Metamath Proof Explorer

Theorem btwnconn1lem4

Description: Lemma for btwnconn1 . Assuming C =/= c , we now attempt to force D = d from here out via a series of congruences. (Contributed by Scott Fenton, 8-Oct-2013)

Ref Expression
Assertion btwnconn1lem4 ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝑑 , 𝑐 ⟩ Cgr ⟨ 𝐷 , 𝐶 ⟩ )

Proof

Step Hyp Ref Expression
1 simp1rl ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ )
2 simp2rl ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ )
3 1 2 jca ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ) )
4 3 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ) )
5 simp11 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑁 ∈ ℕ )
6 simp12 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) )
7 simp13 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) )
8 simp21 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) )
9 simp3l ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) )
10 btwnexch3 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ) → 𝐶 Btwn ⟨ 𝐵 , 𝑑 ⟩ ) )
11 5 6 7 8 9 10 syl122anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ) → 𝐶 Btwn ⟨ 𝐵 , 𝑑 ⟩ ) )
12 11 adantr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ( ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ) → 𝐶 Btwn ⟨ 𝐵 , 𝑑 ⟩ ) )
13 4 12 mpd ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → 𝐶 Btwn ⟨ 𝐵 , 𝑑 ⟩ )
14 simp3lr ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ )
15 14 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ )
16 simp23 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) )
17 simp3r ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) )
18 cgrcomlr ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ↔ ⟨ 𝑏 , 𝑐 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) )
19 5 16 17 8 7 18 syl122anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ↔ ⟨ 𝑏 , 𝑐 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ) )
20 cgrcom ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝑏 , 𝑐 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ↔ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝑏 , 𝑐 ⟩ ) )
21 5 17 16 7 8 20 syl122anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝑏 , 𝑐 ⟩ Cgr ⟨ 𝐵 , 𝐶 ⟩ ↔ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝑏 , 𝑐 ⟩ ) )
22 19 21 bitrd ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ↔ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝑏 , 𝑐 ⟩ ) )
23 22 adantr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ( ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ↔ ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝑏 , 𝑐 ⟩ ) )
24 15 23 mpbid ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝑏 , 𝑐 ⟩ )
25 3simpa ⊢ ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
