Definition df-fppr 47735

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 47734	. 2 class FPPr
2		vn	. . 3 setvar 𝑛
3		cn 12117	. . 3 class ℕ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1540	. . . . . 6 class 𝑥
6		cprime 16574	. . . . . 6 class ℙ
7	5, 6	wnel 3030	. . . . 5 wff 𝑥 ∉ ℙ
8	2	cv 1540	. . . . . . . 8 class 𝑛
9		c1 10999	. . . . . . . . 9 class 1
10		cmin 11336	. . . . . . . . 9 class −
11	5, 9, 10	co 7341	. . . . . . . 8 class (𝑥 − 1)
12		cexp 13960	. . . . . . . 8 class ↑
13	8, 11, 12	co 7341	. . . . . . 7 class (𝑛↑(𝑥 − 1))
14	13, 9, 10	co 7341	. . . . . 6 class ((𝑛↑(𝑥 − 1)) − 1)
15		cdvds 16155	. . . . . 6 class ∥
16	5, 14, 15	wbr 5089	. . . . 5 wff 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)
17	7, 16	wa 395	. . . 4 wff (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))
18		c4 12174	. . . . 5 class 4
19		cuz 12724	. . . . 5 class ℤ≥
20	18, 19	cfv 6477	. . . 4 class (ℤ≥‘4)
21	17, 4, 20	crab 3393	. . 3 class {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}
22	2, 3, 21	cmpt 5170	. 2 class (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
23	1, 22	wceq 1541	1 wff FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})

This definition is referenced by:  fppr  47736  fpprbasnn  47739