Metamath Proof Explorer

Theorem btwnouttr

Description: Outer transitivity law for betweenness. Right-hand side of Theorem 3.7 of Schwabhauser p. 30. (Contributed by Scott Fenton, 14-Jun-2013)

Ref Expression
Assertion btwnouttr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B ≠ C ∧ B Btwn A C ∧ C Btwn B D → B Btwn A D

Proof

Step Hyp Ref Expression
1 simp1 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → N ∈ ℕ
2 simp2r ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B ∈ 𝔼 ⁡ N
3 simp3r ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → D ∈ 𝔼 ⁡ N
4 simp2l ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → A ∈ 𝔼 ⁡ N
5 necom ⊢ B ≠ C ↔ C ≠ B
6 5 a1i ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B ≠ C ↔ C ≠ B
7 simp3l ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → C ∈ 𝔼 ⁡ N
8 btwncom ⊢ N ∈ ℕ ∧ B ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B Btwn A C ↔ B Btwn C A
9 1 2 4 7 8 syl13anc ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B Btwn A C ↔ B Btwn C A
10 btwncom ⊢ N ∈ ℕ ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → C Btwn B D ↔ C Btwn D B
11 1 7 2 3 10 syl13anc ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → C Btwn B D ↔ C Btwn D B
12 6 9 11 3anbi123d ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B ≠ C ∧ B Btwn A C ∧ C Btwn B D ↔ C ≠ B ∧ B Btwn C A ∧ C Btwn D B
13 3ancomb ⊢ C ≠ B ∧ B Btwn C A ∧ C Btwn D B ↔ C ≠ B ∧ C Btwn D B ∧ B Btwn C A
14 12 13 syl6bb ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B ≠ C ∧ B Btwn A C ∧ C Btwn B D ↔ C ≠ B ∧ C Btwn D B ∧ B Btwn C A
15 14 biimpa ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ B ≠ C ∧ B Btwn A C ∧ C Btwn B D → C ≠ B ∧ C Btwn D B ∧ B Btwn C A
16 btwnouttr2 ⊢ N ∈ ℕ ∧ D ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N → C ≠ B ∧ C Btwn D B ∧ B Btwn C A → B Btwn D A
17 1 3 7 2 4 16 syl122anc ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → C ≠ B ∧ C Btwn D B ∧ B Btwn C A → B Btwn D A
18 17 adantr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ B ≠ C ∧ B Btwn A C ∧ C Btwn B D → C ≠ B ∧ C Btwn D B ∧ B Btwn C A → B Btwn D A
19 15 18 mpd ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ B ≠ C ∧ B Btwn A C ∧ C Btwn B D → B Btwn D A
20 1 2 3 4 19 btwncomand ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ B ≠ C ∧ B Btwn A C ∧ C Btwn B D → B Btwn A D
21 20 ex ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B ≠ C ∧ B Btwn A C ∧ C Btwn B D → B Btwn A D