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Theorem broutsideof2 35082
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 35081 . 2 (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝑃 Colinear ⟨𝐴, 𝐡⟩ ∧ Β¬ 𝑃 Btwn ⟨𝐴, 𝐡⟩))
2 btwntriv1 34976 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) β†’ 𝐴 Btwn ⟨𝐴, 𝐡⟩)
323adant3r1 1182 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 Btwn ⟨𝐴, 𝐡⟩)
4 breq1 5150 . . . . . . . 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 2954 . . . . . 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 407 . . . . 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 34972 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) β†’ 𝐡 Btwn ⟨𝐴, 𝐡⟩)
1093adant3r1 1182 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 Btwn ⟨𝐴, 𝐡⟩)
11 breq1 5150 . . . . . . . 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 2954 . . . . . 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 407 . . . . 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 35019 . . . . . 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 1100 . . . . . . . . . 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 34974 . . . . . . . . . 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 594 . . . . . . . . 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, 22syl6bi 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 1428 . . . . . 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 419 . . . 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 1128 . . 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 1138 . . . . . 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 1099 . . . . . . . 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 35030 . . . . . . . 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 594 . . . . . . 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 35029 . . . . . . 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 407 . . . 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 1195 . . . . . . . . . . 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 2945 . . . . . . . . . 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 1218 . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
45 simpr2 1195 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
46 simpr1 1194 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
47 simpr3 1196 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
48 btwnswapid 34977 . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 457 . . . . . . . . . 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 1354 . . . . . . . 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 1196 . . . . . . . . . . 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 2945 . . . . . . . . . 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 1218 . . . . . . . . . . . 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 34975 . . . . . . . . . . . 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 1100 . . . . . . . . . . . . . 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 34977 . . . . . . . . . . . . . 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 595 . . . . . . . . . . . . 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 481 . . . . . . . . . . . 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 457 . . . . . . . . . 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 1354 . . . . . . . 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 1349 . . . 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 512 . . 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 396   ∨ wo 845   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βŸ¨cop 4633   class class class wbr 5147  β€˜cfv 6540  β„•cn 12208  π”Όcee 28135   Btwn cbtwn 28136   Colinear ccolin 34997  OutsideOfcoutsideof 35079
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-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-colinear 34999  df-outsideof 35080
This theorem is referenced by:  outsidene1  35083  outsidene2  35084  btwnoutside  35085  broutsideof3  35086  outsideofcom  35088  outsideoftr  35089  outsideofeq  35090  outsideofeu  35091  lineunray  35107
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