colinbtwnle - Mathbox for Scott Fenton

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Theorem colinbtwnle 35078

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 35077	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 Btwn ⟨𝐴, 𝐶⟩ → ⟨𝐴, 𝐵⟩ Seg_≤ ⟨𝐴, 𝐶⟩))
2		3anrev 1101	. . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)))
3		btwnsegle 35077	. . . . . . 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 1098	. . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)))
6		btwncom 34974	. . . . . . 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 483	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝑁 ∈ ℕ)
9		simpr2 1195	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐵 ∈ (𝔼‘𝑁))
10		simpr3 1196	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁))
11	8, 9, 10	cgrrflx2d 34944	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐵, 𝐶⟩Cgr⟨𝐶, 𝐵⟩)
12		simpr1 1194	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 ∈ (𝔼‘𝑁))
13	8, 12, 10	cgrrflx2d 34944	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ⟨𝐴, 𝐶⟩Cgr⟨𝐶, 𝐴⟩)
14		seglecgr12 35071	. . . . . . . 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 1402	. . . . . . 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 697	. . . . . 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 293	. . . . 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 513	. . . 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 481	. . 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 35019	. . . . 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 769	. . . . . . . . . . . 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 34975	. . . . . . . . . . 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 477	. . . . . . . . . . . . 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 714	. . . . . . . . . . . 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 34974	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐶, 𝐵⟩))
26		3anrot 1100	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)))
27		btwnsegle 35077	. . . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ → ⟨𝐶, 𝐴⟩ Seg_≤ ⟨𝐶, 𝐵⟩))
30	29	imp 407	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐴 Btwn ⟨𝐵, 𝐶⟩) → ⟨𝐶, 𝐴⟩ Seg_≤ ⟨𝐶, 𝐵⟩)
31	30	adantrr 715	. . . . . . . . . . . 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 35075	. . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 697	. . . . . . . . . . 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 35046	. . . . . . . . . 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 34976	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → 𝐴 Btwn ⟨𝐴, 𝐶⟩)
38	37	3adant3r2 1183	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐴 Btwn ⟨𝐴, 𝐶⟩)
39		breq1 5150	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐴 → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐴 Btwn ⟨𝐴, 𝐶⟩))
40	38, 39	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 = 𝐴 → 𝐵 Btwn ⟨𝐴, 𝐶⟩))
41	40	adantr 481	. . . . . . . . . 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 457	. . . . . . . 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 491	. . . . . . 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 413	. . . . . 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 247	. . . . . . 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 769	. . . . . . . . . . 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 771	. . . . . . . . . . . 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 1099	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ↔ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)))
51		btwnsegle 35077	. . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) ∧ 𝐶 Btwn ⟨𝐴, 𝐵⟩) → ⟨𝐴, 𝐶⟩ Seg_≤ ⟨𝐴, 𝐵⟩)
54	53	adantrr 715	. . . . . . . . . . . 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 35075	. . . . . . . . . . . . . 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 1379	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 697	. . . . . . . . . . 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 35046	. . . . . . . . . 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 34972	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → 𝐶 Btwn ⟨𝐴, 𝐶⟩)
61	60	3adant3r2 1183	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → 𝐶 Btwn ⟨𝐴, 𝐶⟩)
62		breq1 5150	. . . . . . . . . . . 12 ⊢ (𝐵 = 𝐶 → (𝐵 Btwn ⟨𝐴, 𝐶⟩ ↔ 𝐶 Btwn ⟨𝐴, 𝐶⟩))
63	61, 62	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → (𝐵 = 𝐶 → 𝐵 Btwn ⟨𝐴, 𝐶⟩))
64	63	adantr 481	. . . . . . . . . 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 457	. . . . . . . 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 492	. . . . . . 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 413	. . . . . 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 239	. . . 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 407	. . 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 211	. 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 413	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 205 ∧ wa 396 ∨ w3o 1086 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⟨cop 4633 class class class wbr 5147 ‘cfv 6540 ℕcn 12208 𝔼cee 28135 Btwn cbtwn 28136 Cgrccgr 28137 Colinear ccolin 34997 Seg_≤ csegle 35066

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-sum 15629 df-ee 28138 df-btwn 28139 df-cgr 28140 df-ofs 34943 df-colinear 34999 df-ifs 35000 df-cgr3 35001 df-segle 35067

This theorem is referenced by: (None)

W3C validator