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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nfermltl8rev | Structured version Visualization version GIF version | ||
| Description: Fermat's little theorem with base 8 reversed is not generally true: There is an integer 𝑝 (for example 9, see 9fppr8 47724) so that "𝑝 is prime" does not follow from 8↑𝑝≡8 (mod 𝑝). (Contributed by AV, 3-Jun-2023.) |
| Ref | Expression |
|---|---|
| nfermltl8rev | ⊢ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 9nn 12364 | . . . 4 ⊢ 9 ∈ ℕ | |
| 2 | 1 | elexi 3503 | . . 3 ⊢ 9 ∈ V |
| 3 | eleq1 2829 | . . . 4 ⊢ (𝑝 = 9 → (𝑝 ∈ (ℤ≥‘3) ↔ 9 ∈ (ℤ≥‘3))) | |
| 4 | oveq2 7439 | . . . . . . . 8 ⊢ (𝑝 = 9 → (8↑𝑝) = (8↑9)) | |
| 5 | id 22 | . . . . . . . 8 ⊢ (𝑝 = 9 → 𝑝 = 9) | |
| 6 | 4, 5 | oveq12d 7449 | . . . . . . 7 ⊢ (𝑝 = 9 → ((8↑𝑝) mod 𝑝) = ((8↑9) mod 9)) |
| 7 | oveq2 7439 | . . . . . . 7 ⊢ (𝑝 = 9 → (8 mod 𝑝) = (8 mod 9)) | |
| 8 | 6, 7 | eqeq12d 2753 | . . . . . 6 ⊢ (𝑝 = 9 → (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) ↔ ((8↑9) mod 9) = (8 mod 9))) |
| 9 | eleq1 2829 | . . . . . 6 ⊢ (𝑝 = 9 → (𝑝 ∈ ℙ ↔ 9 ∈ ℙ)) | |
| 10 | 8, 9 | imbi12d 344 | . . . . 5 ⊢ (𝑝 = 9 → ((((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ) ↔ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ))) |
| 11 | 10 | notbid 318 | . . . 4 ⊢ (𝑝 = 9 → (¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ) ↔ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ))) |
| 12 | 3, 11 | anbi12d 632 | . . 3 ⊢ (𝑝 = 9 → ((𝑝 ∈ (ℤ≥‘3) ∧ ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ)) ↔ (9 ∈ (ℤ≥‘3) ∧ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ)))) |
| 13 | 3z 12650 | . . . . 5 ⊢ 3 ∈ ℤ | |
| 14 | 1 | nnzi 12641 | . . . . 5 ⊢ 9 ∈ ℤ |
| 15 | 3re 12346 | . . . . . 6 ⊢ 3 ∈ ℝ | |
| 16 | 9re 12365 | . . . . . 6 ⊢ 9 ∈ ℝ | |
| 17 | 3lt9 12470 | . . . . . 6 ⊢ 3 < 9 | |
| 18 | 15, 16, 17 | ltleii 11384 | . . . . 5 ⊢ 3 ≤ 9 |
| 19 | eluz2 12884 | . . . . 5 ⊢ (9 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 9 ∈ ℤ ∧ 3 ≤ 9)) | |
| 20 | 13, 14, 18, 19 | mpbir3an 1342 | . . . 4 ⊢ 9 ∈ (ℤ≥‘3) |
| 21 | 8nn 12361 | . . . . . . 7 ⊢ 8 ∈ ℕ | |
| 22 | 8nn0 12549 | . . . . . . 7 ⊢ 8 ∈ ℕ0 | |
| 23 | 0z 12624 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
| 24 | 1nn0 12542 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
| 25 | 8exp8mod9 47723 | . . . . . . . 8 ⊢ ((8↑8) mod 9) = 1 | |
| 26 | 1re 11261 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
| 27 | nnrp 13046 | . . . . . . . . . 10 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
| 28 | 1, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ 9 ∈ ℝ+ |
| 29 | 0le1 11786 | . . . . . . . . 9 ⊢ 0 ≤ 1 | |
| 30 | 1lt9 12472 | . . . . . . . . 9 ⊢ 1 < 9 | |
| 31 | modid 13936 | . . . . . . . . 9 ⊢ (((1 ∈ ℝ ∧ 9 ∈ ℝ+) ∧ (0 ≤ 1 ∧ 1 < 9)) → (1 mod 9) = 1) | |
| 32 | 26, 28, 29, 30, 31 | mp4an 693 | . . . . . . . 8 ⊢ (1 mod 9) = 1 |
| 33 | 25, 32 | eqtr4i 2768 | . . . . . . 7 ⊢ ((8↑8) mod 9) = (1 mod 9) |
| 34 | 8p1e9 12416 | . . . . . . 7 ⊢ (8 + 1) = 9 | |
| 35 | 8cn 12363 | . . . . . . . . 9 ⊢ 8 ∈ ℂ | |
| 36 | 35 | addlidi 11449 | . . . . . . . 8 ⊢ (0 + 8) = 8 |
| 37 | 9cn 12366 | . . . . . . . . . 10 ⊢ 9 ∈ ℂ | |
| 38 | 37 | mul02i 11450 | . . . . . . . . 9 ⊢ (0 · 9) = 0 |
| 39 | 38 | oveq1i 7441 | . . . . . . . 8 ⊢ ((0 · 9) + 8) = (0 + 8) |
| 40 | 35 | mullidi 11266 | . . . . . . . 8 ⊢ (1 · 8) = 8 |
| 41 | 36, 39, 40 | 3eqtr4i 2775 | . . . . . . 7 ⊢ ((0 · 9) + 8) = (1 · 8) |
| 42 | 1, 21, 22, 23, 24, 22, 33, 34, 41 | modxp1i 17108 | . . . . . 6 ⊢ ((8↑9) mod 9) = (8 mod 9) |
| 43 | 9nprm 17150 | . . . . . 6 ⊢ ¬ 9 ∈ ℙ | |
| 44 | 42, 43 | pm3.2i 470 | . . . . 5 ⊢ (((8↑9) mod 9) = (8 mod 9) ∧ ¬ 9 ∈ ℙ) |
| 45 | annim 403 | . . . . 5 ⊢ ((((8↑9) mod 9) = (8 mod 9) ∧ ¬ 9 ∈ ℙ) ↔ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ)) | |
| 46 | 44, 45 | mpbi 230 | . . . 4 ⊢ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ) |
| 47 | 20, 46 | pm3.2i 470 | . . 3 ⊢ (9 ∈ (ℤ≥‘3) ∧ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ)) |
| 48 | 2, 12, 47 | ceqsexv2d 3533 | . 2 ⊢ ∃𝑝(𝑝 ∈ (ℤ≥‘3) ∧ ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ)) |
| 49 | df-rex 3071 | . 2 ⊢ (∃𝑝 ∈ (ℤ≥‘3) ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ) ↔ ∃𝑝(𝑝 ∈ (ℤ≥‘3) ∧ ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ))) | |
| 50 | 48, 49 | mpbir 231 | 1 ⊢ ∃𝑝 ∈ (ℤ≥‘3) ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 ℕcn 12266 3c3 12322 8c8 12327 9c9 12328 ℤcz 12613 ℤ≥cuz 12878 ℝ+crp 13034 mod cmo 13909 ↑cexp 14102 ℙcprime 16708 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-prm 16709 |
| This theorem is referenced by: nfermltlrev 47731 |
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