Definition df-fppr 48641

Description: Define the function that maps a positive integer to the set of Fermat pseudoprimes to the base of this positive integer. Since Fermat pseudoprimes shall be composite (positive) integers, they must be nonprime integers greater than or equal to 4 (we cannot use 𝑥 ∈ ℕ ∧ 𝑥 ∉ ℙ because 𝑥 = 1 would fulfil this requirement, but should not be regarded as "composite" integer). (Contributed by AV, 29-May-2023.)

Assertion

Ref	Expression
df-fppr	⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})

Distinct variable group:   𝑥,𝑛

Detailed syntax breakdown of Definition df-fppr

Step	Hyp	Ref	Expression
1		cfppr 48640	. 2 class FPPr
2		vn	. . 3 setvar 𝑛
3		cn 12257	. . 3 class ℕ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1569	. . . . . 6 class 𝑥
6		cprime 16761	. . . . . 6 class ℙ
7	5, 6	wnel 3061	. . . . 5 wff 𝑥 ∉ ℙ
8	2	cv 1569	. . . . . . . 8 class 𝑛
9		c1 11125	. . . . . . . . 9 class 1
10		cmin 11465	. . . . . . . . 9 class −
11	5, 9, 10	co 7413	. . . . . . . 8 class (𝑥 − 1)
12		cexp 14125	. . . . . . . 8 class ↑
13	8, 11, 12	co 7413	. . . . . . 7 class (𝑛↑(𝑥 − 1))
14	13, 9, 10	co 7413	. . . . . 6 class ((𝑛↑(𝑥 − 1)) − 1)
15		cdvds 16342	. . . . . 6 class ∥
16	5, 14, 15	wbr 5103	. . . . 5 wff 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)
17	7, 16	wa 401	. . . 4 wff (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))
18		c4 12321	. . . . 5 class 4
19		cuz 12887	. . . . 5 class ℤ≥
20	18, 19	cfv 6533	. . . 4 class (ℤ≥‘4)
21	17, 4, 20	crab 3412	. . 3 class {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}
22	2, 3, 21	cmpt 5186	. 2 class (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
23	1, 22	wceq 1570	1 wff FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})

This definition is used by:  fppr  48642  fpprbasnn  48645