Definition df-fppr 45065

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 45064	. 2 class FPPr
2		vn	. . 3 setvar 𝑛
3		cn 11903	. . 3 class ℕ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1538	. . . . . 6 class 𝑥
6		cprime 16304	. . . . . 6 class ℙ
7	5, 6	wnel 3048	. . . . 5 wff 𝑥 ∉ ℙ
8	2	cv 1538	. . . . . . . 8 class 𝑛
9		c1 10803	. . . . . . . . 9 class 1
10		cmin 11135	. . . . . . . . 9 class −
11	5, 9, 10	co 7255	. . . . . . . 8 class (𝑥 − 1)
12		cexp 13710	. . . . . . . 8 class ↑
13	8, 11, 12	co 7255	. . . . . . 7 class (𝑛↑(𝑥 − 1))
14	13, 9, 10	co 7255	. . . . . 6 class ((𝑛↑(𝑥 − 1)) − 1)
15		cdvds 15891	. . . . . 6 class ∥
16	5, 14, 15	wbr 5070	. . . . 5 wff 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)
17	7, 16	wa 395	. . . 4 wff (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))
18		c4 11960	. . . . 5 class 4
19		cuz 12511	. . . . 5 class ℤ≥
20	18, 19	cfv 6418	. . . 4 class (ℤ≥‘4)
21	17, 4, 20	crab 3067	. . 3 class {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}
22	2, 3, 21	cmpt 5153	. 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  45066  fpprbasnn  45069