Definition df-fppr 47730

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 47729	. 2 class FPPr
2		vn	. . 3 setvar 𝑛
3		cn 12193	. . 3 class ℕ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1539	. . . . . 6 class 𝑥
6		cprime 16648	. . . . . 6 class ℙ
7	5, 6	wnel 3030	. . . . 5 wff 𝑥 ∉ ℙ
8	2	cv 1539	. . . . . . . 8 class 𝑛
9		c1 11076	. . . . . . . . 9 class 1
10		cmin 11412	. . . . . . . . 9 class −
11	5, 9, 10	co 7390	. . . . . . . 8 class (𝑥 − 1)
12		cexp 14033	. . . . . . . 8 class ↑
13	8, 11, 12	co 7390	. . . . . . 7 class (𝑛↑(𝑥 − 1))
14	13, 9, 10	co 7390	. . . . . 6 class ((𝑛↑(𝑥 − 1)) − 1)
15		cdvds 16229	. . . . . 6 class ∥
16	5, 14, 15	wbr 5110	. . . . 5 wff 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)
17	7, 16	wa 395	. . . 4 wff (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))
18		c4 12250	. . . . 5 class 4
19		cuz 12800	. . . . 5 class ℤ≥
20	18, 19	cfv 6514	. . . 4 class (ℤ≥‘4)
21	17, 4, 20	crab 3408	. . 3 class {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}
22	2, 3, 21	cmpt 5191	. 2 class (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
23	1, 22	wceq 1540	1 wff FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})

This definition is referenced by:  fppr  47731  fpprbasnn  47734