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

Theorem outsidele 34763
Description: Relate OutsideOf to Seg≀. Theorem 6.13 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 24-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsidele ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ β†’ (βŸ¨π‘ƒ, 𝐴⟩ Seg≀ βŸ¨π‘ƒ, 𝐡⟩ ↔ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)))

Proof of Theorem outsidele
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpl 484 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
2 simpr1 1195 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
3 simpr2 1196 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
4 simpr3 1197 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
5 brsegle2 34740 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘)) ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (βŸ¨π‘ƒ, 𝐴⟩ Seg≀ βŸ¨π‘ƒ, 𝐡⟩ ↔ βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)(𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩)))
61, 2, 3, 2, 4, 5syl122anc 1380 . . . . 5 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (βŸ¨π‘ƒ, 𝐴⟩ Seg≀ βŸ¨π‘ƒ, 𝐡⟩ ↔ βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)(𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩)))
76adantr 482 . . . 4 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑃OutsideOf⟨𝐴, 𝐡⟩) β†’ (βŸ¨π‘ƒ, 𝐴⟩ Seg≀ βŸ¨π‘ƒ, 𝐡⟩ ↔ βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)(𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩)))
8 simprl 770 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝑃OutsideOf⟨𝐴, 𝐡⟩)
9 outsideofcom 34759 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ 𝑃OutsideOf⟨𝐡, 𝐴⟩))
109ad2antrr 725 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ 𝑃OutsideOf⟨𝐡, 𝐴⟩))
118, 10mpbid 231 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝑃OutsideOf⟨𝐡, 𝐴⟩)
12 simpll 766 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
13 simplr1 1216 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
14 simplr3 1218 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
1512, 13, 14cgrrflxd 34619 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ βŸ¨π‘ƒ, 𝐡⟩CgrβŸ¨π‘ƒ, 𝐡⟩)
1615adantr 482 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ βŸ¨π‘ƒ, 𝐡⟩CgrβŸ¨π‘ƒ, 𝐡⟩)
1711, 16jca 513 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ (𝑃OutsideOf⟨𝐡, 𝐴⟩ ∧ βŸ¨π‘ƒ, 𝐡⟩CgrβŸ¨π‘ƒ, 𝐡⟩))
18 simprrl 780 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ©)
19 simpr 486 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
20 simplr2 1217 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
21 btwncolinear1 34700 . . . . . . . . . . . . 13 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© β†’ 𝑃 Colinear βŸ¨π‘¦, 𝐴⟩))
2212, 13, 19, 20, 21syl13anc 1373 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© β†’ 𝑃 Colinear βŸ¨π‘¦, 𝐴⟩))
2322adantr 482 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© β†’ 𝑃 Colinear βŸ¨π‘¦, 𝐴⟩))
2418, 23mpd 15 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝑃 Colinear βŸ¨π‘¦, 𝐴⟩)
25 outsidene1 34754 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ β†’ 𝐴 β‰  𝑃))
2625ad2antrr 725 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ β†’ 𝐴 β‰  𝑃))
278, 26mpd 15 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝐴 β‰  𝑃)
2827neneqd 2945 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ Β¬ 𝐴 = 𝑃)
29 df-3an 1090 . . . . . . . . . . . . 13 ((𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩) ↔ ((𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩)) ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩))
30 simpr2l 1233 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩)) β†’ 𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ©)
3112, 20, 13, 19, 30btwncomand 34646 . . . . . . . . . . . . . 14 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩)) β†’ 𝐴 Btwn βŸ¨π‘¦, π‘ƒβŸ©)
32 simpr3 1197 . . . . . . . . . . . . . 14 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩)) β†’ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩)
33 btwnswapid2 34649 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Btwn βŸ¨π‘¦, π‘ƒβŸ© ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩) β†’ 𝐴 = 𝑃))
3412, 20, 19, 13, 33syl13anc 1373 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ ((𝐴 Btwn βŸ¨π‘¦, π‘ƒβŸ© ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩) β†’ 𝐴 = 𝑃))
