Theorem fpprbasnn 48220

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 48216	. . . 4 ⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
3	2	fvmptndm 6967	. . 3 ⊢ (¬ 𝑁 ∈ ℕ → ( FPPr ‘𝑁) = ∅)
4		eleq2 2828	. . . 4 ⊢ (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ 𝑋 ∈ ∅))
5		noel 4266	. . . . 5 ⊢ ¬ 𝑋 ∈ ∅
6	5	pm2.21i 119	. . . 4 ⊢ (𝑋 ∈ ∅ → 𝑁 ∈ ℕ)
7	4, 6	biimtrdi 254	. . 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 183	1 ⊢ (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ∉ wnel 3038  {crab 3391  ∅c0 4261   class class class wbr 5072  ‘cfv 6485  (class class class)co 7356  1c1 11030   − cmin 11368  ℕcn 12165  4c4 12229  ℤ_≥cuz 12779  ↑cexp 14014   ∥ cdvds 16212  ℙcprime 16631   FPPr cfppr 48215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-dm 5628  df-iota 6441  df-fv 6493  df-fppr 48216

This theorem is referenced by:  fpprnn  48221  fpprwppr  48230  fpprwpprb  48231