Definition df-fppr 48490

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 48489	. 2 class FPPr
2		vn	. . 3 setvar 𝑛
3		cn 12228	. . 3 class ℕ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1569	. . . . . 6 class 𝑥
6		cprime 16724	. . . . . 6 class ℙ
7	5, 6	wnel 3064	. . . . 5 wff 𝑥 ∉ ℙ
8	2	cv 1569	. . . . . . . 8 class 𝑛
9		c1 11096	. . . . . . . . 9 class 1
10		cmin 11436	. . . . . . . . 9 class −
11	5, 9, 10	co 7410	. . . . . . . 8 class (𝑥 − 1)
12		cexp 14093	. . . . . . . 8 class ↑
13	8, 11, 12	co 7410	. . . . . . 7 class (𝑛↑(𝑥 − 1))
14	13, 9, 10	co 7410	. . . . . 6 class ((𝑛↑(𝑥 − 1)) − 1)
15		cdvds 16305	. . . . . 6 class ∥
16	5, 14, 15	wbr 5109	. . . . 5 wff 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)
17	7, 16	wa 400	. . . 4 wff (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))
18		c4 12292	. . . . 5 class 4
19		cuz 12857	. . . . 5 class ℤ≥
20	18, 19	cfv 6536	. . . 4 class (ℤ≥‘4)
21	17, 4, 20	crab 3416	. . 3 class {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}
22	2, 3, 21	cmpt 5192	. 2 class (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
23	1, 22	wceq 1570	1 wff FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})

This definition is referenced by:  fppr  48491  fpprbasnn  48494