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Theorem lineext 35705
Description: Extend a line with a missing point. Theorem 4.14 of [Schwabhauser] p. 37. (Contributed by Scott Fenton, 6-Oct-2013.)
Assertion
Ref Expression
lineext ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Colinear ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
Distinct variable groups:   𝑓,𝑁   𝐴,𝑓   𝐡,𝑓   𝐢,𝑓   𝐷,𝑓   𝑓,𝐸

Proof of Theorem lineext
StepHypRef Expression
1 brcolinear 35688 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Colinear ⟨𝐡, 𝐢⟩ ↔ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∨ 𝐡 Btwn ⟨𝐢, 𝐴⟩ ∨ 𝐢 Btwn ⟨𝐴, 𝐡⟩)))
213adant3 1129 . . 3 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Colinear ⟨𝐡, 𝐢⟩ ↔ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∨ 𝐡 Btwn ⟨𝐢, 𝐴⟩ ∨ 𝐢 Btwn ⟨𝐴, 𝐡⟩)))
32anbi1d 629 . 2 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Colinear ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) ↔ ((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∨ 𝐡 Btwn ⟨𝐢, 𝐴⟩ ∨ 𝐢 Btwn ⟨𝐴, 𝐡⟩) ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)))
4 simp1 1133 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
5 simp3r 1199 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ 𝐸 ∈ (π”Όβ€˜π‘))
6 simp3l 1198 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ 𝐷 ∈ (π”Όβ€˜π‘))
75, 6jca 510 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)))
8 simp21 1203 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
9 simp23 1205 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ 𝐢 ∈ (π”Όβ€˜π‘))
108, 9jca 510 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)))
114, 7, 103jca 1125 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (𝑁 ∈ β„• ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))))
1211adantr 479 . . . . . . 7 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ (𝑁 ∈ β„• ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))))
13 axsegcon 28758 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐷, π‘“βŸ©Cgr⟨𝐴, 𝐢⟩))
1412, 13syl 17 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐷, π‘“βŸ©Cgr⟨𝐴, 𝐢⟩))
15 simprlr 778 . . . . . . . . . . 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‘)) ∧ ((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) ∧ (𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©))) β†’ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)
16 simprrr 780 . . . . . . . . . . 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‘)) ∧ ((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) ∧ (𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©))) β†’ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)
17 an4 654 . . . . . . . . . . . . 13 (((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) ∧ (𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)) ↔ ((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ 𝐷 Btwn ⟨𝐸, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)))
18 simpl1 1188 . . . . . . . . . . . . . . . . 17 (((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„•)
19 simpl21 1248 . . . . . . . . . . . . . . . . 17 (((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‘))
20 simpl22 1249 . . . . . . . . . . . . . . . . 17 (((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‘))
21 simpl3l 1225 . . . . . . . . . . . . . . . . 17 (((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‘))
22 simpl3r 1226 . . . . . . . . . . . . . . . . 17 (((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‘))
23 cgrcomlr 35627 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ↔ ⟨𝐡, 𝐴⟩Cgr⟨𝐸, 𝐷⟩))
2418, 19, 20, 21, 22, 23syl122anc 1376 . . . . . . . . . . . . . . . 16 (((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⟩Cgr⟨𝐷, 𝐸⟩ ↔ ⟨𝐡, 𝐴⟩Cgr⟨𝐸, 𝐷⟩))
2524anbi1d 629 . . . . . . . . . . . . . . 15 (((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⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©) ↔ (⟨𝐡, 𝐴⟩Cgr⟨𝐸, 𝐷⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)))
2625anbi2d 628 . . . . . . . . . . . . . 14 (((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 Btwn ⟨𝐡, 𝐢⟩ ∧ 𝐷 Btwn ⟨𝐸, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)) ↔ ((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ 𝐷 Btwn ⟨𝐸, π‘“βŸ©) ∧ (⟨𝐡, 𝐴⟩Cgr⟨𝐸, 𝐷⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©))))
27 simpl23 1250 . . . . . . . . . . . . . . 15 (((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‘))
28 simpr 483 . . . . . . . . . . . . . . 15 (((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‘))
29 cgrextend 35637 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„• ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝑓 ∈ (π”Όβ€˜π‘))) β†’ (((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ 𝐷 Btwn ⟨𝐸, π‘“βŸ©) ∧ (⟨𝐡, 𝐴⟩Cgr⟨𝐸, 𝐷⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)) β†’ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©))
3018, 20, 19, 27, 22, 21, 28, 29syl133anc 1390 . . . . . . . . . . . . . 14 (((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 Btwn ⟨𝐡, 𝐢⟩ ∧ 𝐷 Btwn ⟨𝐸, π‘“βŸ©) ∧ (⟨𝐡, 𝐴⟩Cgr⟨𝐸, 𝐷⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)) β†’ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©))
3126, 30sylbid 239 . . . . . . . . . . . . 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‘)) β†’ (((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ 𝐷 Btwn ⟨𝐸, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)) β†’ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©))
3217, 31biimtrid 241 . . . . . . . . . . . 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‘)) β†’ (((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) ∧ (𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)) β†’ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©))
3332imp 405 . . . . . . . . . . 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‘)) ∧ ((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) ∧ (𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©))) β†’ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)
