colinbtwnle - Mathbox for Scott Fenton

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Theorem colinbtwnle 36106

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

**Assertion**
Ref	Expression
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 of Theorem colinbtwnle**
Step	Hyp	Ref	Expression
1		btwnsegle 36105	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ → ⟨𝐴, 𝐵⟩ Seg_≤ ⟨𝐴, 𝐶⟩))
2		3anrev 1100	. . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)))
3		btwnsegle 36105	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐶, 𝐴⟩ → ⟨𝐶, 𝐵⟩ Seg_≤ ⟨𝐶, 𝐴⟩))
4	2, 3	sylan2b 594	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐶, 𝐴⟩ → ⟨𝐶, 𝐵⟩ Seg_≤ ⟨𝐶, 𝐴⟩))
5		3ancoma 1097	. . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)))
6		btwncom 36002	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐵 Btwn ⟨𝐶, 𝐴⟩))
7	5, 6	sylan2b 594	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐵 Btwn ⟨𝐶, 𝐴⟩))
8		simpl 482	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
9		simpr2 1196	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
10		simpr3 1197	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁))
11	8, 9, 10	cgrrflx2d 35972	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐵, 𝐶⟩Cgr⟨𝐶, 𝐵⟩)
12		simpr1 1195	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
13	8, 12, 10	cgrrflx2d 35972	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐴, 𝐶⟩Cgr⟨𝐶, 𝐴⟩)
14		seglecgr12 36099	. . . . . . . 8 ⊢ (((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 1404	. . . . . . 7 ⊢ ((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 699	. . . . . 6 ⊢ ((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 294	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ → ⟨𝐵, 𝐶⟩ Seg_≤ ⟨𝐴, 𝐶⟩))
18	1, 17	jcad 512	. . . 4 ⊢ ((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 480	. . 3 ⊢ (((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 36047	. . . . 5 ⊢ ((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 770	. . . . . . . . . . . 12 ⊢ (((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 36003	. . . . . . . . . . 11 ⊢ (((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 476	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ ⟨𝐵, 𝐶⟩ Seg_≤ ⟨𝐴, 𝐶⟩) → ⟨𝐶, 𝐵⟩ Seg_≤ ⟨𝐶, 𝐴⟩)
24	23	adantrl 716	. . . . . . . . . . . 12 ⊢ (((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 36002	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐶, 𝐵⟩))
26		3anrot 1099	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)))
27		btwnsegle 36105	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐶, 𝐵⟩ → ⟨𝐶, 𝐴⟩ Seg_≤ ⟨𝐶, 𝐵⟩))
28	26, 27	sylan2br 595	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐶, 𝐵⟩ → ⟨𝐶, 𝐴⟩ Seg_≤ ⟨𝐶, 𝐵⟩))
29	25, 28	sylbid 240	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ → ⟨𝐶, 𝐴⟩ Seg_≤ ⟨𝐶, 𝐵⟩))
30	29	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐴 Btwn ⟨𝐵, 𝐶⟩) → ⟨𝐶, 𝐴⟩ Seg_≤ ⟨𝐶, 𝐵⟩)
31	30	adantrr 717	. . . . . . . . . . . 12 ⊢ (((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 36103	. . . . . . . . . . . . . 14 ⊢ ((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 1381	. . . . . . . . . . . . 13 ⊢ ((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 480	. . . . . . . . . . . 12 ⊢ (((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 699	. . . . . . . . . . 11 ⊢ (((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 36074	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ∧ ⟨𝐵, 𝐶⟩ Seg_≤ ⟨𝐴, 𝐶⟩)) → 𝐵 = 𝐴)
37		btwntriv1 36004	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → 𝐴 Btwn ⟨𝐴, 𝐶⟩)
38	37	3adant3r2 1184	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 Btwn ⟨𝐴, 𝐶⟩)
39		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐴 → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐴, 𝐶⟩))
40	38, 39	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 = 𝐴 → 𝐵 Btwn ⟨𝐴, 𝐶⟩))
41	40	adantr 480	. . . . . . . . . 10 ⊢ (((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 15	. . . . . . . . 9 ⊢ (((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 456	. . . . . . . 8 ⊢ (((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 490	. . . . . . 7 ⊢ (((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 412	. . . . . 6 ⊢ ((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 248	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐶, 𝐴⟩ → 𝐵 Btwn ⟨𝐴, 𝐶⟩))
47	46	a1dd 50	. . . . . 6 ⊢ ((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 770	. . . . . . . . . . 11 ⊢ (((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 772	. . . . . . . . . . . 12 ⊢ (((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 1098	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)))
51		btwnsegle 36105	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐶⟩ Seg_≤ ⟨𝐴, 𝐵⟩))
52	50, 51	sylan2b 594	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐶⟩ Seg_≤ ⟨𝐴, 𝐵⟩))
53	52	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐶 Btwn ⟨𝐴, 𝐵⟩) → ⟨𝐴, 𝐶⟩ Seg_≤ ⟨𝐴, 𝐵⟩)
54	53	adantrr 717	. . . . . . . . . . . 12 ⊢ (((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 36103	. . . . . . . . . . . . . 14 ⊢ ((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 1381	. . . . . . . . . . . . 13 ⊢ ((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 480	. . . . . . . . . . . 12 ⊢ (((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 699	. . . . . . . . . . 11 ⊢ (((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 36074	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ (𝐶 Btwn ⟨𝐴, 𝐵⟩ ∧ ⟨𝐴, 𝐵⟩ Seg_≤ ⟨𝐴, 𝐶⟩)) → 𝐵 = 𝐶)
60		btwntriv2 36000	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → 𝐶 Btwn ⟨𝐴, 𝐶⟩)
61	60	3adant3r2 1184	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 Btwn ⟨𝐴, 𝐶⟩)
62		breq1 5110	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐶 → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐶 Btwn ⟨𝐴, 𝐶⟩))
63	61, 62	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 = 𝐶 → 𝐵 Btwn ⟨𝐴, 𝐶⟩))
64	63	adantr 480	. . . . . . . . . 10 ⊢ (((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 15	. . . . . . . . 9 ⊢ (((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 456	. . . . . . . 8 ⊢ (((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 491	. . . . . . 7 ⊢ (((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 412	. . . . . 6 ⊢ ((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 1431	. . . . 5 ⊢ ((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 240	. . . 4 ⊢ ((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 406	. . 3 ⊢ (((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 212	. 2 ⊢ (((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 412	1 ⊢ ((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⟩))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ⟨cop 4595 class class class wbr 5107 ‘cfv 6511 ℕcn 12186 𝔼cee 28815 Btwn cbtwn 28816 Cgrccgr 28817 Colinear ccolin 36025 Seg_≤ csegle 36094

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-inf2 9594 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-se 5592 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-isom 6520 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-oi 9463 df-card 9892 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-ico 13312 df-icc 13313 df-fz 13469 df-fzo 13616 df-seq 13967 df-exp 14027 df-hash 14296 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-sum 15653 df-ee 28818 df-btwn 28819 df-cgr 28820 df-ofs 35971 df-colinear 36027 df-ifs 36028 df-cgr3 36029 df-segle 36095

This theorem is referenced by: (None)

W3C validator