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Theorem outsidele 35637
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 482 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
2 simpr1 1191 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
3 simpr2 1192 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
4 simpr3 1193 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
5 brsegle2 35614 . . . . . 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 1376 . . . . 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 480 . . . 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 768 . . . . . . . . . 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 35633 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ 𝑃OutsideOf⟨𝐡, 𝐴⟩))
109ad2antrr 723 . . . . . . . . . 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 764 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
13 simplr1 1212 . . . . . . . . . . 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 1214 . . . . . . . . . . 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 35493 . . . . . . . . . 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 480 . . . . . . . . 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 511 . . . . . . . 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 778 . . . . . . . . . . 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 484 . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . 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 35574 . . . . . . . . . . . . 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 1369 . . . . . . . . . . . 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 480 . . . . . . . . . . 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 35628 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ β†’ 𝐴 β‰  𝑃))
2625ad2antrr 723 . . . . . . . . . . . . 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 2939 . . . . . . . . . . 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 1086 . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . 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 35520 . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . 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 35523 . . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . 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 594 . . . . . . . . . . . 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 456 . . . . . . . . . . 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 35626 . . . . . . . . . 10 (𝑃OutsideOfβŸ¨π‘¦, 𝐴⟩ ↔ (𝑃 Colinear βŸ¨π‘¦, 𝐴⟩ ∧ Β¬ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩))
4124, 39, 40sylanbrc 582 . . . . . . . . 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 779 . . . . . . . . 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 511 . . . . . . . 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 35635 . . . . . . . . . 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 1390 . . . . . . . . 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 480 . . . . . . . 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 696 . . . . . . 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 4869 . . . . . . . . 9 (𝐡 = 𝑦 β†’ βŸ¨π‘ƒ, 𝐡⟩ = βŸ¨π‘ƒ, π‘¦βŸ©)
4948breq2d 5153 . . . . . . . 8 (𝐡 = 𝑦 β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ↔ 𝐴 Btwn βŸ¨π‘ƒ, π‘¦βŸ©))
5018, 49syl5ibrcom 246 . . . . . . 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 657 . . . . 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 35622 . . . 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 480 . . 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 412 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064  βŸ¨cop 4629   class class class wbr 5141  β€˜cfv 6537  β„•cn 12216  π”Όcee 28654   Btwn cbtwn 28655  Cgrccgr 28656   Colinear ccolin 35542   Seg≀ csegle 35611  OutsideOfcoutsideof 35624
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12981  df-ico 13336  df-icc 13337  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14033  df-hash 14296  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-clim 15438  df-sum 15639  df-ee 28657  df-btwn 28658  df-cgr 28659  df-ofs 35488  df-colinear 35544  df-ifs 35545  df-cgr3 35546  df-fs 35547  df-segle 35612  df-outsideof 35625
This theorem is referenced by: (None)
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