Theorem fpprbasnn 46383

Description: The base of a Fermat pseudoprime is a positive integer. (Contributed by AV, 30-May-2023.)

Assertion

Ref	Expression
fpprbasnn	⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)

Proof of Theorem fpprbasnn

Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1 6	. 2 ⊢ (𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ))
2		df-fppr 46379	. . . 4 ⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
3	2	fvmptndm 7025	. . 3 ⊢ (¬ 𝑁 ∈ ℕ → ( FPPr ‘𝑁) = ∅)
4		eleq2 2822	. . . 4 ⊢ (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ 𝑋 ∈ ∅))
5		noel 4329	. . . . 5 ⊢ ¬ 𝑋 ∈ ∅
6	5	pm2.21i 119	. . . 4 ⊢ (𝑋 ∈ ∅ → 𝑁 ∈ ℕ)
7	4, 6	syl6bi 252	. . 3 ⊢ (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ))
8	3, 7	syl 17	. 2 ⊢ (¬ 𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ))
9	1, 8	pm2.61i 182	1 ⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ∉ wnel 3046  {crab 3432  ∅c0 4321   class class class wbr 5147  ‘cfv 6540  (class class class)co 7405  1c1 11107   − cmin 11440  ℕcn 12208  4c4 12265  ℤ_≥cuz 12818  ↑cexp 14023   ∥ cdvds 16193  ℙcprime 16604   FPPr cfppr 46378

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-dm 5685  df-iota 6492  df-fv 6548  df-fppr 46379

This theorem is referenced by:  fpprnn  46384  fpprwppr  46393  fpprwpprb  46394