Theorem fpprbasnn 47661

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 47657	. . . 4 ⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))})
3	2	fvmptndm 7013	. . 3 ⊢ (¬ 𝑁 ∈ ℕ → ( FPPr ‘𝑁) = ∅)
4		eleq2 2822	. . . 4 ⊢ (( FPPr ‘𝑁) = ∅ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ 𝑋 ∈ ∅))
5		noel 4311	. . . . 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 1539   ∈ wcel 2107   ∉ wnel 3035  {crab 3413  ∅c0 4306   class class class wbr 5116  ‘cfv 6527  (class class class)co 7399  1c1 11122   − cmin 11458  ℕcn 12232  4c4 12289  ℤ_≥cuz 12844  ↑cexp 14068   ∥ cdvds 16257  ℙcprime 16675   FPPr cfppr 47656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5263  ax-nul 5273  ax-pr 5399

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-ss 3941  df-nul 4307  df-if 4499  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-br 5117  df-opab 5179  df-mpt 5199  df-dm 5661  df-iota 6480  df-fv 6535  df-fppr 47657

This theorem is referenced by:  fpprnn  47662  fpprwppr  47671  fpprwpprb  47672