Definition df-fppr 45177

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 45176	. 2 class FPPr
2		vn	. . 3 setvar 𝑛
3		cn 11973	. . 3 class ℕ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1538	. . . . . 6 class 𝑥
6		cprime 16376	. . . . . 6 class ℙ
7	5, 6	wnel 3049	. . . . 5 wff 𝑥 ∉ ℙ
8	2	cv 1538	. . . . . . . 8 class 𝑛
9		c1 10872	. . . . . . . . 9 class 1
10		cmin 11205	. . . . . . . . 9 class −
11	5, 9, 10	co 7275	. . . . . . . 8 class (𝑥 − 1)
12		cexp 13782	. . . . . . . 8 class ↑
13	8, 11, 12	co 7275	. . . . . . 7 class (𝑛↑(𝑥 − 1))
14	13, 9, 10	co 7275	. . . . . 6 class ((𝑛↑(𝑥 − 1)) − 1)
15		cdvds 15963	. . . . . 6 class ∥
16	5, 14, 15	wbr 5074	. . . . 5 wff 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)
17	7, 16	wa 396	. . . 4 wff (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))
18		c4 12030	. . . . 5 class 4
19		cuz 12582	. . . . 5 class ℤ≥
20	18, 19	cfv 6433	. . . 4 class (ℤ≥‘4)
21	17, 4, 20	crab 3068	. . 3 class {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}
22	2, 3, 21	cmpt 5157	. 2 class (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
23	1, 22	wceq 1539	1 wff FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})

This definition is referenced by:  fppr  45178  fpprbasnn  45181