Metamath Proof Explorer

Theorem btwnintr

Description: Inner transitivity law for betweenness. Left-hand side of Theorem 3.5 of Schwabhauser p. 30. (Contributed by Scott Fenton, 12-Jun-2013)

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

Proof

Step Hyp Ref Expression
1 simp1 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → N ∈ ℕ
2 simp2l ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → A ∈ 𝔼 ⁡ N
3 simp2r ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B ∈ 𝔼 ⁡ N
4 simp3r ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → D ∈ 𝔼 ⁡ N
5 simp3l ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → C ∈ 𝔼 ⁡ N
6 axpasch ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B Btwn A D ∧ C Btwn B D → ∃ x ∈ 𝔼 ⁡ N x Btwn B B ∧ x Btwn C A
7 1 2 3 4 3 5 6 syl132anc ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B Btwn A D ∧ C Btwn B D → ∃ x ∈ 𝔼 ⁡ N x Btwn B B ∧ x Btwn C A
8 simpl1 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ x ∈ 𝔼 ⁡ N → N ∈ ℕ
9 simpr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ x ∈ 𝔼 ⁡ N → x ∈ 𝔼 ⁡ N
10 simpl2r ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ x ∈ 𝔼 ⁡ N → B ∈ 𝔼 ⁡ N
11 axbtwnid ⊢ N ∈ ℕ ∧ x ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N → x Btwn B B → x = B
12 8 9 10 11 syl3anc ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ x ∈ 𝔼 ⁡ N → x Btwn B B → x = B
13 breq1 ⊢ x = B → x Btwn C A ↔ B Btwn C A
14 13 biimpa ⊢ x = B ∧ x Btwn C A → B Btwn C A
15 simpl3l ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ x ∈ 𝔼 ⁡ N → C ∈ 𝔼 ⁡ N
16 simpl2l ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ x ∈ 𝔼 ⁡ N → A ∈ 𝔼 ⁡ N
17 btwncom ⊢ N ∈ ℕ ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N → B Btwn C A ↔ B Btwn A C
18 8 10 15 16 17 syl13anc ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ x ∈ 𝔼 ⁡ N → B Btwn C A ↔ B Btwn A C
19 14 18 syl5ib ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ x ∈ 𝔼 ⁡ N → x = B ∧ x Btwn C A → B Btwn A C
20 12 19 syland ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N ∧ x ∈ 𝔼 ⁡ N → x Btwn B B ∧ x Btwn C A → B Btwn A C
21 20 rexlimdva ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → ∃ x ∈ 𝔼 ⁡ N x Btwn B B ∧ x Btwn C A → B Btwn A C
22 7 21 syld ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ D ∈ 𝔼 ⁡ N → B Btwn A D ∧ C Btwn B D → B Btwn A C