Metamath Proof Explorer

Theorem lineext

Description: Extend a line with a missing point. Theorem 4.14 of Schwabhauser p. 37. (Contributed by Scott Fenton, 6-Oct-2013)

Ref Expression
Assertion lineext ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐴 Colinear ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )

Proof

Step Hyp Ref Expression
1 brcolinear ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐴 Colinear ⟨ 𝐵 , 𝐶 ⟩ ↔ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∨ 𝐵 Btwn ⟨ 𝐶 , 𝐴 ⟩ ∨ 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ) ) )
2 1 3adant3 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐴 Colinear ⟨ 𝐵 , 𝐶 ⟩ ↔ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∨ 𝐵 Btwn ⟨ 𝐶 , 𝐴 ⟩ ∨ 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ) ) )
3 2 anbi1d ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐴 Colinear ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ↔ ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∨ 𝐵 Btwn ⟨ 𝐶 , 𝐴 ⟩ ∨ 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ) ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) )
4 simp1 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑁 ∈ ℕ )
5 simp3r ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) )
6 simp3l ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
7 5 6 jca ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) )
8 simp21 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
9 simp23 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
10 8 9 jca ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) )
11 4 7 10 3jca ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝑁 ∈ ℕ ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) )
12 11 adantr ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ( 𝑁 ∈ ℕ ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) )
13 axsegcon ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐷 , 𝑓 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ) )
14 12 13 syl ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐷 , 𝑓 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ) )
15 simprlr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ∧ ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) ) → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ )
16 simprrr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ∧ ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) ) → ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ )
17 an4 ⊢ ( ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ∧ ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) ↔ ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) )
18 simpl1 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝑁 ∈ ℕ )
19 simpl21 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
20 simpl22 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
21 simpl3l ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
22 simpl3r ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) )
23 cgrcomlr ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ↔ ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐸 , 𝐷 ⟩ ) )
24 18 19 20 21 22 23 syl122anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ↔ ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐸 , 𝐷 ⟩ ) )
25 24 anbi1d ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ↔ ( ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐸 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) )
26 25 anbi2d ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) ↔ ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐸 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) ) )
27 simpl23 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
28 simpr ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) )
29 cgrextend ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐸 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) → ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) )
30 18 20 19 27 22 21 28 29 syl133anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐵 , 𝐴 ⟩ Cgr ⟨ 𝐸 , 𝐷 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) → ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) )
31 26 30 sylbid ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) → ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) )
32 17 31 biimtrid ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ∧ ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) → ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) )
33 32 imp ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ∧ ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) ) → ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ )
34 15 16 33 3jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ∧ ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) )
35 34 expr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ( ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) )
36 cgrcom ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐷 , 𝑓 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ↔ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) )
37 18 21 28 19 27 36 syl122anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐷 , 𝑓 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ↔ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) )
38 37 anbi2d ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐷 , 𝑓 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ) ↔ ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) )
39 38 adantr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ( ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐷 , 𝑓 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ) ↔ ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) ) )
40 simpl2 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) )
41 brcgr3 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ↔ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) )
42 18 40 21 22 28 41 syl113anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ↔ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) )
43 42 adantr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ↔ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) )
44 35 39 43 3imtr4d ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ( ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐷 , 𝑓 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ) → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
45 44 an32s ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐷 , 𝑓 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ) → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
46 45 reximdva ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ( ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝐷 Btwn ⟨ 𝐸 , 𝑓 ⟩ ∧ ⟨ 𝐷 , 𝑓 ⟩ Cgr ⟨ 𝐴 , 𝐶 ⟩ ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
47 14 46 mpd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ )
48 47 exp32 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) ) )
49 3ancoma ⊢ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ↔ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) )
50 btwncom ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ↔ 𝐵 Btwn ⟨ 𝐶 , 𝐴 ⟩ ) )
51 49 50 sylan2b ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ↔ 𝐵 Btwn ⟨ 𝐶 , 𝐴 ⟩ ) )
52 51 3adant3 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ↔ 𝐵 Btwn ⟨ 𝐶 , 𝐴 ⟩ ) )
53 simp3 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) )
54 simp22 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
55 axsegcon ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) )
56 4 53 54 9 55 syl112anc ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) )
57 56 adantr ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) )
58 cgrextend ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) → ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) )
59 18 40 21 22 28 58 syl113anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) → ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) )
60 simpll ⊢ ( ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ )
61 simpr ⊢ ( ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) → ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ )
62 simplr ⊢ ( ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) → ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ )
63 60 61 62 3jca ⊢ ( ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) )
64 63 ex ⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) → ( ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) )
65 64 adantl ⊢ ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) → ( ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) )
66 59 65 sylcom ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) )
67 an4 ⊢ ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ∧ ( 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) ) ↔ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) ) )
68 cgrcom ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ↔ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) )
69 18 22 28 20 27 68 syl122anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ↔ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) )
70 69 anbi2d ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) ↔ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) )
71 70 anbi2d ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) ) ↔ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) ) )
72 67 71 bitrid ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ∧ ( 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) ) ↔ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝑓 ⟩ ) ) ) )
73 66 72 42 3imtr4d ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ∧ ( 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) ) → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
74 73 expdimp ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ( ( 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
75 74 an32s ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
76 75 reximdva ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ( ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝐸 Btwn ⟨ 𝐷 , 𝑓 ⟩ ∧ ⟨ 𝐸 , 𝑓 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
77 57 76 mpd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ )
78 77 exp32 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) ) )
79 52 78 sylbird ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐵 Btwn ⟨ 𝐶 , 𝐴 ⟩ → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) ) )
80 cgrxfr ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑓 Btwn ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐶 , 𝐵 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑓 , 𝐸 ⟩ ⟩ ) ) )
81 4 8 9 54 53 80 syl131anc ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑓 Btwn ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐶 , 𝐵 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑓 , 𝐸 ⟩ ⟩ ) ) )
82 cgr3permute1 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ↔ ⟨ 𝐴 , ⟨ 𝐶 , 𝐵 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑓 , 𝐸 ⟩ ⟩ ) )
83 18 40 21 22 28 82 syl113anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ↔ ⟨ 𝐴 , ⟨ 𝐶 , 𝐵 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑓 , 𝐸 ⟩ ⟩ ) )
84 83 biimprd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐴 , ⟨ 𝐶 , 𝐵 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑓 , 𝐸 ⟩ ⟩ → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
85 84 adantld ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( 𝑓 Btwn ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐶 , 𝐵 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑓 , 𝐸 ⟩ ⟩ ) → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
86 85 reximdva ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑓 Btwn ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐶 , 𝐵 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑓 , 𝐸 ⟩ ⟩ ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
87 81 86 syld ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
88 87 expd ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) ) )
89 48 79 88 3jaod ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∨ 𝐵 Btwn ⟨ 𝐶 , 𝐴 ⟩ ∨ 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) ) )
90 89 impd ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐴 Btwn ⟨ 𝐵 , 𝐶 ⟩ ∨ 𝐵 Btwn ⟨ 𝐶 , 𝐴 ⟩ ∨ 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ ) ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )
91 3 90 sylbid ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐴 Colinear ⟨ 𝐵 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ) → ∃ 𝑓 ∈ ( 𝔼 ‘ 𝑁 ) ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝑓 ⟩ ⟩ ) )