Definition df-bj-fractemp 37524

Description: Temporary definition: fractional part of a temporary real.

To understand this definition, recall the canonical injection ω⟶R, 𝑛 ↦ [{𝑥 ∈ Q ∣ 𝑥 <_Q ⟨suc 𝑛, 1_o⟩}, 1_P] ~_R where we successively take the successor of 𝑛 to land in positive integers, then take the couple with 1_o 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 1_P 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 = (𝑥 ∈ R ↦ (℩𝑦 ∈ R ((𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)))

Distinct variable group:   𝑥,𝑦,𝑧,𝑛

Detailed syntax breakdown of Definition df-bj-fractemp

Step	Hyp	Ref	Expression
1		cfractemp 37523	. 2 class {R
2		vx	. . 3 setvar 𝑥
3		cnr 10777	. . 3 class R
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1541	. . . . . . 7 class 𝑦
6		c0r 10778	. . . . . . 7 class 0R
7	5, 6	wceq 1542	. . . . . 6 wff 𝑦 = 0R
8		cltr 10783	. . . . . . . 8 class <R
9	6, 5, 8	wbr 5086	. . . . . . 7 wff 0R <R 𝑦
10		c1r 10779	. . . . . . . 8 class 1R
11	5, 10, 8	wbr 5086	. . . . . . 7 wff 𝑦 <R 1R
12	9, 11	wa 395	. . . . . 6 wff (0R <R 𝑦 ∧ 𝑦 <R 1R)
13	7, 12	wo 848	. . . . 5 wff (𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R))
14		vz	. . . . . . . . . . . . 13 setvar 𝑧
15	14	cv 1541	. . . . . . . . . . . 12 class 𝑧
16		vn	. . . . . . . . . . . . . . 15 setvar 𝑛
17	16	cv 1541	. . . . . . . . . . . . . 14 class 𝑛
18	17	csuc 6317	. . . . . . . . . . . . 13 class suc 𝑛
19		c1o 8389	. . . . . . . . . . . . 13 class 1o
20	18, 19	cop 4574	. . . . . . . . . . . 12 class ⟨suc 𝑛, 1o⟩
21		cltq 10770	. . . . . . . . . . . 12 class <Q
22	15, 20, 21	wbr 5086	. . . . . . . . . . 11 wff 𝑧 <Q ⟨suc 𝑛, 1o⟩
23		cnq 10764	. . . . . . . . . . 11 class Q
24	22, 14, 23	crab 3390	. . . . . . . . . 10 class {𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}
25		c1p 10772	. . . . . . . . . 10 class 1P
26	24, 25	cop 4574	. . . . . . . . 9 class ⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩
27		cer 10776	. . . . . . . . 9 class ~R
28	26, 27	cec 8632	. . . . . . . 8 class [⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
29		cplr 10781	. . . . . . . 8 class +R
30	28, 5, 29	co 7358	. . . . . . 7 class ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦)
31	2	cv 1541	. . . . . . 7 class 𝑥
32	30, 31	wceq 1542	. . . . . 6 wff ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥
33		com 7808	. . . . . 6 class ω
34	32, 16, 33	wrex 3062	. . . . 5 wff ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥
35	13, 34	wa 395	. . . 4 wff ((𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)
36	35, 4, 3	crio 7314	. . 3 class (℩𝑦 ∈ R ((𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥))
37	2, 3, 36	cmpt 5167	. 2 class (𝑥 ∈ R ↦ (℩𝑦 ∈ R ((𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)))
38	1, 37	wceq 1542	1 wff {R = (𝑥 ∈ R ↦ (℩𝑦 ∈ R ((𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R)) ∧ ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥)))

This definition is referenced by: (None)