Definition df-fppr 47726

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 47725	. 2 class FPPr
2		vn	. . 3 setvar 𝑛
3		cn 12186	. . 3 class ℕ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1539	. . . . . 6 class 𝑥
6		cprime 16641	. . . . . 6 class ℙ
7	5, 6	wnel 3029	. . . . 5 wff 𝑥 ∉ ℙ
8	2	cv 1539	. . . . . . . 8 class 𝑛
9		c1 11069	. . . . . . . . 9 class 1
10		cmin 11405	. . . . . . . . 9 class −
11	5, 9, 10	co 7387	. . . . . . . 8 class (𝑥 − 1)
12		cexp 14026	. . . . . . . 8 class ↑
13	8, 11, 12	co 7387	. . . . . . 7 class (𝑛↑(𝑥 − 1))
14	13, 9, 10	co 7387	. . . . . 6 class ((𝑛↑(𝑥 − 1)) − 1)
15		cdvds 16222	. . . . . 6 class ∥
16	5, 14, 15	wbr 5107	. . . . 5 wff 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)
17	7, 16	wa 395	. . . 4 wff (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))
18		c4 12243	. . . . 5 class 4
19		cuz 12793	. . . . 5 class ℤ≥
20	18, 19	cfv 6511	. . . 4 class (ℤ≥‘4)
21	17, 4, 20	crab 3405	. . 3 class {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}
22	2, 3, 21	cmpt 5188	. 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  47727  fpprbasnn  47730