colinbtwnle - Mathbox for Scott Fenton

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Theorem colinbtwnle 36100

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 36099	. . . . 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 36099	. . . . . . 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 35996	. . . . . . 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 1194	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
10		simpr3 1195	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁))
11	8, 9, 10	cgrrflx2d 35966	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐵, 𝐶⟩Cgr⟨𝐶, 𝐵⟩)
12		simpr1 1193	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
13	8, 12, 10	cgrrflx2d 35966	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐴, 𝐶⟩Cgr⟨𝐶, 𝐴⟩)
14		seglecgr12 36093	. . . . . . . 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 1401	. . . . . . 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 36041	. . . . 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 771	. . . . . . . . . . . 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 35997	. . . . . . . . . . 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 35996	. . . . . . . . . . . . . . 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 36099	. . . . . . . . . . . . . . . 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 36097	. . . . . . . . . . . . . 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 1378	. . . . . . . . . . . . 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 36068	. . . . . . . . . 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 35998	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → 𝐴 Btwn ⟨𝐴, 𝐶⟩)
38	37	3adant3r2 1182	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 Btwn ⟨𝐴, 𝐶⟩)
39		breq1 5151	. . . . . . . . . . . 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 771	. . . . . . . . . . 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 773	. . . . . . . . . . . 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 36099	. . . . . . . . . . . . . . 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 36097	. . . . . . . . . . . . . 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 1378	. . . . . . . . . . . . 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 36068	. . . . . . . . . 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 35994	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → 𝐶 Btwn ⟨𝐴, 𝐶⟩)
61	60	3adant3r2 1182	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 Btwn ⟨𝐴, 𝐶⟩)
62		breq1 5151	. . . . . . . . . . . 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 1428	. . . . 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 1537 ∈ wcel 2106 ⟨cop 4637 class class class wbr 5148 ‘cfv 6563 ℕcn 12264 𝔼cee 28918 Btwn cbtwn 28919 Cgrccgr 28920 Colinear ccolin 36019 Seg_≤ csegle 36088

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-sum 15720 df-ee 28921 df-btwn 28922 df-cgr 28923 df-ofs 35965 df-colinear 36021 df-ifs 36022 df-cgr3 36023 df-segle 36089

This theorem is referenced by: (None)

W3C validator