colinbtwnle - Mathbox for Scott Fenton

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Theorem colinbtwnle 36610

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 36609	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ → ⟨𝐴, 𝐵⟩ Seg_≤ ⟨𝐴, 𝐶⟩))
2		3anrev 1118	. . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)))
3		btwnsegle 36609	. . . . . . 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 605	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐶, 𝐴⟩ → ⟨𝐶, 𝐵⟩ Seg_≤ ⟨𝐶, 𝐴⟩))
5		3ancoma 1115	. . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)))
6		btwncom 36506	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐵 Btwn ⟨𝐶, 𝐴⟩))
7	5, 6	sylan2b 605	. . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐵 Btwn ⟨𝐶, 𝐴⟩))
8		simpl 487	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
9		simpr2 1214	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
10		simpr3 1215	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁))
11	8, 9, 10	cgrrflx2d 36476	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐵, 𝐶⟩Cgr⟨𝐶, 𝐵⟩)
12		simpr1 1213	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
13	8, 12, 10	cgrrflx2d 36476	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐴, 𝐶⟩Cgr⟨𝐶, 𝐴⟩)
14		seglecgr12 36603	. . . . . . . 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 1429	. . . . . . 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 711	. . . . . 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 297	. . . . 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 521	. . . 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 485	. . 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 36551	. . . . 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 782	. . . . . . . . . . . 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 36507	. . . . . . . . . . 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 481	. . . . . . . . . . . . 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 728	. . . . . . . . . . . 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 36506	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐶, 𝐵⟩))
26		3anrot 1117	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)))
27		btwnsegle 36609	. . . . . . . . . . . . . . . 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 606	. . . . . . . . . . . . . . 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 243	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ → ⟨𝐶, 𝐴⟩ Seg_≤ ⟨𝐶, 𝐵⟩))
30	29	imp 411	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐴 Btwn ⟨𝐵, 𝐶⟩) → ⟨𝐶, 𝐴⟩ Seg_≤ ⟨𝐶, 𝐵⟩)
31	30	adantrr 729	. . . . . . . . . . . 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 36607	. . . . . . . . . . . . . 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 1406	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 711	. . . . . . . . . . 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 36578	. . . . . . . . . 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 36508	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → 𝐴 Btwn ⟨𝐴, 𝐶⟩)
38	37	3adant3r2 1202	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 Btwn ⟨𝐴, 𝐶⟩)
39		breq1 5112	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐴 → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐴, 𝐶⟩))
40	38, 39	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 = 𝐴 → 𝐵 Btwn ⟨𝐴, 𝐶⟩))
41	40	adantr 485	. . . . . . . . . 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 16	. . . . . . . . 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 461	. . . . . . . 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 495	. . . . . . 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 417	. . . . . 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 251	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐶, 𝐴⟩ → 𝐵 Btwn ⟨𝐴, 𝐶⟩))
47	46	a1dd 51	. . . . . 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 782	. . . . . . . . . . 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 784	. . . . . . . . . . . 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 1116	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)))
51		btwnsegle 36609	. . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐶 Btwn ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐶⟩ Seg_≤ ⟨𝐴, 𝐵⟩))
53	52	imp 411	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐶 Btwn ⟨𝐴, 𝐵⟩) → ⟨𝐴, 𝐶⟩ Seg_≤ ⟨𝐴, 𝐵⟩)
54	53	adantrr 729	. . . . . . . . . . . 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 36607	. . . . . . . . . . . . . 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 1406	. . . . . . . . . . . . 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 485	. . . . . . . . . . . 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 711	. . . . . . . . . . 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 36578	. . . . . . . . . 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 36504	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → 𝐶 Btwn ⟨𝐴, 𝐶⟩)
61	60	3adant3r2 1202	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 Btwn ⟨𝐴, 𝐶⟩)
62		breq1 5112	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐶 → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐶 Btwn ⟨𝐴, 𝐶⟩))
63	61, 62	syl5ibrcom 250	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 = 𝐶 → 𝐵 Btwn ⟨𝐴, 𝐶⟩))
64	63	adantr 485	. . . . . . . . . 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 16	. . . . . . . . 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 461	. . . . . . . 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 496	. . . . . . 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 417	. . . . . 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 1456	. . . . 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 243	. . . 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 411	. . 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 215	. 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 417	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 209 ∧ wa 400 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ⟨cop 4595 class class class wbr 5109 ‘cfv 6536 ℕcn 12228 𝔼cee 29237 Btwn cbtwn 29238 Cgrccgr 29239 Colinear ccolin 36529 Seg_≤ csegle 36598

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-n0 12500 df-z 12587 df-uz 12858 df-rp 13012 df-ico 13373 df-icc 13374 df-fz 13531 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-sum 15734 df-ee 29240 df-btwn 29241 df-cgr 29242 df-ofs 36475 df-colinear 36531 df-ifs 36532 df-cgr3 36533 df-segle 36599

This theorem is referenced by: (None)

W3C validator