Metamath Proof Explorer

Theorem colinbtwnle

Description: Given three colinear points A , B , and C , B falls in the middle iff the two segments to B are no longer than A C . Theorem 5.12 of Schwabhauser p. 42. (Contributed by Scott Fenton, 15-Oct-2013) (Revised by Mario Carneiro, 19-Apr-2014)

Ref Expression
Assertion colinbtwnle ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Colinear B C → B Btwn A C ↔ A B Seg ≤ A C ∧ B C Seg ≤ A C

Proof

Step Hyp Ref Expression
1 btwnsegle ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B Btwn A C → A B Seg ≤ A C
2 3anrev ⊢ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ↔ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N
3 btwnsegle ⊢ N ∈ ℕ ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N → B Btwn C A → C B Seg ≤ C A
4 2 3 sylan2b ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B Btwn C A → C B Seg ≤ C A
5 3ancoma ⊢ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ↔ B ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N
6 btwncom ⊢ N ∈ ℕ ∧ B ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B Btwn A C ↔ B Btwn C A
7 5 6 sylan2b ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B Btwn A C ↔ B Btwn C A
8 simpl ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → N ∈ ℕ
9 simpr2 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B ∈ 𝔼 ⁡ N
10 simpr3 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → C ∈ 𝔼 ⁡ N
11 8 9 10 cgrrflx2d ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B C Cgr C B
12 simpr1 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A ∈ 𝔼 ⁡ N
13 8 12 10 cgrrflx2d ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A C Cgr C A
14 seglecgr12 ⊢ N ∈ ℕ ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N → B C Cgr C B ∧ A C Cgr C A → B C Seg ≤ A C ↔ C B Seg ≤ C A
15 8 9 10 12 10 10 9 10 12 14 syl333anc ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B C Cgr C B ∧ A C Cgr C A → B C Seg ≤ A C ↔ C B Seg ≤ C A
16 11 13 15 mp2and ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B C Seg ≤ A C ↔ C B Seg ≤ C A
17 4 7 16 3imtr4d ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B Btwn A C → B C Seg ≤ A C
18 1 17 jcad ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B Btwn A C → A B Seg ≤ A C ∧ B C Seg ≤ A C
19 18 adantr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Colinear B C → B Btwn A C → A B Seg ≤ A C ∧ B C Seg ≤ A C
20 brcolinear ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Colinear B C ↔ A Btwn B C ∨ B Btwn C A ∨ C Btwn A B
21 simprl ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C ∧ B C Seg ≤ A C → A Btwn B C
22 8 12 9 10 21 btwncomand ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C ∧ B C Seg ≤ A C → A Btwn C B
23 16 biimpa ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ B C Seg ≤ A C → C B Seg ≤ C A
24 23 adantrl ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C ∧ B C Seg ≤ A C → C B Seg ≤ C A
25 btwncom ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C ↔ A Btwn C B
26 3anrot ⊢ C ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ↔ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N
27 btwnsegle ⊢ N ∈ ℕ ∧ C ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N → A Btwn C B → C A Seg ≤ C B
28 26 27 sylan2br ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn C B → C A Seg ≤ C B
29 25 28 sylbid ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C → C A Seg ≤ C B
30 29 imp ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C → C A Seg ≤ C B
31 30 adantrr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C ∧ B C Seg ≤ A C → C A Seg ≤ C B
32 segleantisym ⊢ N ∈ ℕ ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N → C B Seg ≤ C A ∧ C A Seg ≤ C B → C B Cgr C A
33 8 10 9 10 12 32 syl122anc ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → C B Seg ≤ C A ∧ C A Seg ≤ C B → C B Cgr C A
34 33 adantr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C ∧ B C Seg ≤ A C → C B Seg ≤ C A ∧ C A Seg ≤ C B → C B Cgr C A
