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Theorem trisegint 34659
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 1192 . . . . 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 1254 . . . . . 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 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†’ 𝐴 ∈ (π”Όβ€˜π‘))
4 simpl31 1255 . . . . . 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 1129 . . . . 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 1256 . . . . . 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 1257 . . . . . 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 513 . . . . 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 1129 . . . 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 1196 . . . . . 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 34645 . . . . . . 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 1373 . . . . . 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 1197 . . . . 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 513 . . . 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 27932 . . . 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 1267 . . . . . . 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 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†’ 𝐸 ∈ (π”Όβ€˜π‘))
2023ad2ant1 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†’ 𝐢 ∈ (π”Όβ€˜π‘))
2133ad2ant1 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†’ 𝐴 ∈ (π”Όβ€˜π‘))
2219, 20, 213jca 1129 . . . . . . 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 1138 . . . . . . . 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 1253 . . . . . . . . 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 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†’ 𝐡 ∈ (π”Όβ€˜π‘))
2623, 25jca 513 . . . . . . 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 1129 . . . . . 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 1202 . . . . . . 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 1270 . . . . . . . 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 34645 . . . . . . . . 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 1373 . . . . . . . 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 513 . . . . . 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 27932 . . . . . 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 1213 . . . . . . . . . . 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 1214 . . . . . . . . . . 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 513 . . . . . . . . . 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 768 . . . . . . . . . . 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 513 . . . . . . . . . 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 1129 . . . . . . . . 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 1230 . . . . . . . . . 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 616 . . . . . . . . 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 34654 . . . . . . . . 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 414 . . . . . . 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 612 . . . . . 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 3162 . . . . 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 3153 . . 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 414 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 397   ∧ w3a 1088   ∈ wcel 2107  βˆƒwrex 3070  βŸ¨cop 4593   class class class wbr 5106  β€˜cfv 6497  β„•cn 12158  π”Όcee 27879   Btwn cbtwn 27880
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
This theorem is referenced by: (None)
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