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Theorem fpprbasnn 46476
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.
StepHypRef Expression
1 ax-1 6 . 2 (𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ))
2 df-fppr 46472 . . . 4 FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
32fvmptndm 7028 . . 3 𝑁 ∈ ℕ → ( FPPr ‘𝑁) = ∅)
4 eleq2 2822 . . . 4 (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ 𝑋 ∈ ∅))
5 noel 4330 . . . . 5 ¬ 𝑋 ∈ ∅
65pm2.21i 119 . . . 4 (𝑋 ∈ ∅ → 𝑁 ∈ ℕ)
74, 6syl6bi 252 . . 3 (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ))
83, 7syl 17 . 2 𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ))
91, 8pm2.61i 182 1 (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wnel 3046  {crab 3432  c0 4322   class class class wbr 5148  cfv 6543  (class class class)co 7411  1c1 11113  cmin 11446  cn 12214  4c4 12271  cuz 12824  cexp 14029  cdvds 16199  cprime 16610   FPPr cfppr 46471
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 5299  ax-nul 5306  ax-pr 5427
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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-dm 5686  df-iota 6495  df-fv 6551  df-fppr 46472
This theorem is referenced by:  fpprnn  46477  fpprwppr  46486  fpprwpprb  46487
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