Theorem fpprbasnn 48086

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 48082	. . . 4 ⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
3	2	fvmptndm 6981	. . 3 ⊢ (¬ 𝑁 ∈ ℕ → ( FPPr ‘𝑁) = ∅)
4		eleq2 2826	. . . 4 ⊢ (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ 𝑋 ∈ ∅))
5		noel 4292	. . . . 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 1542   ∈ wcel 2114   ∉ wnel 3037  {crab 3401  ∅c0 4287   class class class wbr 5100  ‘cfv 6500  (class class class)co 7368  1c1 11039   − cmin 11376  ℕcn 12157  4c4 12214  ℤ_≥cuz 12763  ↑cexp 13996   ∥ cdvds 16191  ℙcprime 16610   FPPr cfppr 48081

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-dm 5642  df-iota 6456  df-fv 6508  df-fppr 48082

This theorem is referenced by:  fpprnn  48087  fpprwppr  48096  fpprwpprb  48097