Definition df-bj-fractemp 34507

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 34506	. 2 class {R
2		vx	. . 3 setvar 𝑥
3		cnr 10280	. . 3 class R
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1535	. . . . . . 7 class 𝑦
6		c0r 10281	. . . . . . 7 class 0R
7	5, 6	wceq 1536	. . . . . 6 wff 𝑦 = 0R
8		cltr 10286	. . . . . . . 8 class <R
9	6, 5, 8	wbr 5059	. . . . . . 7 wff 0R <R 𝑦
10		c1r 10282	. . . . . . . 8 class 1R
11	5, 10, 8	wbr 5059	. . . . . . 7 wff 𝑦 <R 1R
12	9, 11	wa 398	. . . . . 6 wff (0R <R 𝑦 ∧ 𝑦 <R 1R)
13	7, 12	wo 843	. . . . 5 wff (𝑦 = 0R ∨ (0R <R 𝑦 ∧ 𝑦 <R 1R))
14		vz	. . . . . . . . . . . . 13 setvar 𝑧
15	14	cv 1535	. . . . . . . . . . . 12 class 𝑧
16		vn	. . . . . . . . . . . . . . 15 setvar 𝑛
17	16	cv 1535	. . . . . . . . . . . . . 14 class 𝑛
18	17	csuc 6186	. . . . . . . . . . . . 13 class suc 𝑛
19		c1o 8088	. . . . . . . . . . . . 13 class 1o
20	18, 19	cop 4566	. . . . . . . . . . . 12 class ⟨suc 𝑛, 1o⟩
21		cltq 10273	. . . . . . . . . . . 12 class <Q
22	15, 20, 21	wbr 5059	. . . . . . . . . . 11 wff 𝑧 <Q ⟨suc 𝑛, 1o⟩
23		cnq 10267	. . . . . . . . . . 11 class Q
24	22, 14, 23	crab 3141	. . . . . . . . . 10 class {𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}
25		c1p 10275	. . . . . . . . . 10 class 1P
26	24, 25	cop 4566	. . . . . . . . 9 class ⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩
27		cer 10279	. . . . . . . . 9 class ~R
28	26, 27	cec 8280	. . . . . . . 8 class [⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
29		cplr 10284	. . . . . . . 8 class +R
30	28, 5, 29	co 7149	. . . . . . 7 class ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦)
31	2	cv 1535	. . . . . . 7 class 𝑥
32	30, 31	wceq 1536	. . . . . 6 wff ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥
33		com 7573	. . . . . 6 class ω
34	32, 16, 33	wrex 3138	. . . . 5 wff ∃𝑛 ∈ ω ([⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R +R 𝑦) = 𝑥
35	13, 34	wa 398	. . . 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 7106	. . 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 5139	. 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 1536	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)