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Theorem segletr 34745
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 778 . . . . . . 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 781 . . . . . . 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 513 . . . . . 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 1192 . . . . . . . 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 1254 . . . . . . . 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 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†’ 𝑦 ∈ (π”Όβ€˜π‘))
7 simpl31 1255 . . . . . . . 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 1256 . . . . . . . 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 772 . . . . . . . 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 34686 . . . . . . . 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 1389 . . . . . . 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 482 . . . . . 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 470 . . . . . . . . 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 1090 . . . . . . . . . 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 624 . . . . . . . . 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 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†’ 𝑁 ∈ β„•)
19 simpl23 1254 . . . . . . . . . . . 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 1195 . . . . . . . . . . . 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 1255 . . . . . . . . . . . 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 1256 . . . . . . . . . . . 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 1197 . . . . . . . . . . . 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 1196 . . . . . . . . . . . 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 34677 . . . . . . . . . . . 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 1394 . . . . . . . . . . 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 630 . . . . . . . . . 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 482 . . . . . . . . 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 1090 . . . . . . . . . . 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 1257 . . . . . . . . . . . . 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 1235 . . . . . . . . . . . . 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 1233 . . . . . . . . . . . . 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 34657 . . . . . . . . . . . 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 1252 . . . . . . . . . . . . 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 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†’ 𝐡 ∈ (π”Όβ€˜π‘))
36 simpr1r 1232 . . . . . . . . . . . . 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 1282 . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . 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 34624 . . . . . . . . . . . 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 513 . . . . . . . . . . 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 596 . . . . . . . . . 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 458 . . . . . . . . 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 582 . . . . . . 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 651 . . . . . 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 421 . . 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 3201 . 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 1137 . . . . 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 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†’ 𝐴 ∈ (π”Όβ€˜π‘))
52 simp22 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†’ 𝐡 ∈ (π”Όβ€˜π‘))
53 simp23 1209 . . . . 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 1210 . . . . 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 34739 . . . . 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 1380 . . . 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 1211 . . . . 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 1212 . . . . 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 34739 . . . . 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 1380 . . . 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 632 . . 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 3216 . . 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 34739 . . 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 1380 . 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 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  Cgrccgr 27881  Cgr3ccgr3 34667   Seg≀ csegle 34737
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  df-cgr3 34672  df-segle 34738
This theorem is referenced by: (None)
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