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Theorem trisegint 35681
Description: A line segment between two sides of a triange intersects a segment crossing from the remaining side to the opposite vertex. Theorem 3.17 of [Schwabhauser] p. 33. (Contributed by Scott Fenton, 24-Sep-2013.)
Assertion
Ref Expression
trisegint ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩) β†’ βˆƒπ‘ž ∈ (π”Όβ€˜π‘)(π‘ž Btwn βŸ¨π‘ƒ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩)))
Distinct variable groups:   𝐴,π‘ž   𝐡,π‘ž   𝐢,π‘ž   𝐷,π‘ž   𝐸,π‘ž   𝑁,π‘ž   𝑃,π‘ž

Proof of Theorem trisegint
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 simpl1 1188 . . . . 5 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ 𝑁 ∈ β„•)
2 simpl23 1250 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ 𝐢 ∈ (π”Όβ€˜π‘))
3 simpl21 1248 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
4 simpl31 1251 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ 𝐷 ∈ (π”Όβ€˜π‘))
52, 3, 43jca 1125 . . . . 5 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)))
6 simpl32 1252 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ 𝐸 ∈ (π”Όβ€˜π‘))
7 simpl33 1253 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
86, 7jca 510 . . . . 5 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘)))
91, 5, 83jca 1125 . . . 4 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ (𝑁 ∈ β„• ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))))
10 simpr2 1192 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ 𝐸 Btwn ⟨𝐷, 𝐢⟩)
11 btwncom 35667 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ (𝐸 Btwn ⟨𝐷, 𝐢⟩ ↔ 𝐸 Btwn ⟨𝐢, 𝐷⟩))
121, 6, 4, 2, 11syl13anc 1369 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ (𝐸 Btwn ⟨𝐷, 𝐢⟩ ↔ 𝐸 Btwn ⟨𝐢, 𝐷⟩))
1310, 12mpbid 231 . . . . 5 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ 𝐸 Btwn ⟨𝐢, 𝐷⟩)
14 simpr3 1193 . . . . 5 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ 𝑃 Btwn ⟨𝐴, 𝐷⟩)
1513, 14jca 510 . . . 4 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ (𝐸 Btwn ⟨𝐢, 𝐷⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩))
16 axpasch 28796 . . . 4 ((𝑁 ∈ β„• ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝐸 Btwn ⟨𝐢, 𝐷⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩) β†’ βˆƒπ‘Ÿ ∈ (π”Όβ€˜π‘)(π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)))
179, 15, 16sylc 65 . . 3 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ βˆƒπ‘Ÿ ∈ (π”Όβ€˜π‘)(π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩))
18 simp1l1 1263 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝑁 ∈ β„•)
1963ad2ant1 1130 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐸 ∈ (π”Όβ€˜π‘))
2023ad2ant1 1130 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐢 ∈ (π”Όβ€˜π‘))
2133ad2ant1 1130 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
2219, 20, 213jca 1125 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘)))
23 simp2 1134 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ π‘Ÿ ∈ (π”Όβ€˜π‘))
24 simpl22 1249 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
25243ad2ant1 1130 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
2623, 25jca 510 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ (π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)))
2718, 22, 263jca 1125 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ (𝑁 ∈ β„• ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘)) ∧ (π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))))
28 simp3l 1198 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ π‘Ÿ Btwn ⟨𝐸, 𝐴⟩)
29 simp1r1 1266 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐡 Btwn ⟨𝐴, 𝐢⟩)
30 btwncom 35667 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ↔ 𝐡 Btwn ⟨𝐢, 𝐴⟩))
3118, 25, 21, 20, 30syl13anc 1369 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ↔ 𝐡 Btwn ⟨𝐢, 𝐴⟩))
3229, 31mpbid 231 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ 𝐡 Btwn ⟨𝐢, 𝐴⟩)
3328, 32jca 510 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ 𝐡 Btwn ⟨𝐢, 𝐴⟩))
34 axpasch 28796 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘)) ∧ (π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ 𝐡 Btwn ⟨𝐢, 𝐴⟩) β†’ βˆƒπ‘ž ∈ (π”Όβ€˜π‘)(π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩)))
