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Definition df-inp 6314
Description: Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other.

Here we follow the definition of a Dedekind cut from Definition 11.2.1 of [HoTT], p. (varies) with the one exception that we define it over positive rational numbers rather than all rational numbers.

A Dedekind cut is an ordered pair of a lower set 𝑙 and an upper set u which is inhabited (𝑞 Q𝑞 𝑙 𝑟 Q𝑟 u), rounded (𝑞 Q(𝑞 𝑙𝑟 Q(𝑞 <Q 𝑟 𝑟 𝑙)) and likewise for u), disjoint (𝑞 Q¬ (𝑞 𝑙 𝑞 u)) and located (𝑞 Q𝑟 Q(𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))). See HoTT for more discussion of those terms and different ways of defining Dedekind cuts.

(Note: This is a "temporary" definition used in the construction of complex numbers, and is intended to be used only by the construction.) (Contributed by Jim Kingdon, 25-Sep-2019.)

Assertion
Ref Expression
df-inp P = {⟨𝑙, u⟩ ∣ (((𝑙Q uQ) (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u)) ((𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))) 𝑞 Q ¬ (𝑞 𝑙 𝑞 u) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))))}
Distinct variable group:   u,𝑙,𝑞,𝑟

Detailed syntax breakdown of Definition df-inp
StepHypRef Expression
1 cnp 6145 . 2 class P
2 vl . . . . . . . 8 setvar 𝑙
32cv 1225 . . . . . . 7 class 𝑙
4 cnq 6134 . . . . . . 7 class Q
53, 4wss 2890 . . . . . 6 wff 𝑙Q
6 vu . . . . . . . 8 setvar u
76cv 1225 . . . . . . 7 class u
87, 4wss 2890 . . . . . 6 wff uQ
95, 8wa 97 . . . . 5 wff (𝑙Q uQ)
10 vq . . . . . . . 8 setvar 𝑞
1110, 2wel 1371 . . . . . . 7 wff 𝑞 𝑙
1211, 10, 4wrex 2281 . . . . . 6 wff 𝑞 Q 𝑞 𝑙
13 vr . . . . . . . 8 setvar 𝑟
1413, 6wel 1371 . . . . . . 7 wff 𝑟 u
1514, 13, 4wrex 2281 . . . . . 6 wff 𝑟 Q 𝑟 u
1612, 15wa 97 . . . . 5 wff (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u)
179, 16wa 97 . . . 4 wff ((𝑙Q uQ) (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u))
1810cv 1225 . . . . . . . . . . 11 class 𝑞
1913cv 1225 . . . . . . . . . . 11 class 𝑟
20 cltq 6139 . . . . . . . . . . 11 class <Q
2118, 19, 20wbr 3734 . . . . . . . . . 10 wff 𝑞 <Q 𝑟
2213, 2wel 1371 . . . . . . . . . 10 wff 𝑟 𝑙
2321, 22wa 97 . . . . . . . . 9 wff (𝑞 <Q 𝑟 𝑟 𝑙)
2423, 13, 4wrex 2281 . . . . . . . 8 wff 𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)
2511, 24wb 98 . . . . . . 7 wff (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙))
2625, 10, 4wral 2280 . . . . . 6 wff 𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙))
2710, 6wel 1371 . . . . . . . . . 10 wff 𝑞 u
2821, 27wa 97 . . . . . . . . 9 wff (𝑞 <Q 𝑟 𝑞 u)
2928, 10, 4wrex 2281 . . . . . . . 8 wff 𝑞 Q (𝑞 <Q 𝑟 𝑞 u)
3014, 29wb 98 . . . . . . 7 wff (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))
3130, 13, 4wral 2280 . . . . . 6 wff 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))
3226, 31wa 97 . . . . 5 wff (𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u)))
3311, 27wa 97 . . . . . . 7 wff (𝑞 𝑙 𝑞 u)
3433wn 3 . . . . . 6 wff ¬ (𝑞 𝑙 𝑞 u)
3534, 10, 4wral 2280 . . . . 5 wff 𝑞 Q ¬ (𝑞 𝑙 𝑞 u)
3611, 14wo 616 . . . . . . . 8 wff (𝑞 𝑙 𝑟 u)
3721, 36wi 4 . . . . . . 7 wff (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))
3837, 13, 4wral 2280 . . . . . 6 wff 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))
3938, 10, 4wral 2280 . . . . 5 wff 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))
4032, 35, 39w3a 871 . . . 4 wff ((𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))) 𝑞 Q ¬ (𝑞 𝑙 𝑞 u) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u)))
4117, 40wa 97 . . 3 wff (((𝑙Q uQ) (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u)) ((𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))) 𝑞 Q ¬ (𝑞 𝑙 𝑞 u) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))))
4241, 2, 6copab 3787 . 2 class {⟨𝑙, u⟩ ∣ (((𝑙Q uQ) (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u)) ((𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))) 𝑞 Q ¬ (𝑞 𝑙 𝑞 u) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))))}
431, 42wceq 1226 1 wff P = {⟨𝑙, u⟩ ∣ (((𝑙Q uQ) (𝑞 Q 𝑞 𝑙 𝑟 Q 𝑟 u)) ((𝑞 Q (𝑞 𝑙𝑟 Q (𝑞 <Q 𝑟 𝑟 𝑙)) 𝑟 Q (𝑟 u𝑞 Q (𝑞 <Q 𝑟 𝑞 u))) 𝑞 Q ¬ (𝑞 𝑙 𝑞 u) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝑙 𝑟 u))))}
Colors of variables: wff set class
This definition is referenced by:  npsspw  6319  elinp  6322
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