Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  broutsideof2 Structured version   Visualization version   GIF version
Theorem broutsideof2 35771
Description: Alternate form of OutsideOf. Definition 6.1 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 17-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
broutsideof2 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
Proof of Theorem broutsideof2
StepHypRef Expression
1 broutsideof 35770 . 2 (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))
2 btwntriv1 35665 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) β†’ 𝐴 Btwn ⟨𝐴, 𝐡⟩)
323adant3r1 1179 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 Btwn ⟨𝐴, 𝐡⟩)
4 breq1 5147 . . . . . . . 8 (𝐴 = 𝑃 β†’ (𝐴 Btwn ⟨𝐴, 𝐡⟩ ↔ 𝑃 Btwn ⟨𝐴, 𝐡⟩))
53, 4syl5ibcom 244 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 = 𝑃 β†’ 𝑃 Btwn ⟨𝐴, 𝐡⟩))
65necon3bd 2944 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ 𝐴 β‰  𝑃))
76imp 405 . . . . 5 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩) β†’ 𝐴 β‰  𝑃)
87adantrl 714 . . . 4 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ 𝐴 β‰  𝑃)
9 btwntriv2 35661 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) β†’ 𝐡 Btwn ⟨𝐴, 𝐡⟩)
1093adant3r1 1179 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 Btwn ⟨𝐴, 𝐡⟩)
11 breq1 5147 . . . . . . . 8 (𝐡 = 𝑃 β†’ (𝐡 Btwn ⟨𝐴, 𝐡⟩ ↔ 𝑃 Btwn ⟨𝐴, 𝐡⟩))
1210, 11syl5ibcom 244 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐡 = 𝑃 β†’ 𝑃 Btwn ⟨𝐴, 𝐡⟩))
1312necon3bd 2944 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ 𝐡 β‰  𝑃))
1413imp 405 . . . . 5 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩) β†’ 𝐡 β‰  𝑃)
1514adantrl 714 . . . 4 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ 𝐡 β‰  𝑃)
16 brcolinear 35708 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ↔ (𝑃 Btwn ⟨𝐴, 𝐡⟩ ∨ 𝐴 Btwn ⟨𝐡, π‘ƒβŸ© ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
17 pm2.24 124 . . . . . . . 8 (𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ (Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
1817a1i 11 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ (Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
19 3anrot 1097 . . . . . . . . . 10 ((𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ↔ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘)))
20 btwncom 35663 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Btwn ⟨𝐡, π‘ƒβŸ© ↔ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩))
2119, 20sylan2b 592 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Btwn ⟨𝐡, π‘ƒβŸ© ↔ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩))
22 orc 865 . . . . . . . . 9 (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))
2321, 22biimtrdi 252 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Btwn ⟨𝐡, π‘ƒβŸ© β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
2423a1dd 50 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Btwn ⟨𝐡, π‘ƒβŸ© β†’ (Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
25 olc 866 . . . . . . . . 9 (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))
2625a1d 25 . . . . . . . 8 (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ β†’ (Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
2726a1i 11 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ β†’ (Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
2818, 24, 273jaod 1425 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝑃 Btwn ⟨𝐴, 𝐡⟩ ∨ 𝐴 Btwn ⟨𝐡, π‘ƒβŸ© ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) β†’ (Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
2916, 28sylbid 239 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃 Colinear ⟨𝐴, 𝐡⟩ β†’ (Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
3029imp32 417 . . . 4 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))
318, 15, 303jca 1125 . . 3 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
32 simp3 1135 . . . . . 6 ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))
33 3ancomb 1096 . . . . . . . 8 ((𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ↔ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘)))
34 btwncolinear2 35719 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ 𝑃 Colinear ⟨𝐴, 𝐡⟩))
3533, 34sylan2b 592 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ 𝑃 Colinear ⟨𝐴, 𝐡⟩))
36 btwncolinear1 35718 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ β†’ 𝑃 Colinear ⟨𝐴, 𝐡⟩))
3735, 36jaod 857 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) β†’ 𝑃 Colinear ⟨𝐴, 𝐡⟩))
3832, 37syl5 34 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) β†’ 𝑃 Colinear ⟨𝐴, 𝐡⟩))
3938imp 405 . . . 4 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))) β†’ 𝑃 Colinear ⟨𝐴, 𝐡⟩)
40 simpr2 1192 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ 𝐴 β‰  𝑃)