3527, 33, 34sylc 65 . . . . 5 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ βˆƒπ‘ž ∈ (π”Όβ€˜π‘)(π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩))
36 simpll1 1209 . . . . . . . . . . 11 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)))
3736, 1syl 17 . . . . . . . . . 10 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ 𝑁 ∈ β„•)
3836, 7syl 17 . . . . . . . . . . 11 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
39 simpll2 1210 . . . . . . . . . . 11 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ π‘Ÿ ∈ (π”Όβ€˜π‘))
4038, 39jca 510 . . . . . . . . . 10 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ (𝑃 ∈ (π”Όβ€˜π‘) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘)))
41 simplr 767 . . . . . . . . . . 11 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ π‘ž ∈ (π”Όβ€˜π‘))
4236, 2syl 17 . . . . . . . . . . 11 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ 𝐢 ∈ (π”Όβ€˜π‘))
4341, 42jca 510 . . . . . . . . . 10 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ (π‘ž ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)))
4437, 40, 433jca 1125 . . . . . . . . 9 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ (𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘)) ∧ (π‘ž ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))))
45 simpl3r 1226 . . . . . . . . . 10 (((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) β†’ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)
4645anim1i 613 . . . . . . . . 9 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ (π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩ ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩))
47 btwnexch2 35676 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘)) ∧ (π‘ž ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ ((π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩ ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ π‘ž Btwn βŸ¨π‘ƒ, 𝐢⟩))
4844, 46, 47sylc 65 . . . . . . . 8 ((((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) ∧ π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩) β†’ π‘ž Btwn βŸ¨π‘ƒ, 𝐢⟩)
4948ex 411 . . . . . . 7 (((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) β†’ (π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩ β†’ π‘ž Btwn βŸ¨π‘ƒ, 𝐢⟩))
5049anim1d 609 . . . . . 6 (((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) ∧ π‘ž ∈ (π”Όβ€˜π‘)) β†’ ((π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩) β†’ (π‘ž Btwn βŸ¨π‘ƒ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩)))
5150reximdva 3158 . . . . 5 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ (βˆƒπ‘ž ∈ (π”Όβ€˜π‘)(π‘ž Btwn βŸ¨π‘Ÿ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩) β†’ βˆƒπ‘ž ∈ (π”Όβ€˜π‘)(π‘ž Btwn βŸ¨π‘ƒ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩)))
5235, 51mpd 15 . . . 4 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) ∧ π‘Ÿ ∈ (π”Όβ€˜π‘) ∧ (π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩)) β†’ βˆƒπ‘ž ∈ (π”Όβ€˜π‘)(π‘ž Btwn βŸ¨π‘ƒ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩))
5352rexlimdv3a 3149 . . 3 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ (βˆƒπ‘Ÿ ∈ (π”Όβ€˜π‘)(π‘Ÿ Btwn ⟨𝐸, 𝐴⟩ ∧ π‘Ÿ Btwn βŸ¨π‘ƒ, 𝐢⟩) β†’ βˆƒπ‘ž ∈ (π”Όβ€˜π‘)(π‘ž Btwn βŸ¨π‘ƒ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩)))
5417, 53mpd 15 . 2 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩)) β†’ βˆƒπ‘ž ∈ (π”Όβ€˜π‘)(π‘ž Btwn βŸ¨π‘ƒ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩))
5554ex 411 1 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘))) β†’ ((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, 𝐢⟩ ∧ 𝑃 Btwn ⟨𝐴, 𝐷⟩) β†’ βˆƒπ‘ž ∈ (π”Όβ€˜π‘)(π‘ž Btwn βŸ¨π‘ƒ, 𝐢⟩ ∧ π‘ž Btwn ⟨𝐡, 𝐸⟩)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098  βˆƒwrex 3060  βŸ¨cop 4630   class class class wbr 5143  β€˜cfv 6543  β„•cn 12242  π”Όcee 28743   Btwn cbtwn 28744
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
This theorem is referenced by: (None)
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