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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 44862) 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 11928 | . . . 4 ⊢ 9 ∈ ℕ | |
2 | 1 | elexi 3427 | . . 3 ⊢ 9 ∈ V |
3 | eleq1 2825 | . . . 4 ⊢ (𝑝 = 9 → (𝑝 ∈ (ℤ≥‘3) ↔ 9 ∈ (ℤ≥‘3))) | |
4 | oveq2 7221 | . . . . . . . 8 ⊢ (𝑝 = 9 → (8↑𝑝) = (8↑9)) | |
5 | id 22 | . . . . . . . 8 ⊢ (𝑝 = 9 → 𝑝 = 9) | |
6 | 4, 5 | oveq12d 7231 | . . . . . . 7 ⊢ (𝑝 = 9 → ((8↑𝑝) mod 𝑝) = ((8↑9) mod 9)) |
7 | oveq2 7221 | . . . . . . 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 2825 | . . . . . 6 ⊢ (𝑝 = 9 → (𝑝 ∈ ℙ ↔ 9 ∈ ℙ)) | |
10 | 8, 9 | imbi12d 348 | . . . . 5 ⊢ (𝑝 = 9 → ((((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ) ↔ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ))) |
11 | 10 | notbid 321 | . . . 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 634 | . . 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 12210 | . . . . 5 ⊢ 3 ∈ ℤ | |
14 | 1 | nnzi 12201 | . . . . 5 ⊢ 9 ∈ ℤ |
15 | 3re 11910 | . . . . . 6 ⊢ 3 ∈ ℝ | |
16 | 9re 11929 | . . . . . 6 ⊢ 9 ∈ ℝ | |
17 | 3lt9 12034 | . . . . . 6 ⊢ 3 < 9 | |
18 | 15, 16, 17 | ltleii 10955 | . . . . 5 ⊢ 3 ≤ 9 |
19 | eluz2 12444 | . . . . 5 ⊢ (9 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 9 ∈ ℤ ∧ 3 ≤ 9)) | |
20 | 13, 14, 18, 19 | mpbir3an 1343 | . . . 4 ⊢ 9 ∈ (ℤ≥‘3) |
21 | 8nn 11925 | . . . . . . 7 ⊢ 8 ∈ ℕ | |
22 | 8nn0 12113 | . . . . . . 7 ⊢ 8 ∈ ℕ0 | |
23 | 0z 12187 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
24 | 1nn0 12106 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
25 | 8exp8mod9 44861 | . . . . . . . 8 ⊢ ((8↑8) mod 9) = 1 | |
26 | 1re 10833 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
27 | nnrp 12597 | . . . . . . . . . 10 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
28 | 1, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ 9 ∈ ℝ+ |
29 | 0le1 11355 | . . . . . . . . 9 ⊢ 0 ≤ 1 | |
30 | 1lt9 12036 | . . . . . . . . 9 ⊢ 1 < 9 | |
31 | modid 13469 | . . . . . . . . 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 11980 | . . . . . . 7 ⊢ (8 + 1) = 9 | |
35 | 8cn 11927 | . . . . . . . . 9 ⊢ 8 ∈ ℂ | |
36 | 35 | addid2i 11020 | . . . . . . . 8 ⊢ (0 + 8) = 8 |
37 | 9cn 11930 | . . . . . . . . . 10 ⊢ 9 ∈ ℂ | |
38 | 37 | mul02i 11021 | . . . . . . . . 9 ⊢ (0 · 9) = 0 |
39 | 38 | oveq1i 7223 | . . . . . . . 8 ⊢ ((0 · 9) + 8) = (0 + 8) |
40 | 35 | mulid2i 10838 | . . . . . . . 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 16623 | . . . . . 6 ⊢ ((8↑9) mod 9) = (8 mod 9) |
43 | 9nprm 16666 | . . . . . 6 ⊢ ¬ 9 ∈ ℙ | |
44 | 42, 43 | pm3.2i 474 | . . . . 5 ⊢ (((8↑9) mod 9) = (8 mod 9) ∧ ¬ 9 ∈ ℙ) |
45 | annim 407 | . . . . 5 ⊢ ((((8↑9) mod 9) = (8 mod 9) ∧ ¬ 9 ∈ ℙ) ↔ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ)) | |
46 | 44, 45 | mpbi 233 | . . . 4 ⊢ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ) |
47 | 20, 46 | pm3.2i 474 | . . 3 ⊢ (9 ∈ (ℤ≥‘3) ∧ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ)) |
48 | 2, 12, 47 | ceqsexv2d 3457 | . 2 ⊢ ∃𝑝(𝑝 ∈ (ℤ≥‘3) ∧ ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ)) |
49 | df-rex 3067 | . 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 234 | 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 399 = wceq 1543 ∃wex 1787 ∈ wcel 2110 ∃wrex 3062 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 · cmul 10734 < clt 10867 ≤ cle 10868 ℕcn 11830 3c3 11886 8c8 11891 9c9 11892 ℤcz 12176 ℤ≥cuz 12438 ℝ+crp 12586 mod cmo 13442 ↑cexp 13635 ℙcprime 16228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-rp 12587 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-dvds 15816 df-prm 16229 |
This theorem is referenced by: nfermltlrev 44869 |
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