4140neneqd 2935 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ Β¬ 𝐴 = 𝑃)
42 simprl1 1215 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)
43 simprr 771 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ 𝑃 Btwn ⟨𝐴, 𝐡⟩)
44 simpl 481 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
45 simpr2 1192 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
46 simpr1 1191 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
47 simpr3 1193 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
48 btwnswapid 35666 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩) β†’ 𝐴 = 𝑃))
4944, 45, 46, 47, 48syl13anc 1369 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩) β†’ 𝐴 = 𝑃))
5049adantr 479 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩) β†’ 𝐴 = 𝑃))
5142, 43, 50mp2and 697 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ 𝐴 = 𝑃)
5251expr 455 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ (𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ 𝐴 = 𝑃))
5341, 52mtod 197 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩)
54533exp2 1351 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ (𝐴 β‰  𝑃 β†’ (𝐡 β‰  𝑃 β†’ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))))
55 simpr3 1193 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ 𝐡 β‰  𝑃)
5655neneqd 2935 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ Β¬ 𝐡 = 𝑃)
57 simprl1 1215 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)
58 simprr 771 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ 𝑃 Btwn ⟨𝐴, 𝐡⟩)
5944, 46, 45, 47, 58btwncomand 35664 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ 𝑃 Btwn ⟨𝐡, 𝐴⟩)
60 3anrot 1097 . . . . . . . . . . . . . 14 ((𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘)) ↔ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)))
61 btwnswapid 35666 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘))) β†’ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝑃 Btwn ⟨𝐡, 𝐴⟩) β†’ 𝐡 = 𝑃))
6260, 61sylan2br 593 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝑃 Btwn ⟨𝐡, 𝐴⟩) β†’ 𝐡 = 𝑃))
6362adantr 479 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝑃 Btwn ⟨𝐡, 𝐴⟩) β†’ 𝐡 = 𝑃))
6457, 59, 63mp2and 697 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝑃 Btwn ⟨𝐴, 𝐡⟩)) β†’ 𝐡 = 𝑃)
6564expr 455 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ (𝑃 Btwn ⟨𝐴, 𝐡⟩ β†’ 𝐡 = 𝑃))
6656, 65mtod 197 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ ∧ 𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩)
67663exp2 1351 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ β†’ (𝐴 β‰  𝑃 β†’ (𝐡 β‰  𝑃 β†’ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))))
6854, 67jaod 857 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) β†’ (𝐴 β‰  𝑃 β†’ (𝐡 β‰  𝑃 β†’ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))))
6968com12 32 . . . . . 6 ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) β†’ ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 β‰  𝑃 β†’ (𝐡 β‰  𝑃 β†’ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))))
7069com4l 92 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 β‰  𝑃 β†’ (𝐡 β‰  𝑃 β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) β†’ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))))
71703imp2 1346 . . . 4 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))) β†’ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩)
7239, 71jca 510 . . 3 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))) β†’ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))
7331, 72impbida 799 . 2 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩) ↔ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
741, 73bitrid 282 1 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∨ w3o 1083   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βŸ¨cop 4631   class class class wbr 5144  β€˜cfv 6543  β„•cn 12237  π”Όcee 28738   Btwn cbtwn 28739   Colinear ccolin 35686  OutsideOfcoutsideof 35768
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 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-inf2 9659  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9460  df-oi 9528  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-nn 12238  df-2 12300  df-3 12301  df-n0 12498  df-z 12584  df-uz 12848  df-rp 13002  df-ico 13357  df-icc 13358  df-fz 13512  df-fzo 13655  df-seq 13994  df-exp 14054  df-hash 14317  df-cj 15073  df-re 15074  df-im 15075  df-sqrt 15209  df-abs 15210  df-clim 15459  df-sum 15660  df-ee 28741  df-btwn 28742  df-cgr 28743  df-colinear 35688  df-outsideof 35769
This theorem is referenced by:  outsidene1  35772  outsidene2  35773  btwnoutside  35774  broutsideof3  35775  outsideofcom  35777  outsideoftr  35778  outsideofeq  35779  outsideofeu  35780  lineunray  35796
  Copyright terms: Public domain W3C validator