3415, 16, 333jca 1125 . . . . . . . . . 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‘)) ∧ ((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) ∧ (𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©))) β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©))
3534expr 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‘)) ∧ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ ((𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©) β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)))
36 cgrcom 35619 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝑓 ∈ (π”Όβ€˜π‘)) ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐷, π‘“βŸ©Cgr⟨𝐴, 𝐢⟩ ↔ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©))
3718, 21, 28, 19, 27, 36syl122anc 1376 . . . . . . . . . . 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‘)) β†’ (⟨𝐷, π‘“βŸ©Cgr⟨𝐴, 𝐢⟩ ↔ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©))
3837anbi2d 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‘)) β†’ ((𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐷, π‘“βŸ©Cgr⟨𝐴, 𝐢⟩) ↔ (𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)))
3938adantr 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‘)) ∧ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ ((𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐷, π‘“βŸ©Cgr⟨𝐴, 𝐢⟩) ↔ (𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)))
40 simpl2 1189 . . . . . . . . . . 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‘)) β†’ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)))
41 brcgr3 35675 . . . . . . . . . . 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‘))) β†’ (⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ© ↔ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)))
4218, 40, 21, 22, 28, 41syl113anc 1379 . . . . . . . . . 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‘)) β†’ (⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ© ↔ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)))
4342adantr 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‘)) ∧ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ (⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ© ↔ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)))
4435, 39, 433imtr4d 293 . . . . . . . 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‘)) ∧ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ ((𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐷, π‘“βŸ©Cgr⟨𝐴, 𝐢⟩) β†’ ⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
4544an32s 650 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) ∧ 𝑓 ∈ (π”Όβ€˜π‘)) β†’ ((𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐷, π‘“βŸ©Cgr⟨𝐴, 𝐢⟩) β†’ ⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
4645reximdva 3165 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ (βˆƒπ‘“ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐸, π‘“βŸ© ∧ ⟨𝐷, π‘“βŸ©Cgr⟨𝐴, 𝐢⟩) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
4714, 46mpd 15 . . . . 5 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 Btwn ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©)
4847exp32 419 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 Btwn ⟨𝐡, 𝐢⟩ β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©)))
49 3ancoma 1095 . . . . . . 7 ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ↔ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)))
50 btwncom 35643 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ↔ 𝐡 Btwn ⟨𝐢, 𝐴⟩))
5149, 50sylan2b 592 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ↔ 𝐡 Btwn ⟨𝐢, 𝐴⟩))
52513adant3 1129 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ↔ 𝐡 Btwn ⟨𝐢, 𝐴⟩))
53 simp3 1135 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘)))
54 simp22 1204 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
55 axsegcon 28758 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)(𝐸 Btwn ⟨𝐷, π‘“βŸ© ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩))
564, 53, 54, 9, 55syl112anc 1371 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)(𝐸 Btwn ⟨𝐷, π‘“βŸ© ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩))
5756adantr 479 . . . . . . 7 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)(𝐸 Btwn ⟨𝐷, π‘“βŸ© ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩))
58 cgrextend 35637 . . . . . . . . . . . . 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‘))) β†’ (((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)) β†’ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©))
5918, 40, 21, 22, 28, 58syl113anc 1379 . . . . . . . . . . . 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‘)) β†’ (((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)) β†’ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©))
60 simpll 765 . . . . . . . . . . . . . . 15 (((⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©) ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©) β†’ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)
61 simpr 483 . . . . . . . . . . . . . . 15 (((⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©) ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©) β†’ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©)
62 simplr 767 . . . . . . . . . . . . . . 15 (((⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©) ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©) β†’ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)
6360, 61, 623jca 1125 . . . . . . . . . . . . . 14 (((⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©) ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ©) β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©))
6463ex 411 . . . . . . . . . . . . 13 ((⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©) β†’ (⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)))
6564adantl 480 . . . . . . . . . . . 12 (((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)) β†’ (⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)))
