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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 7412 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛↑(𝑥 − 1)) = (𝑁↑(𝑥 − 1))) | |
2 | 1 | oveq1d 7420 | . . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑛↑(𝑥 − 1)) − 1) = ((𝑁↑(𝑥 − 1)) − 1)) |
3 | 2 | breq2d 5159 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1) ↔ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))) |
4 | 3 | anbi2d 629 | . . 3 ⊢ (𝑛 = 𝑁 → ((𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)) ↔ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1)))) |
5 | 4 | rabbidv 3440 | . 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 46379 | . 2 ⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}) | |
7 | fvex 6901 | . . 3 ⊢ (ℤ≥‘4) ∈ V | |
8 | 7 | rabex 5331 | . 2 ⊢ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))} ∈ V |
9 | 5, 6, 8 | fvmpt 6995 | 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 396 = wceq 1541 ∈ wcel 2106 ∉ wnel 3046 {crab 3432 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 1c1 11107 − cmin 11440 ℕcn 12208 4c4 12265 ℤ≥cuz 12818 ↑cexp 14023 ∥ cdvds 16193 ℙcprime 16604 FPPr cfppr 46378 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-fppr 46379 |
This theorem is referenced by: fpprmod 46381 |
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