Definition df-fppr 47599

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 47598	. 2 class FPPr
2		vn	. . 3 setvar 𝑛
3		cn 12293	. . 3 class ℕ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1536	. . . . . 6 class 𝑥
6		cprime 16718	. . . . . 6 class ℙ
7	5, 6	wnel 3052	. . . . 5 wff 𝑥 ∉ ℙ
8	2	cv 1536	. . . . . . . 8 class 𝑛
9		c1 11185	. . . . . . . . 9 class 1
10		cmin 11520	. . . . . . . . 9 class −
11	5, 9, 10	co 7448	. . . . . . . 8 class (𝑥 − 1)
12		cexp 14112	. . . . . . . 8 class ↑
13	8, 11, 12	co 7448	. . . . . . 7 class (𝑛↑(𝑥 − 1))
14	13, 9, 10	co 7448	. . . . . 6 class ((𝑛↑(𝑥 − 1)) − 1)
15		cdvds 16302	. . . . . 6 class ∥
16	5, 14, 15	wbr 5166	. . . . 5 wff 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)
17	7, 16	wa 395	. . . 4 wff (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))
18		c4 12350	. . . . 5 class 4
19		cuz 12903	. . . . 5 class ℤ≥
20	18, 19	cfv 6573	. . . 4 class (ℤ≥‘4)
21	17, 4, 20	crab 3443	. . 3 class {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}
22	2, 3, 21	cmpt 5249	. 2 class (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
23	1, 22	wceq 1537	1 wff FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})

This definition is referenced by:  fppr  47600  fpprbasnn  47603