Definition df-bj-fractemp 35368

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 35367	. 2 class {R
2		vx	. . 3 setvar 𝑥
3		cnr 10621	. . 3 class R
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1538	. . . . . . 7 class 𝑦
6		c0r 10622	. . . . . . 7 class 0R
7	5, 6	wceq 1539	. . . . . 6 wff 𝑦 = 0R
8		cltr 10627	. . . . . . . 8 class <R
9	6, 5, 8	wbr 5074	. . . . . . 7 wff 0R <R 𝑦
10		c1r 10623	. . . . . . . 8 class 1R
11	5, 10, 8	wbr 5074	. . . . . . 7 wff 𝑦 <R 1R
12	9, 11	wa 396	. . . . . 6 wff (0R <R 𝑦 ∧ 𝑦 <R 1R)
13	7, 12	wo 844	. . . . 5 wff (𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R))
14		vz	. . . . . . . . . . . . 13 setvar 𝑧
15	14	cv 1538	. . . . . . . . . . . 12 class 𝑧
16		vn	. . . . . . . . . . . . . . 15 setvar 𝑛
17	16	cv 1538	. . . . . . . . . . . . . 14 class 𝑛
18	17	csuc 6268	. . . . . . . . . . . . 13 class suc 𝑛
19		c1o 8290	. . . . . . . . . . . . 13 class 1o
20	18, 19	cop 4567	. . . . . . . . . . . 12 class ⟨suc 𝑛, 1o⟩
21		cltq 10614	. . . . . . . . . . . 12 class <Q
22	15, 20, 21	wbr 5074	. . . . . . . . . . 11 wff 𝑧 <Q ⟨suc 𝑛, 1o⟩
23		cnq 10608	. . . . . . . . . . 11 class Q
24	22, 14, 23	crab 3068	. . . . . . . . . 10 class {𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}
25		c1p 10616	. . . . . . . . . 10 class 1P
26	24, 25	cop 4567	. . . . . . . . 9 class ⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩
27		cer 10620	. . . . . . . . 9 class ~R
28	26, 27	cec 8496	. . . . . . . 8 class [⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
29		cplr 10625	. . . . . . . 8 class +R
30	28, 5, 29	co 7275	. . . . . . 7 class ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦)
31	2	cv 1538	. . . . . . 7 class 𝑥
32	30, 31	wceq 1539	. . . . . 6 wff ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥
33		com 7712	. . . . . 6 class ω
34	32, 16, 33	wrex 3065	. . . . 5 wff ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥
35	13, 34	wa 396	. . . 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 7231	. . 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 5157	. 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 1539	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)