Definition df-bj-fractemp 35741

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 35740	. 2 class {R
2		vx	. . 3 setvar 𝑥
3		cnr 10810	. . 3 class R
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1540	. . . . . . 7 class 𝑦
6		c0r 10811	. . . . . . 7 class 0R
7	5, 6	wceq 1541	. . . . . 6 wff 𝑦 = 0R
8		cltr 10816	. . . . . . . 8 class <R
9	6, 5, 8	wbr 5110	. . . . . . 7 wff 0R <R 𝑦
10		c1r 10812	. . . . . . . 8 class 1R
11	5, 10, 8	wbr 5110	. . . . . . 7 wff 𝑦 <R 1R
12	9, 11	wa 396	. . . . . 6 wff (0R <R 𝑦 ∧ 𝑦 <R 1R)
13	7, 12	wo 845	. . . . 5 wff (𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R))
14		vz	. . . . . . . . . . . . 13 setvar 𝑧
15	14	cv 1540	. . . . . . . . . . . 12 class 𝑧
16		vn	. . . . . . . . . . . . . . 15 setvar 𝑛
17	16	cv 1540	. . . . . . . . . . . . . 14 class 𝑛
18	17	csuc 6324	. . . . . . . . . . . . 13 class suc 𝑛
19		c1o 8410	. . . . . . . . . . . . 13 class 1o
20	18, 19	cop 4597	. . . . . . . . . . . 12 class ⟨suc 𝑛, 1o⟩
21		cltq 10803	. . . . . . . . . . . 12 class <Q
22	15, 20, 21	wbr 5110	. . . . . . . . . . 11 wff 𝑧 <Q ⟨suc 𝑛, 1o⟩
23		cnq 10797	. . . . . . . . . . 11 class Q
24	22, 14, 23	crab 3405	. . . . . . . . . 10 class {𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}
25		c1p 10805	. . . . . . . . . 10 class 1P
26	24, 25	cop 4597	. . . . . . . . 9 class ⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩
27		cer 10809	. . . . . . . . 9 class ~R
28	26, 27	cec 8653	. . . . . . . 8 class [⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
29		cplr 10814	. . . . . . . 8 class +R
30	28, 5, 29	co 7362	. . . . . . 7 class ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦)
31	2	cv 1540	. . . . . . 7 class 𝑥
32	30, 31	wceq 1541	. . . . . 6 wff ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥
33		com 7807	. . . . . 6 class ω
34	32, 16, 33	wrex 3069	. . . . 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 7317	. . 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 5193	. 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 1541	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)