Definition df-fppr 48201

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 48200	. 2 class FPPr
2		vn	. . 3 setvar 𝑛
3		cn 12174	. . 3 class ℕ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1541	. . . . . 6 class 𝑥
6		cprime 16640	. . . . . 6 class ℙ
7	5, 6	wnel 3036	. . . . 5 wff 𝑥 ∉ ℙ
8	2	cv 1541	. . . . . . . 8 class 𝑛
9		c1 11039	. . . . . . . . 9 class 1
10		cmin 11377	. . . . . . . . 9 class −
11	5, 9, 10	co 7367	. . . . . . . 8 class (𝑥 − 1)
12		cexp 14023	. . . . . . . 8 class ↑
13	8, 11, 12	co 7367	. . . . . . 7 class (𝑛↑(𝑥 − 1))
14	13, 9, 10	co 7367	. . . . . 6 class ((𝑛↑(𝑥 − 1)) − 1)
15		cdvds 16221	. . . . . 6 class ∥
16	5, 14, 15	wbr 5085	. . . . 5 wff 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)
17	7, 16	wa 395	. . . 4 wff (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))
18		c4 12238	. . . . 5 class 4
19		cuz 12788	. . . . 5 class ℤ≥
20	18, 19	cfv 6498	. . . 4 class (ℤ≥‘4)
21	17, 4, 20	crab 3389	. . 3 class {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}
22	2, 3, 21	cmpt 5166	. 2 class (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
23	1, 22	wceq 1542	1 wff FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})

This definition is referenced by:  fppr  48202  fpprbasnn  48205