Definition df-bj-fractemp

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

To understand this definition, recall the canonical injection _om --> R. , n |-> [ { x e. Q. | x . } , 1P ] ~R where we successively take the successor of n to land in positive integers, then take the couple with 1o 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 1P 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.)

	Ref	Expression
Assertion	df-bj-fractemp	$$⊢{\{}_R=\left(x\in 𝑹⟼\left(\iota y\in 𝑹|\left(\left(y=0_{𝑹}\vee \left(0_{𝑹}{<}_{𝑹}y\wedge y{<}_{𝑹}{1}_{𝑹}\right)\right)\wedge \exists n\in \mathrm{\omega }\left[⟨\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\},1_{𝑷}⟩\right]{~}_{𝑹}{+}_{𝑹}y=x\right)\right)\right)$$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cfractemp	$$class{\{}_R$$
1		vx	$$setvarx$$
2		cnr	$$class𝑹$$
3		vy	$$setvary$$
4	3	cv	$$setvary$$
5		c0r	$$class{0}_{𝑹}$$
6	4 5	wceq	$$wffy=0_{𝑹}$$
7		cltr	$$class{<}_{𝑹}$$
8	5 4 7	wbr	$$wff{0}_{𝑹}{<}_{𝑹}y$$
9		c1r	$$class{1}_{𝑹}$$
10	4 9 7	wbr	$$wffy{<}_{𝑹}{1}_{𝑹}$$
11	8 10	wa	$$wff\left(0_{𝑹}{<}_{𝑹}y\wedge y{<}_{𝑹}{1}_{𝑹}\right)$$
12	6 11	wo	$$wff\left(y=0_{𝑹}\vee \left(0_{𝑹}{<}_{𝑹}y\wedge y{<}_{𝑹}{1}_{𝑹}\right)\right)$$
13		vn	$$setvarn$$
14		com	$$class\mathrm{\omega }$$
15		vz	$$setvarz$$
16		cnq	$$class𝑸$$
17	15	cv	$$setvarz$$
18		cltq	$$class{<}_{𝑸}$$
19	13	cv	$$setvarn$$
20	19	csuc	$$class\mathrm{suc}n$$
21		c1o	$$class{1}_{𝑜}$$
22	20 21	cop	$$class⟨\mathrm{suc}n,1_{𝑜}⟩$$
23	17 22 18	wbr	$$wffz{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩$$
24	23 15 16	crab	$$class\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\}$$
25		c1p	$$class{1}_{𝑷}$$
26	24 25	cop	$$class⟨\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\},1_{𝑷}⟩$$
27		cer	$$class{~}_{𝑹}$$
28	26 27	cec	$$class\left[⟨\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\},1_{𝑷}⟩\right]{~}_{𝑹}$$
29		cplr	$$class{+}_{𝑹}$$
30	28 4 29	co	$$class\left(\left[⟨\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\},1_{𝑷}⟩\right]{~}_{𝑹}{+}_{𝑹}y\right)$$
31	1	cv	$$setvarx$$
32	30 31	wceq	$$wff\left[⟨\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\},1_{𝑷}⟩\right]{~}_{𝑹}{+}_{𝑹}y=x$$
33	32 13 14	wrex	$$wff\exists n\in \mathrm{\omega }\left[⟨\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\},1_{𝑷}⟩\right]{~}_{𝑹}{+}_{𝑹}y=x$$
34	12 33	wa	$$wff\left(\left(y=0_{𝑹}\vee \left(0_{𝑹}{<}_{𝑹}y\wedge y{<}_{𝑹}{1}_{𝑹}\right)\right)\wedge \exists n\in \mathrm{\omega }\left[⟨\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\},1_{𝑷}⟩\right]{~}_{𝑹}{+}_{𝑹}y=x\right)$$
35	34 3 2	crio	$$class\left(\iota y\in 𝑹|\left(\left(y=0_{𝑹}\vee \left(0_{𝑹}{<}_{𝑹}y\wedge y{<}_{𝑹}{1}_{𝑹}\right)\right)\wedge \exists n\in \mathrm{\omega }\left[⟨\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\},1_{𝑷}⟩\right]{~}_{𝑹}{+}_{𝑹}y=x\right)\right)$$
36	1 2 35	cmpt	$$class\left(x\in 𝑹⟼\left(\iota y\in 𝑹|\left(\left(y=0_{𝑹}\vee \left(0_{𝑹}{<}_{𝑹}y\wedge y{<}_{𝑹}{1}_{𝑹}\right)\right)\wedge \exists n\in \mathrm{\omega }\left[⟨\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\},1_{𝑷}⟩\right]{~}_{𝑹}{+}_{𝑹}y=x\right)\right)\right)$$
37	0 36	wceq	$$wff{\{}_R=\left(x\in 𝑹⟼\left(\iota y\in 𝑹|\left(\left(y=0_{𝑹}\vee \left(0_{𝑹}{<}_{𝑹}y\wedge y{<}_{𝑹}{1}_{𝑹}\right)\right)\wedge \exists n\in \mathrm{\omega }\left[⟨\left\{z\in 𝑸|z{<}_{𝑸}⟨\mathrm{suc}n,1_{𝑜}⟩\right\},1_{𝑷}⟩\right]{~}_{𝑹}{+}_{𝑹}y=x\right)\right)\right)$$