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Theorem outsidele 35785
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 481 . . . . . 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 35762 . . . . . 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 479 . . . 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 769 . . . . . . . . . 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 35781 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ 𝑃OutsideOf⟨𝐡, 𝐴⟩))
109ad2antrr 724 . . . . . . . . . 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 765 . . . . . . . . . . 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 35641 . . . . . . . . . 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 479 . . . . . . . . 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 510 . . . . . . . 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 779 . . . . . . . . . . 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 483 . . . . . . . . . . . . 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 35722 . . . . . . . . . . . . 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 479 . . . . . . . . . . 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 35776 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ β†’ 𝐴 β‰  𝑃))
2625ad2antrr 724 . . . . . . . . . . . . 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 2935 . . . . . . . . . . 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 35668 . . . . . . . . . . . . . 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 35671 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . 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 697 . . . . . . . . . . . . 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 593 . . . . . . . . . . . 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 455 . . . . . . . . . . 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 35774 . . . . . . . . . 10 (𝑃OutsideOfβŸ¨π‘¦, 𝐴⟩ ↔ (𝑃 Colinear βŸ¨π‘¦, 𝐴⟩ ∧ Β¬ 𝑃 Btwn βŸ¨π‘¦, 𝐴⟩))
4124, 39, 40sylanbrc 581 . . . . . . . . 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 780 . . . . . . . . 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 510 . . . . . . . 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 35783 . . . . . . . . . 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 479 . . . . . . . 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 697 . . . . . . 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 4870 . . . . . . . . 9 (𝐡 = 𝑦 β†’ βŸ¨π‘ƒ, 𝐡⟩ = βŸ¨π‘ƒ, π‘¦βŸ©)
4948breq2d 5155 . . . . . . . 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 658 . . . . 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 3146 . . . 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 35770 . . . 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 479 . . 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 411 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆƒwrex 3060  βŸ¨cop 4630   class class class wbr 5143  β€˜cfv 6543  β„•cn 12242  π”Όcee 28743   Btwn cbtwn 28744  Cgrccgr 28745   Colinear ccolin 35690   Seg≀ csegle 35759  OutsideOfcoutsideof 35772
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 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216
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 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  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 5144  df-opab 5206  df-mpt 5227  df-tr 5261  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-sup 9465  df-oi 9533  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-n0 12503  df-z 12589  df-uz 12853  df-rp 13007  df-ico 13362  df-icc 13363  df-fz 13517  df-fzo 13660  df-seq 13999  df-exp 14059  df-hash 14322  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-clim 15464  df-sum 15665  df-ee 28746  df-btwn 28747  df-cgr 28748  df-ofs 35636  df-colinear 35692  df-ifs 35693  df-cgr3 35694  df-fs 35695  df-segle 35760  df-outsideof 35773
This theorem is referenced by: (None)
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