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Definition df-bj-fractemp 36078
Description: Temporary definition: fractional part of a temporary real.

To understand this definition, recall the canonical injection ω⟶R, 𝑛 ↦ [{𝑥Q𝑥 <Q ⟨suc 𝑛, 1o⟩}, 1P] ~R where we successively take the successor of 𝑛 to land in positive integers, then take the couple with 1o as second component to land in positive rationals, then take the Dedekind cut that positive rational forms, and finally take the equivalence class of the couple with 1P as second component. Adding one at the beginning and subtracting it at the end is necessary since the constructions used in set.mm use the positive integers, positive rationals, and positive reals as intermediate number systems. (Contributed by BJ, 22-Jan-2023.) The precise definition is irrelevant and should generally not be used. One could even inline it. The definitive fractional part of an extended or projective complex number will be defined later. (New usage is discouraged.)

Assertion
Ref Expression
df-bj-fractemp {R = (𝑥R ↦ (𝑦R ((𝑦 = 0R ∨ (0R <R 𝑦𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑛

Detailed syntax breakdown of Definition df-bj-fractemp
StepHypRef Expression
1 cfractemp 36077 . 2 class {R
2 vx . . 3 setvar 𝑥
3 cnr 10860 . . 3 class R
4 vy . . . . . . . 8 setvar 𝑦
54cv 1541 . . . . . . 7 class 𝑦
6 c0r 10861 . . . . . . 7 class 0R
75, 6wceq 1542 . . . . . 6 wff 𝑦 = 0R
8 cltr 10866 . . . . . . . 8 class <R
96, 5, 8wbr 5149 . . . . . . 7 wff 0R <R 𝑦
10 c1r 10862 . . . . . . . 8 class 1R
115, 10, 8wbr 5149 . . . . . . 7 wff 𝑦 <R 1R
129, 11wa 397 . . . . . 6 wff (0R <R 𝑦𝑦 <R 1R)
137, 12wo 846 . . . . 5 wff (𝑦 = 0R ∨ (0R <R 𝑦𝑦 <R 1R))
14 vz . . . . . . . . . . . . 13 setvar 𝑧
1514cv 1541 . . . . . . . . . . . 12 class 𝑧
16 vn . . . . . . . . . . . . . . 15 setvar 𝑛
1716cv 1541 . . . . . . . . . . . . . 14 class 𝑛
1817csuc 6367 . . . . . . . . . . . . 13 class suc 𝑛
19 c1o 8459 . . . . . . . . . . . . 13 class 1o
2018, 19cop 4635 . . . . . . . . . . . 12 class ⟨suc 𝑛, 1o
21 cltq 10853 . . . . . . . . . . . 12 class <Q
2215, 20, 21wbr 5149 . . . . . . . . . . 11 wff 𝑧 <Q ⟨suc 𝑛, 1o
23 cnq 10847 . . . . . . . . . . 11 class Q
2422, 14, 23crab 3433 . . . . . . . . . 10 class {𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}
25 c1p 10855 . . . . . . . . . 10 class 1P
2624, 25cop 4635 . . . . . . . . 9 class ⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P
27 cer 10859 . . . . . . . . 9 class ~R
2826, 27cec 8701 . . . . . . . 8 class [⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
29 cplr 10864 . . . . . . . 8 class +R
3028, 5, 29co 7409 . . . . . . 7 class ([⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦)
312cv 1541 . . . . . . 7 class 𝑥
3230, 31wceq 1542 . . . . . 6 wff ([⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥
33 com 7855 . . . . . 6 class ω
3432, 16, 33wrex 3071 . . . . 5 wff 𝑛 ∈ ω ([⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥
3513, 34wa 397 . . . 4 wff ((𝑦 = 0R ∨ (0R <R 𝑦𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)
3635, 4, 3crio 7364 . . 3 class (𝑦R ((𝑦 = 0R ∨ (0R <R 𝑦𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥))
372, 3, 36cmpt 5232 . 2 class (𝑥R ↦ (𝑦R ((𝑦 = 0R ∨ (0R <R 𝑦𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)))
381, 37wceq 1542 1 wff {R = (𝑥R ↦ (𝑦R ((𝑦 = 0R ∨ (0R <R 𝑦𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧Q𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)))
Colors of variables: wff setvar class
This definition is referenced by: (None)
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