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Theorem segletr 35618
Description: Segment less than is transitive. Theorem 5.8 of [Schwabhauser] p. 42. (Contributed by Scott Fenton, 11-Oct-2013.)
Assertion
Ref Expression
segletr ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ ((⟨𝐴, 𝐡⟩ Seg≀ ⟨𝐢, 𝐷⟩ ∧ ⟨𝐢, 𝐷⟩ Seg≀ ⟨𝐸, 𝐹⟩) β†’ ⟨𝐴, 𝐡⟩ Seg≀ ⟨𝐸, 𝐹⟩))

Proof of Theorem segletr
Dummy variables 𝑦 𝑧 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprll 776 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ 𝑦 Btwn ⟨𝐢, 𝐷⟩)
2 simprrr 779 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©)
31, 2jca 511 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ (𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))
4 simpl1 1188 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
5 simpl23 1250 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) β†’ 𝐢 ∈ (π”Όβ€˜π‘))
6 simprl 768 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
7 simpl31 1251 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) β†’ 𝐷 ∈ (π”Όβ€˜π‘))
8 simpl32 1252 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) β†’ 𝐸 ∈ (π”Όβ€˜π‘))
9 simprr 770 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) β†’ 𝑧 ∈ (π”Όβ€˜π‘))
10 cgrxfr 35559 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) β†’ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) β†’ βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ ⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ©)))
114, 5, 6, 7, 8, 9, 10syl132anc 1385 . . . . . . 7 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) β†’ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) β†’ βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ ⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ©)))
1211adantr 480 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) β†’ βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ ⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ©)))
133, 12mpd 15 . . . . 5 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ ⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ©))
14 anass 468 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ 𝑀 ∈ (π”Όβ€˜π‘)) ↔ ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) ∧ 𝑀 ∈ (π”Όβ€˜π‘))))
15 df-3an 1086 . . . . . . . . . 10 ((𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘)) ↔ ((𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) ∧ 𝑀 ∈ (π”Όβ€˜π‘)))
1615anbi2i 622 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ↔ ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) ∧ 𝑀 ∈ (π”Όβ€˜π‘))))
1714, 16bitr4i 278 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ 𝑀 ∈ (π”Όβ€˜π‘)) ↔ ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))))
18 simpl1 1188 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
19 simpl23 1250 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ 𝐢 ∈ (π”Όβ€˜π‘))
20 simpr1 1191 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
21 simpl31 1251 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ 𝐷 ∈ (π”Όβ€˜π‘))
22 simpl32 1252 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ 𝐸 ∈ (π”Όβ€˜π‘))
23 simpr3 1193 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ 𝑀 ∈ (π”Όβ€˜π‘))
24 simpr2 1192 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ 𝑧 ∈ (π”Όβ€˜π‘))
25 brcgr3 35550 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ© ↔ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)))
2618, 19, 20, 21, 22, 23, 24, 25syl133anc 1390 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ© ↔ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)))
2726anbi2d 628 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ ((𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ ⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ©) ↔ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©))))
2827adantr 480 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ ((𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ ⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ©) ↔ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©))))
29 df-3an 1086 . . . . . . . . . . 11 (((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©))) ↔ (((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©)) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©))))
30 simpl33 1253 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ 𝐹 ∈ (π”Όβ€˜π‘))
31 simpr3l 1231 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)))) β†’ 𝑀 Btwn ⟨𝐸, π‘§βŸ©)
32 simpr2l 1229 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)))) β†’ 𝑧 Btwn ⟨𝐸, 𝐹⟩)
3318, 22, 23, 24, 30, 31, 32btwnexchand 35530 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)))) β†’ 𝑀 Btwn ⟨𝐸, 𝐹⟩)
34 simpl21 1248 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
35 simpl22 1249 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
36 simpr1r 1228 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)))) β†’ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©)
37 simp3r1 1278 . . . . . . . . . . . . . 14 (((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©))) β†’ ⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ©)
3837adantl 481 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)))) β†’ ⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ©)
3918, 34, 35, 19, 20, 22, 23, 36, 38cgrtrand 35497 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)))) β†’ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©)
4033, 39jca 511 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)))) β†’ (𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©))
4129, 40sylan2br 594 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ (((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©)) ∧ (𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)))) β†’ (𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©))
4241expr 456 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ ((𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ (⟨𝐢, π‘¦βŸ©Cgr⟨𝐸, π‘€βŸ© ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ© ∧ βŸ¨π‘¦, 𝐷⟩CgrβŸ¨π‘€, π‘§βŸ©)) β†’ (𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©)))
