Metamath Proof Explorer

Theorem btwnxfr

Description: A condition for extending betweenness to a new set of points based on congruence with another set of points. Theorem 4.6 of Schwabhauser p. 36. (Contributed by Scott Fenton, 4-Oct-2013)

Ref Expression
Assertion btwnxfr ( ( 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 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) → 𝐸 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) )

Proof

Step Hyp Ref Expression
1 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 ⟩ ) ) )
2 simp2 ⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ) → ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ )
3 1 2 biimtrdi ⊢ ( ( 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 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ) )
4 simp1 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑁 ∈ ℕ )
5 simp21 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
6 simp22 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
7 simp23 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
8 simp31 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
9 simp33 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) )
10 cgrxfr ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ) → ∃ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) )
11 4 5 6 7 8 9 10 syl132anc ⊢ ( ( 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 , 𝐶 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ) → ∃ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) )
12 3 11 sylan2d ⊢ ( ( 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 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) → ∃ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) )
13 12 imp ⊢ ( ( ( 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 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ) → ∃ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) )
14 simprrl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) ) → 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ )
15 14 14 jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) ) → ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) )
16 simpl1 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝑁 ∈ ℕ )
17 simpl31 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
18 simpl33 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) )
19 16 17 18 cgrrflxd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ )
20 simpr ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) )
21 16 20 18 cgrrflxd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝑒 , 𝐹 ⟩ )
22 19 21 jca ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝑒 , 𝐹 ⟩ ) )
23 22 adantr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) ) → ( ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝑒 , 𝐹 ⟩ ) )
24 simpr ⊢ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ )
25 simpr ⊢ ( ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ )
26 simpl2 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) )
27 simpl3 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) )
28 17 20 18 3jca ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) )
29 cgr3tr4 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ) → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) → ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) )
30 16 26 27 28 29 syl13anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) → ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) )
31 cgr3com ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ↔ ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) )
32 16 27 17 20 18 31 syl113anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ↔ ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) )
33 simpl32 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) )
34 brcgr3 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ) ) )
35 16 17 20 18 17 33 18 34 syl133anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ↔ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ) ) )
36 simpr1 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ) ) → ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ )
37 simpr3 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ) ) → ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ )
38 16 20 18 33 18 37 cgrcomlrand ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ) ) → ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ )
39 36 38 jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ) ) → ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) )
40 39 ex ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ) → ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) ) )
41 35 40 sylbid ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ → ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) ) )
42 32 41 sylbid ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ → ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) ) )
43 30 42 syld ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) → ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) ) )
44 24 25 43 syl2ani ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) → ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) ) )
45 44 imp ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) ) → ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) )
46 15 23 45 3jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) ) → ( ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) ∧ ( ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝑒 , 𝐹 ⟩ ) ∧ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) ) )
47 46 ex ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) → ( ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) ∧ ( ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝑒 , 𝐹 ⟩ ) ∧ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) ) ) )
48 brifs ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ ⟨ 𝐷 , 𝑒 ⟩ , ⟨ 𝐹 , 𝑒 ⟩ ⟩ InnerFiveSeg ⟨ ⟨ 𝐷 , 𝑒 ⟩ , ⟨ 𝐹 , 𝐸 ⟩ ⟩ ↔ ( ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) ∧ ( ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝑒 , 𝐹 ⟩ ) ∧ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) ) ) )
49 ifscgr ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ ⟨ 𝐷 , 𝑒 ⟩ , ⟨ 𝐹 , 𝑒 ⟩ ⟩ InnerFiveSeg ⟨ ⟨ 𝐷 , 𝑒 ⟩ , ⟨ 𝐹 , 𝐸 ⟩ ⟩ → ⟨ 𝑒 , 𝑒 ⟩ Cgr ⟨ 𝑒 , 𝐸 ⟩ ) )
50 48 49 sylbird ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) ∧ ( ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝑒 , 𝐹 ⟩ ) ∧ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) ) → ⟨ 𝑒 , 𝑒 ⟩ Cgr ⟨ 𝑒 , 𝐸 ⟩ ) )
51 16 17 20 18 20 17 20 18 33 50 syl333anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) ∧ ( ⟨ 𝐷 , 𝐹 ⟩ Cgr ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝑒 , 𝐹 ⟩ Cgr ⟨ 𝑒 , 𝐹 ⟩ ) ∧ ( ⟨ 𝐷 , 𝑒 ⟩ Cgr ⟨ 𝐷 , 𝐸 ⟩ ∧ ⟨ 𝐹 , 𝑒 ⟩ Cgr ⟨ 𝐹 , 𝐸 ⟩ ) ) → ⟨ 𝑒 , 𝑒 ⟩ Cgr ⟨ 𝑒 , 𝐸 ⟩ ) )
52 47 51 syld ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) → ⟨ 𝑒 , 𝑒 ⟩ Cgr ⟨ 𝑒 , 𝐸 ⟩ ) )
53 cgrid2 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝑒 , 𝑒 ⟩ Cgr ⟨ 𝑒 , 𝐸 ⟩ → 𝑒 = 𝐸 ) )
54 16 20 20 33 53 syl13anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ⟨ 𝑒 , 𝑒 ⟩ Cgr ⟨ 𝑒 , 𝐸 ⟩ → 𝑒 = 𝐸 ) )
55 52 54 syld ⊢ ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) → 𝑒 = 𝐸 ) )
56 55 imp ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) ) → 𝑒 = 𝐸 )
57 56 14 eqbrtrrd ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ∧ ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) ) ) → 𝐸 Btwn ⟨ 𝐷 , 𝐹 ⟩ )
58 57 expr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ) → ( ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) → 𝐸 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) )
59 58 an32s ⊢ ( ( ( ( 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 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ) ∧ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) → 𝐸 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) )
60 59 rexlimdva ⊢ ( ( ( 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 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ) → ( ∃ 𝑒 ∈ ( 𝔼 ‘ 𝑁 ) ( 𝑒 Btwn ⟨ 𝐷 , 𝐹 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝑒 , 𝐹 ⟩ ⟩ ) → 𝐸 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) )
61 13 60 mpd ⊢ ( ( ( 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 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) ) → 𝐸 Btwn ⟨ 𝐷 , 𝐹 ⟩ )
62 61 ex ⊢ ( ( 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 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐷 , ⟨ 𝐸 , 𝐹 ⟩ ⟩ ) → 𝐸 Btwn ⟨ 𝐷 , 𝐹 ⟩ ) )