Definition df-fppr 47713

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 47712	. 2 class FPPr
2		vn	. . 3 setvar 𝑛
3		cn 12146	. . 3 class ℕ
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1539	. . . . . 6 class 𝑥
6		cprime 16600	. . . . . 6 class ℙ
7	5, 6	wnel 3029	. . . . 5 wff 𝑥 ∉ ℙ
8	2	cv 1539	. . . . . . . 8 class 𝑛
9		c1 11029	. . . . . . . . 9 class 1
10		cmin 11365	. . . . . . . . 9 class −
11	5, 9, 10	co 7353	. . . . . . . 8 class (𝑥 − 1)
12		cexp 13986	. . . . . . . 8 class ↑
13	8, 11, 12	co 7353	. . . . . . 7 class (𝑛↑(𝑥 − 1))
14	13, 9, 10	co 7353	. . . . . 6 class ((𝑛↑(𝑥 − 1)) − 1)
15		cdvds 16181	. . . . . 6 class ∥
16	5, 14, 15	wbr 5095	. . . . 5 wff 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)
17	7, 16	wa 395	. . . 4 wff (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))
18		c4 12203	. . . . 5 class 4
19		cuz 12753	. . . . 5 class ℤ≥
20	18, 19	cfv 6486	. . . 4 class (ℤ≥‘4)
21	17, 4, 20	crab 3396	. . 3 class {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}
22	2, 3, 21	cmpt 5176	. 2 class (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
23	1, 22	wceq 1540	1 wff FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})

This definition is referenced by:  fppr  47714  fpprbasnn  47717