35 24 31 34 mp2and ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C ∧ B C Seg ≤ A C → C B Cgr C A
36 8 10 9 12 22 35 endofsegidand ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C ∧ B C Seg ≤ A C → B = A
37 btwntriv1 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn A C
38 37 3adant3r2 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn A C
39 breq1 ⊢ B = A → B Btwn A C ↔ A Btwn A C
40 38 39 syl5ibrcom ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B = A → B Btwn A C
41 40 adantr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C ∧ B C Seg ≤ A C → B = A → B Btwn A C
42 36 41 mpd ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C ∧ B C Seg ≤ A C → B Btwn A C
43 42 expr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C → B C Seg ≤ A C → B Btwn A C
44 43 adantld ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Btwn B C → A B Seg ≤ A C ∧ B C Seg ≤ A C → B Btwn A C
45 44 ex ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C → A B Seg ≤ A C ∧ B C Seg ≤ A C → B Btwn A C
46 7 biimprd ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B Btwn C A → B Btwn A C
47 46 a1dd ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B Btwn C A → A B Seg ≤ A C ∧ B C Seg ≤ A C → B Btwn A C
48 simprl ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B ∧ A B Seg ≤ A C → C Btwn A B
49 simprr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B ∧ A B Seg ≤ A C → A B Seg ≤ A C
50 3ancomb ⊢ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ↔ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N
51 btwnsegle ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N → C Btwn A B → A C Seg ≤ A B
52 50 51 sylan2b ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → C Btwn A B → A C Seg ≤ A B
53 52 imp ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B → A C Seg ≤ A B
54 53 adantrr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B ∧ A B Seg ≤ A C → A C Seg ≤ A B
55 segleantisym ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A B Seg ≤ A C ∧ A C Seg ≤ A B → A B Cgr A C
56 8 12 9 12 10 55 syl122anc ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A B Seg ≤ A C ∧ A C Seg ≤ A B → A B Cgr A C
57 56 adantr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B ∧ A B Seg ≤ A C → A B Seg ≤ A C ∧ A C Seg ≤ A B → A B Cgr A C
58 49 54 57 mp2and ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B ∧ A B Seg ≤ A C → A B Cgr A C
59 8 12 9 10 48 58 endofsegidand ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B ∧ A B Seg ≤ A C → B = C
60 btwntriv2 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → C Btwn A C
61 60 3adant3r2 ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → C Btwn A C
62 breq1 ⊢ B = C → B Btwn A C ↔ C Btwn A C
63 61 62 syl5ibrcom ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → B = C → B Btwn A C
64 63 adantr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B ∧ A B Seg ≤ A C → B = C → B Btwn A C
65 59 64 mpd ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B ∧ A B Seg ≤ A C → B Btwn A C
66 65 expr ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B → A B Seg ≤ A C → B Btwn A C
67 66 adantrd ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ C Btwn A B → A B Seg ≤ A C ∧ B C Seg ≤ A C → B Btwn A C
68 67 ex ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → C Btwn A B → A B Seg ≤ A C ∧ B C Seg ≤ A C → B Btwn A C
69 45 47 68 3jaod ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Btwn B C ∨ B Btwn C A ∨ C Btwn A B → A B Seg ≤ A C ∧ B C Seg ≤ A C → B Btwn A C
70 20 69 sylbid ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Colinear B C → A B Seg ≤ A C ∧ B C Seg ≤ A C → B Btwn A C
71 70 imp ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Colinear B C → A B Seg ≤ A C ∧ B C Seg ≤ A C → B Btwn A C
72 19 71 impbid ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N ∧ A Colinear B C → B Btwn A C ↔ A B Seg ≤ A C ∧ B C Seg ≤ A C
73 72 ex ⊢ N ∈ ℕ ∧ A ∈ 𝔼 ⁡ N ∧ B ∈ 𝔼 ⁡ N ∧ C ∈ 𝔼 ⁡ N → A Colinear B C → B Btwn A C ↔ A B Seg ≤ A C ∧ B C Seg ≤ A C