Theorem btwncolinear5

Description: Betweenness implies colinearity. (Contributed by Scott Fenton, 15-Oct-2013) (Revised by Mario Carneiro, 19-Apr-2014)

	Ref	Expression
Assertion	btwncolinear5	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ → 𝐶 Colinear ⟨ 𝐴 , 𝐵 ⟩ ) )

Proof

Step	Hyp	Ref	Expression
1		btwncolinear1	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ → 𝐴 Colinear ⟨ 𝐵 , 𝐶 ⟩ ) )
2		colinearperm4	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐴 Colinear ⟨ 𝐵 , 𝐶 ⟩ ↔ 𝐶 Colinear ⟨ 𝐴 , 𝐵 ⟩ ) )
3	1 2	sylibd	⊢ ( ( 𝑁 ∈ ℕ ∧ ( 𝐴 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐵 ∈ ( 𝔼 ‘ 𝑁 ) ∧ 𝐶 ∈ ( 𝔼 ‘ 𝑁 ) ) ) → ( 𝐶 Btwn ⟨ 𝐴 , 𝐵 ⟩ → 𝐶 Colinear ⟨ 𝐴 , 𝐵 ⟩ ) )