26 25 anim1i ⊢ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) → ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) )
27 btwnconn1lem3 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐵 , 𝑑 ⟩ Cgr ⟨ 𝑏 , 𝐷 ⟩ )
28 26 27 syl3anr1 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐵 , 𝑑 ⟩ Cgr ⟨ 𝑏 , 𝐷 ⟩ )
29 simp22 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) )
30 simp2rr ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
31 30 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
32 simp2lr ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
33 32 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
34 5 29 16 8 29 33 cgrcomland ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝑐 , 𝐷 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ )
35 5 8 9 16 29 8 29 31 34 cgrtr3and ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝑐 , 𝐷 ⟩ )
36 brcgr3 ⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐵 , ⟨ 𝐶 , 𝑑 ⟩ ⟩ Cgr3 ⟨ 𝑏 , ⟨ 𝑐 , 𝐷 ⟩ ⟩ ↔ ( ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝑏 , 𝑐 ⟩ ∧ ⟨ 𝐵 , 𝑑 ⟩ Cgr ⟨ 𝑏 , 𝐷 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝑐 , 𝐷 ⟩ ) ) )
37 5 7 8 9 17 16 29 36 syl133anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ 𝐵 , ⟨ 𝐶 , 𝑑 ⟩ ⟩ Cgr3 ⟨ 𝑏 , ⟨ 𝑐 , 𝐷 ⟩ ⟩ ↔ ( ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝑏 , 𝑐 ⟩ ∧ ⟨ 𝐵 , 𝑑 ⟩ Cgr ⟨ 𝑏 , 𝐷 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝑐 , 𝐷 ⟩ ) ) )
38 37 adantr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ( ⟨ 𝐵 , ⟨ 𝐶 , 𝑑 ⟩ ⟩ Cgr3 ⟨ 𝑏 , ⟨ 𝑐 , 𝐷 ⟩ ⟩ ↔ ( ⟨ 𝐵 , 𝐶 ⟩ Cgr ⟨ 𝑏 , 𝑐 ⟩ ∧ ⟨ 𝐵 , 𝑑 ⟩ Cgr ⟨ 𝑏 , 𝐷 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝑐 , 𝐷 ⟩ ) ) )
39 24 28 35 38 mpbir3and ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐵 , ⟨ 𝐶 , 𝑑 ⟩ ⟩ Cgr3 ⟨ 𝑏 , ⟨ 𝑐 , 𝐷 ⟩ ⟩ )
40 simpl ⊢ ( ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) )
41 simpr ⊢ ( ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) → 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) )
42 40 41 41 3jca ⊢ ( ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) → ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) )
43 42 3anim3i ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) )
44 26 3anim1i ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) )
45 btwnconn1lem1 ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑏 , 𝐶 ⟩ )
46 43 44 45 syl2an ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑏 , 𝐶 ⟩ )
47 5 8 16 cgrrflx2d ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝑐 , 𝐶 ⟩ )
48 47 adantr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝑐 , 𝐶 ⟩ )
49 46 48 jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ( ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑏 , 𝐶 ⟩ ∧ ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝑐 , 𝐶 ⟩ ) )
