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Theorem fpprbasnn 44110
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 44106 . . . 4 FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
32fvmptndm 6786 . . 3 𝑁 ∈ ℕ → ( FPPr ‘𝑁) = ∅)
4 eleq2 2904 . . . 4 (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ 𝑋 ∈ ∅))
5 noel 4279 . . . . 5 ¬ 𝑋 ∈ ∅
65pm2.21i 119 . . . 4 (𝑋 ∈ ∅ → 𝑁 ∈ ℕ)
74, 6syl6bi 256 . . 3 (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ))
83, 7syl 17 . 2 𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ))
91, 8pm2.61i 185 1 (𝑋 ∈ ( FPPr ‘𝑁) → 𝑁 ∈ ℕ)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wnel 3118  {crab 3137  c0 4275   class class class wbr 5052  cfv 6343  (class class class)co 7145  1c1 10530  cmin 10862  cn 11630  4c4 11687  cuz 12236  cexp 13430  cdvds 15603  cprime 16009   FPPr cfppr 44105
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-dm 5552  df-iota 6302  df-fv 6351  df-fppr 44106
This theorem is referenced by:  fpprnn  44111  fpprwppr  44120  fpprwpprb  44121
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