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Mirrors > Home > MPE Home > Th. List > Mathboxes > fppr | Structured version Visualization version GIF version |
Description: The set of Fermat pseudoprimes to the base 𝑁. (Contributed by AV, 29-May-2023.) |
Ref | Expression |
---|---|
fppr | ⊢ (𝑁 ∈ ℕ → ( FPPr ‘𝑁) = {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7420 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛↑(𝑥 − 1)) = (𝑁↑(𝑥 − 1))) | |
2 | 1 | oveq1d 7428 | . . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑛↑(𝑥 − 1)) − 1) = ((𝑁↑(𝑥 − 1)) − 1)) |
3 | 2 | breq2d 5161 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1) ↔ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))) |
4 | 3 | anbi2d 627 | . . 3 ⊢ (𝑛 = 𝑁 → ((𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)) ↔ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1)))) |
5 | 4 | rabbidv 3438 | . 2 ⊢ (𝑛 = 𝑁 → {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))} = {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))}) |
6 | df-fppr 46693 | . 2 ⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}) | |
7 | fvex 6905 | . . 3 ⊢ (ℤ≥‘4) ∈ V | |
8 | 7 | rabex 5333 | . 2 ⊢ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))} ∈ V |
9 | 5, 6, 8 | fvmpt 6999 | 1 ⊢ (𝑁 ∈ ℕ → ( FPPr ‘𝑁) = {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∉ wnel 3044 {crab 3430 class class class wbr 5149 ‘cfv 6544 (class class class)co 7413 1c1 11115 − cmin 11450 ℕcn 12218 4c4 12275 ℤ≥cuz 12828 ↑cexp 14033 ∥ cdvds 16203 ℙcprime 16614 FPPr cfppr 46692 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7416 df-fppr 46693 |
This theorem is referenced by: fpprmod 46695 |
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