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Theorem segletr 35743
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 777 . . . . . . 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 780 . . . . . . 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 510 . . . . . 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 769 . . . . . . . 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 771 . . . . . . . 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 35684 . . . . . . . 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 479 . . . . . 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 467 . . . . . . . . 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 621 . . . . . . . . 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 277 . . . . . . . 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 35675 . . . . . . . . . . . 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 479 . . . . . . . . 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 35655 . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 35622 . . . . . . . . . . . 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 510 . . . . . . . . . . 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 593 . . . . . . . . . 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 455 . . . . . . . . 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 579 . . . . . . 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 650 . . . . . 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 3165 . . . . 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 418 . . 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 3208 . 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 35737 . . . . 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 35737 . . . . 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 3224 . . 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 288 . 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 35737 . . 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 293 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 394   ∧ w3a 1084   ∈ wcel 2098  βˆƒwrex 3067  βŸ¨cop 4638   class class class wbr 5152  β€˜cfv 6553  β„•cn 12250  π”Όcee 28719   Btwn cbtwn 28720  Cgrccgr 28721  Cgr3ccgr3 35665   Seg≀ csegle 35735
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-sup 9473  df-oi 9541  df-card 9970  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12251  df-2 12313  df-3 12314  df-n0 12511  df-z 12597  df-uz 12861  df-rp 13015  df-ico 13370  df-icc 13371  df-fz 13525  df-fzo 13668  df-seq 14007  df-exp 14067  df-hash 14330  df-cj 15086  df-re 15087  df-im 15088  df-sqrt 15222  df-abs 15223  df-clim 15472  df-sum 15673  df-ee 28722  df-btwn 28723  df-cgr 28724  df-ofs 35612  df-cgr3 35670  df-segle 35736
This theorem is referenced by: (None)
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