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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
StepHypRef Expression
1 csegle 31383 . 2 class Seg
2 vp . . . . . . . . . . 11 setvar 𝑝
32cv 1474 . . . . . . . . . 10 class 𝑝
4 va . . . . . . . . . . . 12 setvar 𝑎
54cv 1474 . . . . . . . . . . 11 class 𝑎
6 vb . . . . . . . . . . . 12 setvar 𝑏
76cv 1474 . . . . . . . . . . 11 class 𝑏
85, 7cop 4131 . . . . . . . . . 10 class 𝑎, 𝑏
93, 8wceq 1475 . . . . . . . . 9 wff 𝑝 = ⟨𝑎, 𝑏
10 vq . . . . . . . . . . 11 setvar 𝑞
1110cv 1474 . . . . . . . . . 10 class 𝑞
12 vc . . . . . . . . . . . 12 setvar 𝑐
1312cv 1474 . . . . . . . . . . 11 class 𝑐
14 vd . . . . . . . . . . . 12 setvar 𝑑
1514cv 1474 . . . . . . . . . . 11 class 𝑑
1613, 15cop 4131 . . . . . . . . . 10 class 𝑐, 𝑑
1711, 16wceq 1475 . . . . . . . . 9 wff 𝑞 = ⟨𝑐, 𝑑
18 vy . . . . . . . . . . . . 13 setvar 𝑦
1918cv 1474 . . . . . . . . . . . 12 class 𝑦
20 cbtwn 25569 . . . . . . . . . . . 12 class Btwn
2119, 16, 20wbr 4583 . . . . . . . . . . 11 wff 𝑦 Btwn ⟨𝑐, 𝑑
2213, 19cop 4131 . . . . . . . . . . . 12 class 𝑐, 𝑦
23 ccgr 25570 . . . . . . . . . . . 12 class Cgr
248, 22, 23wbr 4583 . . . . . . . . . . 11 wff 𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦
2521, 24wa 383 . . . . . . . . . 10 wff (𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)
26 vn . . . . . . . . . . . 12 setvar 𝑛
2726cv 1474 . . . . . . . . . . 11 class 𝑛
28 cee 25568 . . . . . . . . . . 11 class 𝔼
2927, 28cfv 5804 . . . . . . . . . 10 class (𝔼‘𝑛)
3025, 18, 29wrex 2897 . . . . . . . . 9 wff 𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩)
319, 17, 30w3a 1031 . . . . . . . 8 wff (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
3231, 14, 29wrex 2897 . . . . . . 7 wff 𝑑 ∈ (𝔼‘𝑛)(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ ∃𝑦 ∈ (𝔼‘𝑛)(𝑦 Btwn ⟨𝑐, 𝑑⟩ ∧ ⟨𝑎, 𝑏⟩Cgr⟨𝑐, 𝑦⟩))
3332, 12, 29wrex 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⟩))
3433, 6, 29wrex 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⟩))
3534, 4, 29wrex 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
3735, 26, 36wrex 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⟩))
3837, 2, 10copab 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⟩))}
391, 38wceq 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⟩))}
 Colors of variables: wff setvar class This definition is referenced by:  brsegle  31385
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