Definition df-segle 36089

Description: Define the segment length comparison relationship. This relationship expresses that the segment 𝐴𝐵 is no longer than 𝐶𝐷. In this section, we establish various properties of this relationship showing that it is a transitive, reflexive relationship on pairs of points that is substitutive under congruence. Definition 5.4 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 11-Oct-2013.)

Assertion

Ref	Expression
df-segle	⊢ Seg≤ = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))}

Distinct variable group:   𝑞,𝑝,𝑛,𝑎,𝑏,𝑐,𝑑,𝑦

Detailed syntax breakdown of Definition df-segle

Step	Hyp	Ref	Expression
1		csegle 36088	. 2 class Seg≤
2		vp	. . . . . . . . . . 11 setvar 𝑝
3	2	cv 1536	. . . . . . . . . 10 class 𝑝
4		va	. . . . . . . . . . . 12 setvar 𝑎
5	4	cv 1536	. . . . . . . . . . 11 class 𝑎
6		vb	. . . . . . . . . . . 12 setvar 𝑏
7	6	cv 1536	. . . . . . . . . . 11 class 𝑏
8	5, 7	cop 4637	. . . . . . . . . 10 class ⟨𝑎, 𝑏⟩
9	3, 8	wceq 1537	. . . . . . . . 9 wff 𝑝 = ⟨𝑎, 𝑏⟩
10		vq	. . . . . . . . . . 11 setvar 𝑞
11	10	cv 1536	. . . . . . . . . 10 class 𝑞
12		vc	. . . . . . . . . . . 12 setvar 𝑐
13	12	cv 1536	. . . . . . . . . . 11 class 𝑐
14		vd	. . . . . . . . . . . 12 setvar 𝑑
15	14	cv 1536	. . . . . . . . . . 11 class 𝑑
16	13, 15	cop 4637	. . . . . . . . . 10 class ⟨𝑐, 𝑑⟩
17	11, 16	wceq 1537	. . . . . . . . 9 wff 𝑞 = ⟨𝑐, 𝑑⟩
18		vy	. . . . . . . . . . . . 13 setvar 𝑦
19	18	cv 1536	. . . . . . . . . . . 12 class 𝑦
20		cbtwn 28919	. . . . . . . . . . . 12 class Btwn
21	19, 16, 20	wbr 5148	. . . . . . . . . . 11 wff 𝑦 Btwn ⟨𝑐, 𝑑⟩
22	13, 19	cop 4637	. . . . . . . . . . . 12 class ⟨𝑐, 𝑦⟩
23		ccgr 28920	. . . . . . . . . . . 12 class Cgr
24	8, 22, 23	wbr 5148	. . . . . . . . . . 11 wff ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩
25	21, 24	wa 395	. . . . . . . . . 10 wff (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)
26		vn	. . . . . . . . . . . 12 setvar 𝑛
27	26	cv 1536	. . . . . . . . . . 11 class 𝑛
28		cee 28918	. . . . . . . . . . 11 class 𝔼
29	27, 28	cfv 6563	. . . . . . . . . 10 class (𝔼‘𝑛)
30	25, 18, 29	wrex 3068	. . . . . . . . 9 wff ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)
31	9, 17, 30	w3a 1086	. . . . . . . 8 wff (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
32	31, 14, 29	wrex 3068	. . . . . . 7 wff ∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
33	32, 12, 29	wrex 3068	. . . . . 6 wff ∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
34	33, 6, 29	wrex 3068	. . . . 5 wff ∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
35	34, 4, 29	wrex 3068	. . . 4 wff ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
36		cn 12264	. . . 4 class ℕ
37	35, 26, 36	wrex 3068	. . 3 wff ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
38	37, 2, 10	copab 5210	. 2 class {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))}
39	1, 38	wceq 1537	1 wff Seg≤ = {⟨𝑝, 𝑞⟩ ∣ ∃𝑛 ∈ ℕ ∃𝑎 ∈ (𝔼‘𝑛)∃𝑏 ∈ (𝔼‘𝑛)∃𝑐 ∈ (𝔼‘𝑛)∃𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))}

This definition is referenced by:  brsegle  36090