . . 7 (((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩) ∨ π‘₯ = 𝑃) ↔ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃) ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩))
109107, 108bitri 274 . . . . . 6 ((π‘₯ = 𝑃 ∨ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)) ↔ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃) ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩))
110106, 109bitrdi 286 . . . . 5 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© ↔ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃) ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)))
111110an32s 650 . . . 4 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ© ↔ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃) ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)))
112111rabbidva 3439 . . 3 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©} = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃) ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)})
113 simp1 1136 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ 𝑁 ∈ β„•)
114 simp21 1206 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
115 simp22 1207 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ 𝑄 ∈ (π”Όβ€˜π‘))
116 simp3l 1201 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ 𝑃 β‰  𝑄)
117 fvline2 35106 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝑄)) β†’ (𝑃Line𝑄) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©})
118113, 114, 115, 116, 117syl13anc 1372 . . . 4 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ (𝑃Line𝑄) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©})
119118adantr 481 . . 3 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (𝑃Line𝑄) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ Colinear βŸ¨π‘ƒ, π‘„βŸ©})
120 fvray 35101 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝑄)) β†’ (𝑃Ray𝑄) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘„, π‘₯⟩})
121113, 114, 115, 116, 120syl13anc 1372 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ (𝑃Ray𝑄) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘„, π‘₯⟩})
122 rabsn 4724 . . . . . . . . 9 (𝑃 ∈ (π”Όβ€˜π‘) β†’ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ = 𝑃} = {𝑃})
123114, 122syl 17 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ = 𝑃} = {𝑃})
124123eqcomd 2738 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ {𝑃} = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ = 𝑃})
125121, 124uneq12d 4163 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ ((𝑃Ray𝑄) βˆͺ {𝑃}) = ({π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘„, π‘₯⟩} βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ = 𝑃}))
126 simp23 1208 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ 𝑅 ∈ (π”Όβ€˜π‘))
127 simp3r 1202 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ 𝑃 β‰  𝑅)
128 fvray 35101 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘) ∧ 𝑃 β‰  𝑅)) β†’ (𝑃Ray𝑅) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩})
129113, 114, 126, 127, 128syl13anc 1372 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ (𝑃Ray𝑅) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩})
130125, 129uneq12d 4163 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ (((𝑃Ray𝑄) βˆͺ {𝑃}) βˆͺ (𝑃Ray𝑅)) = (({π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘„, π‘₯⟩} βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ = 𝑃}) βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩}))
131130adantr 481 . . . 4 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (((𝑃Ray𝑄) βˆͺ {𝑃}) βˆͺ (𝑃Ray𝑅)) = (({π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘„, π‘₯⟩} βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ = 𝑃}) βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩}))
132 unrab 4304 . . . . . 6 ({π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘„, π‘₯⟩} βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ = 𝑃}) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃)}
133132uneq1i 4158 . . . . 5 (({π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘„, π‘₯⟩} βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ = 𝑃}) βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩}) = ({π‘₯ ∈ (π”Όβ€˜π‘) ∣ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃)} βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩})
134 unrab 4304 . . . . 5 ({π‘₯ ∈ (π”Όβ€˜π‘) ∣ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃)} βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩}) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃) ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)}
135133, 134eqtri 2760 . . . 4 (({π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘„, π‘₯⟩} βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ π‘₯ = 𝑃}) βˆͺ {π‘₯ ∈ (π”Όβ€˜π‘) ∣ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩}) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃) ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)}
136131, 135eqtrdi 2788 . . 3 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (((𝑃Ray𝑄) βˆͺ {𝑃}) βˆͺ (𝑃Ray𝑅)) = {π‘₯ ∈ (π”Όβ€˜π‘) ∣ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃) ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)})
137112, 119, 1363eqtr4d 2782 . 2 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) ∧ 𝑃 Btwn βŸ¨π‘„, π‘…βŸ©) β†’ (𝑃Line𝑄) = (((𝑃Ray𝑄) βˆͺ {𝑃}) βˆͺ (𝑃Ray𝑅)))
138137ex 413 1 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 β‰  𝑄 ∧ 𝑃 β‰  𝑅)) β†’ (𝑃 Btwn βŸ¨π‘„, π‘…βŸ© β†’ (𝑃Line𝑄) = (((𝑃Ray𝑄) βˆͺ {𝑃}) βˆͺ (𝑃Ray𝑅))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432   βˆͺ cun 3945  {csn 4627  βŸ¨cop 4633   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  β„•cn 12208  π”Όcee 28135   Btwn cbtwn 28136   Colinear ccolin 34997  OutsideOfcoutsideof 35079  Linecline2 35094  Raycray 35095
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-ec 8701  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629  df-ee 28138  df-btwn 28139  df-cgr 28140  df-ofs 34943  df-colinear 34999  df-ifs 35000  df-cgr3 35001  df-fs 35002  df-outsideof 35080  df-line2 35097  df-ray 35098
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator