Definition df-bj-fractemp 35295

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 35294	. 2 class {R
2		vx	. . 3 setvar 𝑥
3		cnr 10552	. . 3 class R
4		vy	. . . . . . . 8 setvar 𝑦
5	4	cv 1538	. . . . . . 7 class 𝑦
6		c0r 10553	. . . . . . 7 class 0R
7	5, 6	wceq 1539	. . . . . 6 wff 𝑦 = 0R
8		cltr 10558	. . . . . . . 8 class <R
9	6, 5, 8	wbr 5070	. . . . . . 7 wff 0R <R 𝑦
10		c1r 10554	. . . . . . . 8 class 1R
11	5, 10, 8	wbr 5070	. . . . . . 7 wff 𝑦 <R 1R
12	9, 11	wa 395	. . . . . 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 1538	. . . . . . . . . . . 12 class 𝑧
16		vn	. . . . . . . . . . . . . . 15 setvar 𝑛
17	16	cv 1538	. . . . . . . . . . . . . 14 class 𝑛
18	17	csuc 6253	. . . . . . . . . . . . 13 class suc 𝑛
19		c1o 8260	. . . . . . . . . . . . 13 class 1o
20	18, 19	cop 4564	. . . . . . . . . . . 12 class ⟨suc 𝑛, 1o⟩
21		cltq 10545	. . . . . . . . . . . 12 class <Q
22	15, 20, 21	wbr 5070	. . . . . . . . . . 11 wff 𝑧 <Q ⟨suc 𝑛, 1o⟩
23		cnq 10539	. . . . . . . . . . 11 class Q
24	22, 14, 23	crab 3067	. . . . . . . . . 10 class {𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}
25		c1p 10547	. . . . . . . . . 10 class 1P
26	24, 25	cop 4564	. . . . . . . . 9 class ⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩
27		cer 10551	. . . . . . . . 9 class ~R
28	26, 27	cec 8454	. . . . . . . 8 class [⟨{𝑧 ∈ Q ∣ 𝑧 <Q ⟨suc 𝑛, 1o⟩}, 1P⟩] ~R
29		cplr 10556	. . . . . . . 8 class +R
30	28, 5, 29	co 7255	. . . . . . 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 7687	. . . . . 6 class ω
34	32, 16, 33	wrex 3064	. . . . 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 7211	. . 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 5153	. 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)