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Mirrors > Home > MPE Home > Th. List > Mathboxes > fpprmod | Structured version Visualization version GIF version |
Description: The set of Fermat pseudoprimes to the base 𝑁, expressed by a modulo operation instead of the divisibility relation. (Contributed by AV, 30-May-2023.) |
Ref | Expression |
---|---|
fpprmod | ⊢ (𝑁 ∈ ℕ → ( FPPr ‘𝑁) = {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fppr 46904 | . 2 ⊢ (𝑁 ∈ ℕ → ( FPPr ‘𝑁) = {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))}) | |
2 | eluz4eluz2 12867 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘4) → 𝑥 ∈ (ℤ≥‘2)) | |
3 | nnz 12577 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
4 | eluz4nn 12868 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘4) → 𝑥 ∈ ℕ) | |
5 | nnm1nn0 12511 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (𝑥 − 1) ∈ ℕ0) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘4) → (𝑥 − 1) ∈ ℕ0) |
7 | zexpcl 14040 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ (𝑥 − 1) ∈ ℕ0) → (𝑁↑(𝑥 − 1)) ∈ ℤ) | |
8 | 3, 6, 7 | syl2an 595 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℤ≥‘4)) → (𝑁↑(𝑥 − 1)) ∈ ℤ) |
9 | modm1div 16208 | . . . . . 6 ⊢ ((𝑥 ∈ (ℤ≥‘2) ∧ (𝑁↑(𝑥 − 1)) ∈ ℤ) → (((𝑁↑(𝑥 − 1)) mod 𝑥) = 1 ↔ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))) | |
10 | 2, 8, 9 | syl2an2 683 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℤ≥‘4)) → (((𝑁↑(𝑥 − 1)) mod 𝑥) = 1 ↔ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))) |
11 | 10 | bicomd 222 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℤ≥‘4)) → (𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1) ↔ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1)) |
12 | 11 | anbi2d 628 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℤ≥‘4)) → ((𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1)) ↔ (𝑥 ∉ ℙ ∧ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1))) |
13 | 12 | rabbidva 3431 | . 2 ⊢ (𝑁 ∈ ℕ → {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))} = {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1)}) |
14 | 1, 13 | eqtrd 2764 | 1 ⊢ (𝑁 ∈ ℕ → ( FPPr ‘𝑁) = {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∉ wnel 3038 {crab 3424 class class class wbr 5139 ‘cfv 6534 (class class class)co 7402 1c1 11108 − cmin 11442 ℕcn 12210 2c2 12265 4c4 12267 ℕ0cn0 12470 ℤcz 12556 ℤ≥cuz 12820 mod cmo 13832 ↑cexp 14025 ∥ cdvds 16196 ℙcprime 16607 FPPr cfppr 46902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-n0 12471 df-z 12557 df-uz 12821 df-rp 12973 df-fl 13755 df-mod 13833 df-seq 13965 df-exp 14026 df-dvds 16197 df-fppr 46903 |
This theorem is referenced by: fpprel 46906 |
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