50 13 39 49 3jca ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ( 𝐶 Btwn ⟨ 𝐵 , 𝑑 ⟩ ∧ ⟨ 𝐵 , ⟨ 𝐶 , 𝑑 ⟩ ⟩ Cgr3 ⟨ 𝑏 , ⟨ 𝑐 , 𝐷 ⟩ ⟩ ∧ ( ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑏 , 𝐶 ⟩ ∧ ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝑐 , 𝐶 ⟩ ) ) )
51 simp1l2 ⊢ ( ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) → 𝐵 ≠ 𝐶 )
52 51 adantl ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → 𝐵 ≠ 𝐶 )
53 brofs2 ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ⟨ ⟨ 𝐵 , 𝐶 ⟩ , ⟨ 𝑑 , 𝑐 ⟩ ⟩ OuterFiveSeg ⟨ ⟨ 𝑏 , 𝑐 ⟩ , ⟨ 𝐷 , 𝐶 ⟩ ⟩ ↔ ( 𝐶 Btwn ⟨ 𝐵 , 𝑑 ⟩ ∧ ⟨ 𝐵 , ⟨ 𝐶 , 𝑑 ⟩ ⟩ Cgr3 ⟨ 𝑏 , ⟨ 𝑐 , 𝐷 ⟩ ⟩ ∧ ( ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑏 , 𝐶 ⟩ ∧ ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝑐 , 𝐶 ⟩ ) ) ) )
54 53 anbi1d ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ ⟨ 𝐵 , 𝐶 ⟩ , ⟨ 𝑑 , 𝑐 ⟩ ⟩ OuterFiveSeg ⟨ ⟨ 𝑏 , 𝑐 ⟩ , ⟨ 𝐷 , 𝐶 ⟩ ⟩ ∧ 𝐵 ≠ 𝐶 ) ↔ ( ( 𝐶 Btwn ⟨ 𝐵 , 𝑑 ⟩ ∧ ⟨ 𝐵 , ⟨ 𝐶 , 𝑑 ⟩ ⟩ Cgr3 ⟨ 𝑏 , ⟨ 𝑐 , 𝐷 ⟩ ⟩ ∧ ( ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑏 , 𝐶 ⟩ ∧ ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝑐 , 𝐶 ⟩ ) ) ∧ 𝐵 ≠ 𝐶 ) ) )
55 5segofs ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ⟨ ⟨ 𝐵 , 𝐶 ⟩ , ⟨ 𝑑 , 𝑐 ⟩ ⟩ OuterFiveSeg ⟨ ⟨ 𝑏 , 𝑐 ⟩ , ⟨ 𝐷 , 𝐶 ⟩ ⟩ ∧ 𝐵 ≠ 𝐶 ) → ⟨ 𝑑 , 𝑐 ⟩ Cgr ⟨ 𝐷 , 𝐶 ⟩ ) )
56 54 55 sylbird ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐶 Btwn ⟨ 𝐵 , 𝑑 ⟩ ∧ ⟨ 𝐵 , ⟨ 𝐶 , 𝑑 ⟩ ⟩ Cgr3 ⟨ 𝑏 , ⟨ 𝑐 , 𝐷 ⟩ ⟩ ∧ ( ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑏 , 𝐶 ⟩ ∧ ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝑐 , 𝐶 ⟩ ) ) ∧ 𝐵 ≠ 𝐶 ) → ⟨ 𝑑 , 𝑐 ⟩ Cgr ⟨ 𝐷 , 𝐶 ⟩ ) )
57 5 7 8 9 16 17 16 29 8 56 syl333anc ⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( ( ( 𝐶 Btwn ⟨ 𝐵 , 𝑑 ⟩ ∧ ⟨ 𝐵 , ⟨ 𝐶 , 𝑑 ⟩ ⟩ Cgr3 ⟨ 𝑏 , ⟨ 𝑐 , 𝐷 ⟩ ⟩ ∧ ( ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑏 , 𝐶 ⟩ ∧ ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝑐 , 𝐶 ⟩ ) ) ∧ 𝐵 ≠ 𝐶 ) → ⟨ 𝑑 , 𝑐 ⟩ Cgr ⟨ 𝐷 , 𝐶 ⟩ ) )
58 57 adantr ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ( ( ( 𝐶 Btwn ⟨ 𝐵 , 𝑑 ⟩ ∧ ⟨ 𝐵 , ⟨ 𝐶 , 𝑑 ⟩ ⟩ Cgr3 ⟨ 𝑏 , ⟨ 𝑐 , 𝐷 ⟩ ⟩ ∧ ( ⟨ 𝐵 , 𝑐 ⟩ Cgr ⟨ 𝑏 , 𝐶 ⟩ ∧ ⟨ 𝐶 , 𝑐 ⟩ Cgr ⟨ 𝑐 , 𝐶 ⟩ ) ) ∧ 𝐵 ≠ 𝐶 ) → ⟨ 𝑑 , 𝑐 ⟩ Cgr ⟨ 𝐷 , 𝐶 ⟩ ) )
59 50 52 58 mp2and ⊢ ( ( ( ( 𝑁 ∈ ℕ ∧ 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐷 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑐 ∈ ( 𝔼 ‘ 𝑁 ) ) ∧ ( 𝑑 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝑏 ∈ ( 𝔼 ‘ 𝑁 ) ) ) ∧ ( ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ∧ 𝐶 ≠ 𝑐 ) ∧ ( 𝐵 Btwn ⟨ 𝐴 , 𝐶 ⟩ ∧ 𝐵 Btwn ⟨ 𝐴 , 𝐷 ⟩ ) ) ∧ ( ( 𝐷 Btwn ⟨ 𝐴 , 𝑐 ⟩ ∧ ⟨ 𝐷 , 𝑐 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ∧ ( 𝐶 Btwn ⟨ 𝐴 , 𝑑 ⟩ ∧ ⟨ 𝐶 , 𝑑 ⟩ Cgr ⟨ 𝐶 , 𝐷 ⟩ ) ) ∧ ( ( 𝑐 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑐 , 𝑏 ⟩ Cgr ⟨ 𝐶 , 𝐵 ⟩ ) ∧ ( 𝑑 Btwn ⟨ 𝐴 , 𝑏 ⟩ ∧ ⟨ 𝑑 , 𝑏 ⟩ Cgr ⟨ 𝐷 , 𝐵 ⟩ ) ) ) ) → ⟨ 𝑑 , 𝑐 ⟩ Cgr ⟨ 𝐷 , 𝐶 ⟩ )