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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 7369 | . . . . . 6 ⊢ (𝑛 = 𝑁 → (𝑛↑(𝑥 − 1)) = (𝑁↑(𝑥 − 1))) | |
2 | 1 | oveq1d 7377 | . . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑛↑(𝑥 − 1)) − 1) = ((𝑁↑(𝑥 − 1)) − 1)) |
3 | 2 | breq2d 5122 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1) ↔ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))) |
4 | 3 | anbi2d 630 | . . 3 ⊢ (𝑛 = 𝑁 → ((𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1)) ↔ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1)))) |
5 | 4 | rabbidv 3418 | . 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 45991 | . 2 ⊢ FPPr = (𝑛 ∈ ℕ ↦ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑛↑(𝑥 − 1)) − 1))}) | |
7 | fvex 6860 | . . 3 ⊢ (ℤ≥‘4) ∈ V | |
8 | 7 | rabex 5294 | . 2 ⊢ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))} ∈ V |
9 | 5, 6, 8 | fvmpt 6953 | 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 397 = wceq 1542 ∈ wcel 2107 ∉ wnel 3050 {crab 3410 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 1c1 11059 − cmin 11392 ℕcn 12160 4c4 12217 ℤ≥cuz 12770 ↑cexp 13974 ∥ cdvds 16143 ℙcprime 16554 FPPr cfppr 45990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-fppr 45991 |
This theorem is referenced by: fpprmod 45993 |
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