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Theorem lineunray 35580
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 1188 . . . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 1248 . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . 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 35492 . . . . . . . . . . . . 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 1369 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 865 . . . . . . . . . . . . . 14 (π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ© β†’ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))
98orcd 870 . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . . . . . 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 2988 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . 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 1250 . . . . . . . . . . . . . . . . 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 35535 . . . . . . . . . . . . . . . . 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 1376 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 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 871 . . . . . . . . . . . . 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 456 . . . . . . . . . . . 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 35447 . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . 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 864 . . . . . . . . . . . . . . 15 (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ β†’ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))
2726orcd 870 . . . . . . . . . . . . . 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 253 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 1425 . . . . . . . . . . 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 865 . . . . . . . . . 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 35500 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ 𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑄 ∈ (π”Όβ€˜π‘)) β†’ 𝑃 Colinear βŸ¨π‘ƒ, π‘„βŸ©)
351, 3, 4, 34syl3anc 1368 . . . . . . . . . . . 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 5141 . . . . . . . . . . . 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 480 . . . . . . . . . 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 35504 . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . 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 35506 . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . 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 856 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 1226 . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . 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 1125 . . . . . . . . . . . . . . 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 35457 . . . . . . . . . . . . . . . . 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 1376 . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 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 35505 . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 456 . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . 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 35448 . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 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 35448 . . . . . . . . . . . . . . 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 35454 . . . . . . . . . . . . . 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 35503 . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 456 . . . . . . . . . . . 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 856 . . . . . . . . . . 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 856 . . . . . . . . . 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 856 . . . . . . . . 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 1016 . . . . . . . . 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 2933 . . . . . . . . . . . 12 (π‘₯ β‰  𝑃 ↔ Β¬ π‘₯ = 𝑃)
7574anbi1i 623 . . . . . . . . . . 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 1004 . . . . . . . . . . 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 277 . . . . . . . . . 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 909 . . . . . . . . 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 275 . . . . . . . 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 287 . . . . . . 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 35555 . . . . . . . . . . . 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 1369 . . . . . . . . . . 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 1147 . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . . 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 2988 . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . 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 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†’ (𝑄 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘„βŸ©))
8885, 86, 873jca 1125 . . . . . . . . . . . . 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 456 . . . . . . . . . . . 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 279 . . . . . . . . . 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 35555 . . . . . . . . . . . 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 1369 . . . . . . . . . . 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 1147 . . . . . . . . . . . 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 1226 . . . . . . . . . . . . . . 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 2988 . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . 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 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†’ (𝑅 Btwn βŸ¨π‘ƒ, π‘₯⟩ ∨ π‘₯ Btwn βŸ¨π‘ƒ, π‘…βŸ©))
9996, 97, 983jca 1125 . . . . . . . . . . . . 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 456 . . . . . . . . . . . 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 279 . . . . . . . . . 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 915 . . . . . . . . 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 480 . . . . . . . 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 912 . . . . . . 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 282 . . . . . 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 867 . . . . . . 7 ((π‘₯ = 𝑃 ∨ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)) ↔ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩) ∨ π‘₯ = 𝑃))
108 or32 922 . . . . . . 7 (((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩) ∨ π‘₯ = 𝑃) ↔ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃) ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩))
109107, 108bitri 275 . . . . . 6 ((π‘₯ = 𝑃 ∨ (𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩)) ↔ ((𝑃OutsideOfβŸ¨π‘„, π‘₯⟩ ∨ π‘₯ = 𝑃) ∨ 𝑃OutsideOfβŸ¨π‘…, π‘₯⟩))
110106, 109bitrdi 287 . . . . 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 649 . . . 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 3431 . . 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 1133 . . . . 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 1203 . . . . 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 1204 . . . . 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 1198 . . . . 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 35579 . . . . 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 1369 . . . 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 480 . . 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 35574 . . . . . . . 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 1369 . . . . . . 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 4717 . . . . . . . . 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 2730 . . . . . . 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 4156 . . . . . 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 1205 . . . . . . 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 1199 . . . . . . 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 35574 . . . . . . 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 1369 . . . . . 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 4156 . . . . 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 480 . . . 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 4297 . . . . . 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 4151 . . . . 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 4297 . . . . 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 2752 . . . 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 2780 . . 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 2774 . 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 412 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 395   ∨ wo 844   ∨ w3o 1083   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2932  {crab 3424   βˆͺ cun 3938  {csn 4620  βŸ¨cop 4626   class class class wbr 5138  β€˜cfv 6533  (class class class)co 7401  β„•cn 12208  π”Όcee 28581   Btwn cbtwn 28582   Colinear ccolin 35470  OutsideOfcoutsideof 35552  Linecline2 35567  Raycray 35568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9631  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8698  df-ec 8700  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-sup 9432  df-oi 9500  df-card 9929  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 28584  df-btwn 28585  df-cgr 28586  df-ofs 35416  df-colinear 35472  df-ifs 35473  df-cgr3 35474  df-fs 35475  df-outsideof 35553  df-line2 35570  df-ray 35571
This theorem is referenced by: (None)
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