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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 43961 | . 2 ⊢ (𝑁 ∈ ℕ → ( FPPr ‘𝑁) = {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))}) | |
2 | eluz4eluz2 12279 | . . . . . 6 ⊢ (𝑥 ∈ (ℤ≥‘4) → 𝑥 ∈ (ℤ≥‘2)) | |
3 | nnz 11998 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
4 | eluz4nn 12280 | . . . . . . . 8 ⊢ (𝑥 ∈ (ℤ≥‘4) → 𝑥 ∈ ℕ) | |
5 | nnm1nn0 11932 | . . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (𝑥 − 1) ∈ ℕ0) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝑥 ∈ (ℤ≥‘4) → (𝑥 − 1) ∈ ℕ0) |
7 | zexpcl 13441 | . . . . . . 7 ⊢ ((𝑁 ∈ ℤ ∧ (𝑥 − 1) ∈ ℕ0) → (𝑁↑(𝑥 − 1)) ∈ ℤ) | |
8 | 3, 6, 7 | syl2an 597 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℤ≥‘4)) → (𝑁↑(𝑥 − 1)) ∈ ℤ) |
9 | modm1div 15612 | . . . . . 6 ⊢ ((𝑥 ∈ (ℤ≥‘2) ∧ (𝑁↑(𝑥 − 1)) ∈ ℤ) → (((𝑁↑(𝑥 − 1)) mod 𝑥) = 1 ↔ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))) | |
10 | 2, 8, 9 | syl2an2 684 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℤ≥‘4)) → (((𝑁↑(𝑥 − 1)) mod 𝑥) = 1 ↔ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1))) |
11 | 10 | bicomd 225 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℤ≥‘4)) → (𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1) ↔ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1)) |
12 | 11 | anbi2d 630 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℤ≥‘4)) → ((𝑥 ∉ ℙ ∧ 𝑥 ∥ ((𝑁↑(𝑥 − 1)) − 1)) ↔ (𝑥 ∉ ℙ ∧ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1))) |
13 | 12 | rabbidva 3475 | . 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 2855 | 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∉ wnel 3122 {crab 3141 class class class wbr 5059 ‘cfv 6348 (class class class)co 7149 1c1 10531 − cmin 10863 ℕcn 11631 2c2 11686 4c4 11688 ℕ0cn0 11891 ℤcz 11975 ℤ≥cuz 12237 mod cmo 13234 ↑cexp 13426 ∥ cdvds 15600 ℙcprime 16008 FPPr cfppr 43959 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-sup 8899 df-inf 8900 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fl 13159 df-mod 13235 df-seq 13367 df-exp 13427 df-dvds 15601 df-fppr 43960 |
This theorem is referenced by: fpprel 43963 |
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