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Theorem broutsideof2 35641
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 35640 . 2 (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))
2 btwntriv1 35535 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) β†’ 𝐴 Btwn ⟨𝐴, 𝐡⟩)
323adant3r1 1180 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 Btwn ⟨𝐴, 𝐡⟩)
4 breq1 5145 . . . . . . . 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 2949 . . . . . 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 406 . . . . 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 715 . . . 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 35531 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) β†’ 𝐡 Btwn ⟨𝐴, 𝐡⟩)
1093adant3r1 1180 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 Btwn ⟨𝐴, 𝐡⟩)
11 breq1 5145 . . . . . . . 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 2949 . . . . . 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 406 . . . . 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 715 . . . 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 35578 . . . . . 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 1098 . . . . . . . . . 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 35533 . . . . . . . . . 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 593 . . . . . . . . 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 866 . . . . . . . . 9 (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))
2321, 22syl6bi 253 . . . . . . . 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 867 . . . . . . . . 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 1426 . . . . . 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 418 . . . 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 1126 . . 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 1136 . . . . . 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 1097 . . . . . . . 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 35589 . . . . . . . 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 593 . . . . . . 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 35588 . . . . . . 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 858 . . . . . 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 406 . . . 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 1193 . . . . . . . . . . 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 2940 . . . . . . . . . 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 1216 . . . . . . . . . . . 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 772 . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
45 simpr2 1193 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
46 simpr1 1192 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
47 simpr3 1194 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
48 btwnswapid 35536 . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 698 . . . . . . . . . . 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 456 . . . . . . . . . 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 1352 . . . . . . . 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 1194 . . . . . . . . . . 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 2940 . . . . . . . . . 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 1216 . . . . . . . . . . . 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 772 . . . . . . . . . . . . 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 35534 . . . . . . . . . . . 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 1098 . . . . . . . . . . . . . 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 35536 . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . 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 480 . . . . . . . . . . . 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 698 . . . . . . . . . . 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 456 . . . . . . . . . 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 1352 . . . . . . . 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 858 . . . . . . 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 1347 . . . 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 511 . . 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 800 . 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 283 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 395   ∨ wo 846   ∨ w3o 1084   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2935  βŸ¨cop 4630   class class class wbr 5142  β€˜cfv 6542  β„•cn 12228  π”Όcee 28673   Btwn cbtwn 28674   Colinear ccolin 35556  OutsideOfcoutsideof 35638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-inf2 9650  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9451  df-oi 9519  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-nn 12229  df-2 12291  df-3 12292  df-n0 12489  df-z 12575  df-uz 12839  df-rp 12993  df-ico 13348  df-icc 13349  df-fz 13503  df-fzo 13646  df-seq 13985  df-exp 14045  df-hash 14308  df-cj 15064  df-re 15065  df-im 15066  df-sqrt 15200  df-abs 15201  df-clim 15450  df-sum 15651  df-ee 28676  df-btwn 28677  df-cgr 28678  df-colinear 35558  df-outsideof 35639
This theorem is referenced by:  outsidene1  35642  outsidene2  35643  btwnoutside  35644  broutsideof3  35645  outsideofcom  35647  outsideoftr  35648  outsideofeq  35649  outsideofeu  35650  lineunray  35666
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