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Mirrors > Home > MPE Home > Th. List > Mathboxes > fpprel | Structured version Visualization version GIF version |
Description: A Fermat pseudoprime to the base 𝑁. (Contributed by AV, 30-May-2023.) |
Ref | Expression |
---|---|
fpprel | ⊢ (𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ (𝑋 ∈ (ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((𝑁↑(𝑋 − 1)) mod 𝑋) = 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fpprmod 46948 | . . . 4 ⊢ (𝑁 ∈ ℕ → ( FPPr ‘𝑁) = {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1)}) | |
2 | 1 | eleq2d 2813 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ 𝑋 ∈ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1)})) |
3 | neleq1 3046 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 ∉ ℙ ↔ 𝑋 ∉ ℙ)) | |
4 | oveq1 7411 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 − 1) = (𝑋 − 1)) | |
5 | 4 | oveq2d 7420 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑁↑(𝑥 − 1)) = (𝑁↑(𝑋 − 1))) |
6 | id 22 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
7 | 5, 6 | oveq12d 7422 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑁↑(𝑥 − 1)) mod 𝑥) = ((𝑁↑(𝑋 − 1)) mod 𝑋)) |
8 | 7 | eqeq1d 2728 | . . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑁↑(𝑥 − 1)) mod 𝑥) = 1 ↔ ((𝑁↑(𝑋 − 1)) mod 𝑋) = 1)) |
9 | 3, 8 | anbi12d 630 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 ∉ ℙ ∧ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1) ↔ (𝑋 ∉ ℙ ∧ ((𝑁↑(𝑋 − 1)) mod 𝑋) = 1))) |
10 | 9 | elrab 3678 | . . 3 ⊢ (𝑋 ∈ {𝑥 ∈ (ℤ≥‘4) ∣ (𝑥 ∉ ℙ ∧ ((𝑁↑(𝑥 − 1)) mod 𝑥) = 1)} ↔ (𝑋 ∈ (ℤ≥‘4) ∧ (𝑋 ∉ ℙ ∧ ((𝑁↑(𝑋 − 1)) mod 𝑋) = 1))) |
11 | 2, 10 | bitrdi 287 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑋 ∈ ( FPPr ‘𝑁) ↔ (𝑋 ∈ (ℤ≥‘4) ∧ (𝑋 ∉ ℙ ∧ ((𝑁↑(𝑋 − 1)) mod 𝑋) = 1)))) |
12 | 3anass 1092 | . 2 ⊢ ((𝑋 ∈ (ℤ≥‘4) ∧ 𝑋 ∉ ℙ ∧ ((𝑁↑(𝑋 − 1)) mod 𝑋) = 1) ↔ (𝑋 ∈ (ℤ≥‘4) ∧ (𝑋 ∉ ℙ ∧ ((𝑁↑(𝑋 − 1)) mod 𝑋) = 1))) | |
13 | 11, 12 | bitr4di 289 | 1 ⊢ (𝑁 ∈ ℕ → (𝑋 ∈ ( 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 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∉ wnel 3040 {crab 3426 ‘cfv 6536 (class class class)co 7404 1c1 11110 − cmin 11445 ℕcn 12213 4c4 12270 ℤ≥cuz 12823 mod cmo 13837 ↑cexp 14030 ℙcprime 16613 FPPr cfppr 46945 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-n0 12474 df-z 12560 df-uz 12824 df-rp 12978 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-dvds 16203 df-fppr 46946 |
This theorem is referenced by: fpprnn 46951 fppr2odd 46952 341fppr2 46955 4fppr1 46956 9fppr8 46958 fpprwppr 46960 fpprwpprb 46961 fpprel2 46962 |
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