Definition df-segle 33568

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 33567	. 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 4573	. . . . . . . . . 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 4573	. . . . . . . . . 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 26675	. . . . . . . . . . . 12 class Btwn
21	19, 16, 20	wbr 5066	. . . . . . . . . . 11 wff 𝑦 Btwn ⟨𝑐, 𝑑⟩
22	13, 19	cop 4573	. . . . . . . . . . . 12 class ⟨𝑐, 𝑦⟩
23		ccgr 26676	. . . . . . . . . . . 12 class Cgr
24	8, 22, 23	wbr 5066	. . . . . . . . . . 11 wff ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩
25	21, 24	wa 398	. . . . . . . . . 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 26674	. . . . . . . . . . 11 class 𝔼
29	27, 28	cfv 6355	. . . . . . . . . 10 class (𝔼‘𝑛)
30	25, 18, 29	wrex 3139	. . . . . . . . 9 wff ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)
31	9, 17, 30	w3a 1083	. . . . . . . 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 3139	. . . . . . 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 3139	. . . . . 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 3139	. . . . 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 3139	. . . 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 11638	. . . 4 class ℕ
37	35, 26, 36	wrex 3139	. . 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 5128	. 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  33569