4328, 42sylbid 239 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘) ∧ 𝑀 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ ((𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ ⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ©) β†’ (𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©)))
4417, 43sylanb 580 . . . . . . 7 (((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ 𝑀 ∈ (π”Όβ€˜π‘)) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ ((𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ ⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ©) β†’ (𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©)))
4544an32s 649 . . . . . 6 (((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) ∧ 𝑀 ∈ (π”Όβ€˜π‘)) β†’ ((𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ ⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ©) β†’ (𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©)))
4645reximdva 3162 . . . . 5 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ (βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, π‘§βŸ© ∧ ⟨𝐢, βŸ¨π‘¦, 𝐷⟩⟩Cgr3⟨𝐸, βŸ¨π‘€, π‘§βŸ©βŸ©) β†’ βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©)))
4713, 46mpd 15 . . . 4 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) ∧ (𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘))) ∧ ((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))) β†’ βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©))
4847exp31 419 . . 3 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ ((𝑦 ∈ (π”Όβ€˜π‘) ∧ 𝑧 ∈ (π”Όβ€˜π‘)) β†’ (((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©)) β†’ βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©))))
4948rexlimdvv 3204 . 2 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ (βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)βˆƒπ‘§ ∈ (π”Όβ€˜π‘)((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©)) β†’ βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©)))
50 simp1 1133 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
51 simp21 1203 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
52 simp22 1204 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
53 simp23 1205 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ 𝐢 ∈ (π”Όβ€˜π‘))
54 simp31 1206 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ 𝐷 ∈ (π”Όβ€˜π‘))
55 brsegle 35612 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐴, 𝐡⟩ Seg≀ ⟨𝐢, 𝐷⟩ ↔ βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)(𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©)))
5650, 51, 52, 53, 54, 55syl122anc 1376 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐴, 𝐡⟩ Seg≀ ⟨𝐢, 𝐷⟩ ↔ βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)(𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©)))
57 simp32 1207 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ 𝐸 ∈ (π”Όβ€˜π‘))
58 simp33 1208 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ 𝐹 ∈ (π”Όβ€˜π‘))
59 brsegle 35612 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐢, 𝐷⟩ Seg≀ ⟨𝐸, 𝐹⟩ ↔ βˆƒπ‘§ ∈ (π”Όβ€˜π‘)(𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©)))
6050, 53, 54, 57, 58, 59syl122anc 1376 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐢, 𝐷⟩ Seg≀ ⟨𝐸, 𝐹⟩ ↔ βˆƒπ‘§ ∈ (π”Όβ€˜π‘)(𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©)))
6156, 60anbi12d 630 . . 3 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ ((⟨𝐴, 𝐡⟩ Seg≀ ⟨𝐢, 𝐷⟩ ∧ ⟨𝐢, 𝐷⟩ Seg≀ ⟨𝐸, 𝐹⟩) ↔ (βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)(𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ βˆƒπ‘§ ∈ (π”Όβ€˜π‘)(𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))))
62 reeanv 3220 . . 3 (βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)βˆƒπ‘§ ∈ (π”Όβ€˜π‘)((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©)) ↔ (βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)(𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ βˆƒπ‘§ ∈ (π”Όβ€˜π‘)(𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©)))
6361, 62bitr4di 289 . 2 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ ((⟨𝐴, 𝐡⟩ Seg≀ ⟨𝐢, 𝐷⟩ ∧ ⟨𝐢, 𝐷⟩ Seg≀ ⟨𝐸, 𝐹⟩) ↔ βˆƒπ‘¦ ∈ (π”Όβ€˜π‘)βˆƒπ‘§ ∈ (π”Όβ€˜π‘)((𝑦 Btwn ⟨𝐢, 𝐷⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐢, π‘¦βŸ©) ∧ (𝑧 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐢, 𝐷⟩Cgr⟨𝐸, π‘§βŸ©))))
64 brsegle 35612 . . 3 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐴, 𝐡⟩ Seg≀ ⟨𝐸, 𝐹⟩ ↔ βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©)))
6550, 51, 52, 57, 58, 64syl122anc 1376 . 2 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐴, 𝐡⟩ Seg≀ ⟨𝐸, 𝐹⟩ ↔ βˆƒπ‘€ ∈ (π”Όβ€˜π‘)(𝑀 Btwn ⟨𝐸, 𝐹⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐸, π‘€βŸ©)))
6649, 63, 653imtr4d 294 1 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐹 ∈ (π”Όβ€˜π‘))) β†’ ((⟨𝐴, 𝐡⟩ Seg≀ ⟨𝐢, 𝐷⟩ ∧ ⟨𝐢, 𝐷⟩ Seg≀ ⟨𝐸, 𝐹⟩) β†’ ⟨𝐴, 𝐡⟩ Seg≀ ⟨𝐸, 𝐹⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   ∈ wcel 2098  βˆƒwrex 3064  βŸ¨cop 4629   class class class wbr 5141  β€˜cfv 6536  β„•cn 12213  π”Όcee 28649   Btwn cbtwn 28650  Cgrccgr 28651  Cgr3ccgr3 35540   Seg≀ csegle 35610
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 7721  ax-inf2 9635  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
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 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-sup 9436  df-oi 9504  df-card 9933  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-div 11873  df-nn 12214  df-2 12276  df-3 12277  df-n0 12474  df-z 12560  df-uz 12824  df-rp 12978  df-ico 13333  df-icc 13334  df-fz 13488  df-fzo 13631  df-seq 13970  df-exp 14030  df-hash 14293  df-cj 15049  df-re 15050  df-im 15051  df-sqrt 15185  df-abs 15186  df-clim 15435  df-sum 15636  df-ee 28652  df-btwn 28653  df-cgr 28654  df-ofs 35487  df-cgr3 35545  df-segle 35611
This theorem is referenced by: (None)
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