6659, 65sylcom 30 . . . . . . . . . . 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‘)) β†’ (((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)) β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, 𝐢⟩Cgr⟨𝐷, π‘“βŸ© ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)))
67 an4 654 . . . . . . . . . . . 12 (((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) ∧ (𝐸 Btwn ⟨𝐷, π‘“βŸ© ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩)) ↔ ((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩)))
68 cgrcom 35619 . . . . . . . . . . . . . . 15 ((𝑁 ∈ β„• ∧ (𝐸 ∈ (π”Όβ€˜π‘) ∧ 𝑓 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩ ↔ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©))
6918, 22, 28, 20, 27, 68syl122anc 1376 . . . . . . . . . . . . . 14 (((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‘“βŸ©Cgr⟨𝐡, 𝐢⟩ ↔ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©))
7069anbi2d 628 . . . . . . . . . . . . 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‘)) β†’ ((⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩) ↔ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©)))
7170anbi2d 628 . . . . . . . . . . . 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‘)) β†’ (((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩)) ↔ ((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©))))
7267, 71bitrid 282 . . . . . . . . . . 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‘)) β†’ (((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) ∧ (𝐸 Btwn ⟨𝐷, π‘“βŸ© ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩)) ↔ ((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ 𝐸 Btwn ⟨𝐷, π‘“βŸ©) ∧ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐡, 𝐢⟩Cgr⟨𝐸, π‘“βŸ©))))
7366, 72, 423imtr4d 293 . . . . . . . . . 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‘)) β†’ (((𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) ∧ (𝐸 Btwn ⟨𝐷, π‘“βŸ© ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩)) β†’ ⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
7473expdimp 451 . . . . . . . . 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‘)) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ ((𝐸 Btwn ⟨𝐷, π‘“βŸ© ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩) β†’ ⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
7574an32s 650 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) ∧ 𝑓 ∈ (π”Όβ€˜π‘)) β†’ ((𝐸 Btwn ⟨𝐷, π‘“βŸ© ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩) β†’ ⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
7675reximdva 3165 . . . . . . 7 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ (βˆƒπ‘“ ∈ (π”Όβ€˜π‘)(𝐸 Btwn ⟨𝐷, π‘“βŸ© ∧ ⟨𝐸, π‘“βŸ©Cgr⟨𝐡, 𝐢⟩) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
7757, 76mpd 15 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) ∧ (𝐡 Btwn ⟨𝐴, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩)) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©)
7877exp32 419 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (𝐡 Btwn ⟨𝐴, 𝐢⟩ β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©)))
7952, 78sylbird 259 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (𝐡 Btwn ⟨𝐢, 𝐴⟩ β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©)))
80 cgrxfr 35684 . . . . . . 7 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ ((𝐢 Btwn ⟨𝐴, 𝐡⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)(𝑓 Btwn ⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, ⟨𝐢, 𝐡⟩⟩Cgr3⟨𝐷, βŸ¨π‘“, 𝐸⟩⟩)))
814, 8, 9, 54, 53, 80syl131anc 1380 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ ((𝐢 Btwn ⟨𝐴, 𝐡⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)(𝑓 Btwn ⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, ⟨𝐢, 𝐡⟩⟩Cgr3⟨𝐷, βŸ¨π‘“, 𝐸⟩⟩)))
82 cgr3permute1 35677 . . . . . . . . . 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‘))) β†’ (⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ© ↔ ⟨𝐴, ⟨𝐢, 𝐡⟩⟩Cgr3⟨𝐷, βŸ¨π‘“, 𝐸⟩⟩))
8318, 40, 21, 22, 28, 82syl113anc 1379 . . . . . . . . 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‘)) β†’ (⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ© ↔ ⟨𝐴, ⟨𝐢, 𝐡⟩⟩Cgr3⟨𝐷, βŸ¨π‘“, 𝐸⟩⟩))
8483biimprd 247 . . . . . . . 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‘)) β†’ (⟨𝐴, ⟨𝐢, 𝐡⟩⟩Cgr3⟨𝐷, βŸ¨π‘“, 𝐸⟩⟩ β†’ ⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
8584adantld 489 . . . . . . 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‘)) β†’ ((𝑓 Btwn ⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, ⟨𝐢, 𝐡⟩⟩Cgr3⟨𝐷, βŸ¨π‘“, 𝐸⟩⟩) β†’ ⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
8685reximdva 3165 . . . . . 6 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (βˆƒπ‘“ ∈ (π”Όβ€˜π‘)(𝑓 Btwn ⟨𝐷, 𝐸⟩ ∧ ⟨𝐴, ⟨𝐢, 𝐡⟩⟩Cgr3⟨𝐷, βŸ¨π‘“, 𝐸⟩⟩) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
8781, 86syld 47 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ ((𝐢 Btwn ⟨𝐴, 𝐡⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
8887expd 414 . . . 4 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (𝐢 Btwn ⟨𝐴, 𝐡⟩ β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©)))
8948, 79, 883jaod 1425 . . 3 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∨ 𝐡 Btwn ⟨𝐢, 𝐴⟩ ∨ 𝐢 Btwn ⟨𝐴, 𝐡⟩) β†’ (⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩ β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©)))
9089impd 409 . 2 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ (((𝐴 Btwn ⟨𝐡, 𝐢⟩ ∨ 𝐡 Btwn ⟨𝐢, 𝐴⟩ ∨ 𝐢 Btwn ⟨𝐴, 𝐡⟩) ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
913, 90sylbid 239 1 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)) ∧ (𝐷 ∈ (π”Όβ€˜π‘) ∧ 𝐸 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 Colinear ⟨𝐡, 𝐢⟩ ∧ ⟨𝐴, 𝐡⟩Cgr⟨𝐷, 𝐸⟩) β†’ βˆƒπ‘“ ∈ (π”Όβ€˜π‘)⟨𝐴, ⟨𝐡, 𝐢⟩⟩Cgr3⟨𝐷, ⟨𝐸, π‘“βŸ©βŸ©))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ w3o 1083   ∧ 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   Colinear ccolin 35666
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-colinear 35668  df-cgr3 35670
This theorem is referenced by:  brsegle2  35738
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