3534adantr 482 . . . . . . . . . . . . . 14 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩)) β†’ ((𝐴 Btwn βŸ¨π‘¦, π‘ƒβŸ© ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩) β†’ 𝐴 = 𝑃))
3631, 32, 35mp2and 698 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩)) β†’ 𝐴 = 𝑃)
3729, 36sylan2br 596 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ ((𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩)) ∧ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩)) β†’ 𝐴 = 𝑃)
3837expr 458 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ (𝑃 Btwn βŸ¨π‘¦, 𝐴⟩ β†’ 𝐴 = 𝑃))
3928, 38mtod 197 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ Β¬ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩)
40 broutsideof 34752 . . . . . . . . . 10 (𝑃OutsideOfβŸ¨π‘¦, 𝐴⟩ ↔ (𝑃 Colinear βŸ¨π‘¦, 𝐴⟩ ∧ Β¬ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩))
4124, 39, 40sylanbrc 584 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝑃OutsideOfβŸ¨π‘¦, 𝐴⟩)
42 simprrr 781 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩)
4341, 42jca 513 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ (𝑃OutsideOfβŸ¨π‘¦, 𝐴⟩ ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))
44 outsideofeq 34761 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ (((𝑃OutsideOf⟨𝐡, 𝐴⟩ ∧ βŸ¨π‘ƒ, 𝐡⟩CgrβŸ¨π‘ƒ, 𝐡⟩) ∧ (𝑃OutsideOfβŸ¨π‘¦, 𝐴⟩ ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩)) β†’ 𝐡 = 𝑦))
4512, 13, 20, 13, 14, 14, 19, 44syl133anc 1394 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ (((𝑃OutsideOf⟨𝐡, 𝐴⟩ ∧ βŸ¨π‘ƒ, 𝐡⟩CgrβŸ¨π‘ƒ, 𝐡⟩) ∧ (𝑃OutsideOfβŸ¨π‘¦, 𝐴⟩ ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩)) β†’ 𝐡 = 𝑦))
4645adantr 482 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ (((𝑃OutsideOf⟨𝐡, 𝐴⟩ ∧ βŸ¨π‘ƒ, 𝐡⟩CgrβŸ¨π‘ƒ, 𝐡⟩) ∧ (𝑃OutsideOfβŸ¨π‘¦, 𝐴⟩ ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩)) β†’ 𝐡 = 𝑦))
4717, 43, 46mp2and 698 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝐡 = 𝑦)
48 opeq2 4832 . . . . . . . . 9 (𝐡 = 𝑦 β†’ βŸ¨π‘ƒ, 𝐡⟩ = βŸ¨π‘ƒ, π‘¦βŸ©)
4948breq2d 5118 . . . . . . . 8 (𝐡 = 𝑦 β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ↔ 𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ©))
5018, 49syl5ibrcom 247 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ (𝐡 = 𝑦 β†’ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩))
5147, 50mpd 15 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)
5251an4s 659 . . . . 5 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑃OutsideOf⟨𝐴, 𝐡⟩) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ (𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩))) β†’ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)
5352rexlimdvaa 3150 . . . 4 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑃OutsideOf⟨𝐴, 𝐡⟩) β†’ (βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)(𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ© ∧ βŸ¨π‘ƒ, π‘¦βŸ©CgrβŸ¨π‘ƒ, 𝐡⟩) β†’ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩))
547, 53sylbid 239 . . 3 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑃OutsideOf⟨𝐴, 𝐡⟩) β†’ (βŸ¨π‘ƒ, 𝐴⟩ Seg≀ βŸ¨π‘ƒ, 𝐡⟩ β†’ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩))
55 btwnsegle 34748 . . . 4 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ βŸ¨π‘ƒ, 𝐴⟩ Seg≀ βŸ¨π‘ƒ, 𝐡⟩))
5655adantr 482 . . 3 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑃OutsideOf⟨𝐴, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ βŸ¨π‘ƒ, 𝐴⟩ Seg≀ βŸ¨π‘ƒ, 𝐡⟩))
5754, 56impbid 211 . 2 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑃OutsideOf⟨𝐴, 𝐡⟩) β†’ (βŸ¨π‘ƒ, 𝐴⟩ Seg≀ βŸ¨π‘ƒ, 𝐡⟩ ↔ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩))
5857ex 414 1 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ β†’ (βŸ¨π‘ƒ, 𝐴⟩ Seg≀ βŸ¨π‘ƒ, 𝐡⟩ ↔ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆƒwrex 3070  βŸ¨cop 4593   class class class wbr 5106  β€˜cfv 6497  β„•cn 12158  π”Όcee 27879   Btwn cbtwn 27880  Cgrccgr 27881   Colinear ccolin 34668   Seg≀ csegle 34737  OutsideOfcoutsideof 34750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-ico 13276  df-icc 13277  df-fz 13431  df-fzo 13574  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-sum 15577  df-ee 27882  df-btwn 27883  df-cgr 27884  df-ofs 34614  df-colinear 34670  df-ifs 34671  df-cgr3 34672  df-fs 34673  df-segle 34738  df-outsideof 34751
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator