Metamath Proof Explorer

Theorem brofs2

Description: Change some conditions for outer five segment predicate. (Contributed by Scott Fenton, 6-Oct-2013)

Ref Expression
Assertion brofs2 ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ ⟩ OuterFiveSeg ⟨ ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ ⟩ ↔ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) )

Proof

Step Hyp Ref Expression
1 brofs ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ ⟩ OuterFiveSeg ⟨ ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ ⟩ ↔ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) )
2 simpr1l ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ )
3 simpr2l ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ )
4 simpr1 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) )
5 simpr2 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) )
6 4 5 jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ) )
7 6 ex ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) → ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ) ) )
8 simp11 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑁 ∈ ℕ )
9 simp12 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
10 simp13 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
11 simp21 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
12 simp23 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) )
13 simp31 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) )
14 simp32 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) )
15 cgrextend ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ) → ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝐺 ⟩ ) )
16 8 9 10 11 12 13 14 15 syl133anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ) → ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝐺 ⟩ ) )
17 7 16 syld ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) → ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝐺 ⟩ ) )
18 17 imp ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝐺 ⟩ )
19 simpr2r ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ )
20 3 18 19 3jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝐺 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) )
21 20 ex ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝐺 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ) )
22 brcgr3 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ↔ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝐺 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ) )
23 8 9 10 11 12 13 14 22 syl133anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ↔ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝐺 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ) )
24 21 23 sylibrd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ) )
25 24 imp ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ )
26 simpr3 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) )
27 2 25 26 3jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) )
28 simpr1 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ )
29 3simpa ⊢ ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) → ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ) )
30 btwnxfr ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ) → 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) )
31 8 9 10 11 12 13 14 30 syl133anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ) → 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) )
32 29 31 syl5 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) → 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) )
33 32 imp ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ )
34 28 33 jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) )
35 3simpb ⊢ ( ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐴 , 𝐶 ⟩ Cgr ⟨ 𝐸 , 𝐺 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) )
36 23 35 syl6bi ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ) )
37 36 imp ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) )
38 37 3ad2antr2 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) )
39 simpr3 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) )
40 34 38 39 3jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) → ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) )
41 27 40 impbida ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐹 Btwn ⟨ 𝐸 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐵 ⟩ Cgr ⟨ 𝐸 , 𝐹 ⟩ ∧ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝐹 , 𝐺 ⟩ ) ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ↔ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) )
42 1 41 bitrd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐸 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐹 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐺 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐻 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ ⟨ 𝐴 , 𝐵 ⟩ , ⟨ 𝐶 , 𝐷 ⟩ ⟩ OuterFiveSeg ⟨ ⟨ 𝐸 , 𝐹 ⟩ , ⟨ 𝐺 , 𝐻 ⟩ ⟩ ↔ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ ⟨ 𝐴 , ⟨ 𝐵 , 𝐶 ⟩ ⟩ Cgr3 ⟨ 𝐸 , ⟨ 𝐹 , 𝐺 ⟩ ⟩ ∧ ( ⟨ 𝐴 , 𝐷 ⟩ Cgr ⟨ 𝐸 , 𝐻 ⟩ ∧ ⟨ 𝐵 , 𝐷 ⟩ Cgr ⟨ 𝐹 , 𝐻 ⟩ ) ) ) )