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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nfermltlrev | Structured version Visualization version GIF version | ||
| Description: Fermat's little theorem reversed is not generally true: There are integers 𝑎 and 𝑝 so that "𝑝 is prime" does not follow from 𝑎↑𝑝≡𝑎 (mod 𝑝). (Contributed by AV, 3-Jun-2023.) |
| Ref | Expression |
|---|---|
| nfermltlrev | ⊢ ∃𝑎 ∈ ℤ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 8nn 12314 | . . . 4 ⊢ 8 ∈ ℕ | |
| 2 | 1 | elexi 3477 | . . 3 ⊢ 8 ∈ V |
| 3 | eleq1 2851 | . . . 4 ⊢ (𝑎 = 8 → (𝑎 ∈ ℤ ↔ 8 ∈ ℤ)) | |
| 4 | oveq1 7404 | . . . . . . . . 9 ⊢ (𝑎 = 8 → (𝑎↑𝑝) = (8↑𝑝)) | |
| 5 | 4 | oveq1d 7412 | . . . . . . . 8 ⊢ (𝑎 = 8 → ((𝑎↑𝑝) mod 𝑝) = ((8↑𝑝) mod 𝑝)) |
| 6 | oveq1 7404 | . . . . . . . 8 ⊢ (𝑎 = 8 → (𝑎 mod 𝑝) = (8 mod 𝑝)) | |
| 7 | 5, 6 | eqeq12d 2779 | . . . . . . 7 ⊢ (𝑎 = 8 → (((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) ↔ ((8↑𝑝) mod 𝑝) = (8 mod 𝑝))) |
| 8 | 7 | imbi1d 343 | . . . . . 6 ⊢ (𝑎 = 8 → ((((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ) ↔ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ))) |
| 9 | 8 | notbid 320 | . . . . 5 ⊢ (𝑎 = 8 → (¬ (((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ) ↔ ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ))) |
| 10 | 9 | rexbidv 3187 | . . . 4 ⊢ (𝑎 = 8 → (∃𝑝 ∈ (ℤ≥‘3) ¬ (((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ) ↔ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ))) |
| 11 | 3, 10 | anbi12d 641 | . . 3 ⊢ (𝑎 = 8 → ((𝑎 ∈ ℤ ∧ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ)) ↔ (8 ∈ ℤ ∧ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ)))) |
| 12 | 1 | nnzi 12596 | . . . 4 ⊢ 8 ∈ ℤ |
| 13 | nfermltl8rev 48365 | . . . 4 ⊢ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ) | |
| 14 | 12, 13 | pm3.2i 474 | . . 3 ⊢ (8 ∈ ℤ ∧ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ)) |
| 15 | 2, 11, 14 | ceqsexv2d 3504 | . 2 ⊢ ∃𝑎(𝑎 ∈ ℤ ∧ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ)) |
| 16 | df-rex 3088 | . 2 ⊢ (∃𝑎 ∈ ℤ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ) ↔ ∃𝑎(𝑎 ∈ ℤ ∧ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ))) | |
| 17 | 15, 16 | mpbir 233 | 1 ⊢ ∃𝑎 ∈ ℤ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((𝑎↑𝑝) mod 𝑝) = (𝑎 mod 𝑝) → 𝑝 ∈ ℙ) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1561 ∃wex 1800 ∈ wcel 2143 ∃wrex 3087 ‘cfv 6522 (class class class)co 7397 ℕcn 12211 3c3 12274 8c8 12279 ℤcz 12569 ℤ≥cuz 12840 mod cmo 13880 ↑cexp 14075 ℙcprime 16706 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 |
| This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-om 7848 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-1o 8438 df-2o 8439 df-er 8679 df-en 8929 df-dom 8930 df-sdom 8931 df-fin 8932 df-sup 9389 df-inf 9390 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-div 11846 df-nn 12212 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-7 12286 df-8 12287 df-9 12288 df-n0 12483 df-z 12570 df-dec 12690 df-uz 12841 df-rp 12995 df-fl 13803 df-mod 13881 df-seq 14016 df-exp 14076 df-cj 15127 df-re 15128 df-im 15129 df-sqrt 15263 df-abs 15264 df-dvds 16288 df-prm 16707 |
| This theorem is referenced by: (None) |
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