Definition df-segle 34907

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 34906	. 2 class Seg≤
2		vp	. . . . . . . . . . 11 setvar 𝑝
3	2	cv 1540	. . . . . . . . . 10 class 𝑝
4		va	. . . . . . . . . . . 12 setvar 𝑎
5	4	cv 1540	. . . . . . . . . . 11 class 𝑎
6		vb	. . . . . . . . . . . 12 setvar 𝑏
7	6	cv 1540	. . . . . . . . . . 11 class 𝑏
8	5, 7	cop 4628	. . . . . . . . . 10 class ⟨𝑎, 𝑏⟩
9	3, 8	wceq 1541	. . . . . . . . 9 wff 𝑝 = ⟨𝑎, 𝑏⟩
10		vq	. . . . . . . . . . 11 setvar 𝑞
11	10	cv 1540	. . . . . . . . . 10 class 𝑞
12		vc	. . . . . . . . . . . 12 setvar 𝑐
13	12	cv 1540	. . . . . . . . . . 11 class 𝑐
14		vd	. . . . . . . . . . . 12 setvar 𝑑
15	14	cv 1540	. . . . . . . . . . 11 class 𝑑
16	13, 15	cop 4628	. . . . . . . . . 10 class ⟨𝑐, 𝑑⟩
17	11, 16	wceq 1541	. . . . . . . . 9 wff 𝑞 = ⟨𝑐, 𝑑⟩
18		vy	. . . . . . . . . . . . 13 setvar 𝑦
19	18	cv 1540	. . . . . . . . . . . 12 class 𝑦
20		cbtwn 28012	. . . . . . . . . . . 12 class Btwn
21	19, 16, 20	wbr 5141	. . . . . . . . . . 11 wff 𝑦 Btwn ⟨𝑐, 𝑑⟩
22	13, 19	cop 4628	. . . . . . . . . . . 12 class ⟨𝑐, 𝑦⟩
23		ccgr 28013	. . . . . . . . . . . 12 class Cgr
24	8, 22, 23	wbr 5141	. . . . . . . . . . 11 wff ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩
25	21, 24	wa 396	. . . . . . . . . 10 wff (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)
26		vn	. . . . . . . . . . . 12 setvar 𝑛
27	26	cv 1540	. . . . . . . . . . 11 class 𝑛
28		cee 28011	. . . . . . . . . . 11 class 𝔼
29	27, 28	cfv 6532	. . . . . . . . . 10 class (𝔼‘𝑛)
30	25, 18, 29	wrex 3069	. . . . . . . . 9 wff ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)
31	9, 17, 30	w3a 1087	. . . . . . . 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 3069	. . . . . . 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 3069	. . . . . 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 3069	. . . . 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 3069	. . . 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 12194	. . . 4 class ℕ
37	35, 26, 36	wrex 3069	. . 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 5203	. 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 1541	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  34908