Definition df-segle 31384

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 31383	. 2 class Seg≤
2		vp	. . . . . . . . . . 11 setvar 𝑝
3	2	cv 1474	. . . . . . . . . 10 class 𝑝
4		va	. . . . . . . . . . . 12 setvar 𝑎
5	4	cv 1474	. . . . . . . . . . 11 class 𝑎
6		vb	. . . . . . . . . . . 12 setvar 𝑏
7	6	cv 1474	. . . . . . . . . . 11 class 𝑏
8	5, 7	cop 4131	. . . . . . . . . 10 class ⟨𝑎, 𝑏⟩
9	3, 8	wceq 1475	. . . . . . . . 9 wff 𝑝 = ⟨𝑎, 𝑏⟩
10		vq	. . . . . . . . . . 11 setvar 𝑞
11	10	cv 1474	. . . . . . . . . 10 class 𝑞
12		vc	. . . . . . . . . . . 12 setvar 𝑐
13	12	cv 1474	. . . . . . . . . . 11 class 𝑐
14		vd	. . . . . . . . . . . 12 setvar 𝑑
15	14	cv 1474	. . . . . . . . . . 11 class 𝑑
16	13, 15	cop 4131	. . . . . . . . . 10 class ⟨𝑐, 𝑑⟩
17	11, 16	wceq 1475	. . . . . . . . 9 wff 𝑞 = ⟨𝑐, 𝑑⟩
18		vy	. . . . . . . . . . . . 13 setvar 𝑦
19	18	cv 1474	. . . . . . . . . . . 12 class 𝑦
20		cbtwn 25569	. . . . . . . . . . . 12 class Btwn
21	19, 16, 20	wbr 4583	. . . . . . . . . . 11 wff 𝑦 Btwn ⟨𝑐, 𝑑⟩
22	13, 19	cop 4131	. . . . . . . . . . . 12 class ⟨𝑐, 𝑦⟩
23		ccgr 25570	. . . . . . . . . . . 12 class Cgr
24	8, 22, 23	wbr 4583	. . . . . . . . . . 11 wff ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩
25	21, 24	wa 383	. . . . . . . . . 10 wff (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)
26		vn	. . . . . . . . . . . 12 setvar 𝑛
27	26	cv 1474	. . . . . . . . . . 11 class 𝑛
28		cee 25568	. . . . . . . . . . 11 class 𝔼
29	27, 28	cfv 5804	. . . . . . . . . 10 class (𝔼‘𝑛)
30	25, 18, 29	wrex 2897	. . . . . . . . 9 wff ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)
31	9, 17, 30	w3a 1031	. . . . . . . 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 2897	. . . . . . 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 2897	. . . . . 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 2897	. . . . 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 2897	. . . 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 10897	. . . 4 class ℕ
37	35, 26, 36	wrex 2897	. . 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 4642	. 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 1475	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  31385