Theorem fpprbasnn 47717

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 47713	. . . 4 ⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
3	2	fvmptndm 6965	. . 3 ⊢ (¬ 𝑁 ∈ ℕ → ( FPPr ‘𝑁) = ∅)
4		eleq2 2817	. . . 4 ⊢ (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ 𝑋 ∈ ∅))
5		noel 4291	. . . . 5 ⊢ ¬ 𝑋 ∈ ∅
6	5	pm2.21i 119	. . . 4 ⊢ (𝑋 ∈ ∅ → 𝑁 ∈ ℕ)
7	4, 6	biimtrdi 253	. . 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 395   = wceq 1540   ∈ wcel 2109   ∉ wnel 3029  {crab 3396  ∅c0 4286   class class class wbr 5095  ‘cfv 6486  (class class class)co 7353  1c1 11029   − cmin 11365  ℕcn 12146  4c4 12203  ℤ_≥cuz 12753  ↑cexp 13986   ∥ cdvds 16181  ℙcprime 16600   FPPr cfppr 47712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-dm 5633  df-iota 6442  df-fv 6494  df-fppr 47713

This theorem is referenced by:  fpprnn  47718  fpprwppr  47727  fpprwpprb  47728