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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 43384) 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 11583 | . . . 4 ⊢ 9 ∈ ℕ | |
2 | 1 | elexi 3456 | . . 3 ⊢ 9 ∈ V |
3 | eleq1 2870 | . . . 4 ⊢ (𝑝 = 9 → (𝑝 ∈ (ℤ≥‘3) ↔ 9 ∈ (ℤ≥‘3))) | |
4 | oveq2 7024 | . . . . . . . 8 ⊢ (𝑝 = 9 → (8↑𝑝) = (8↑9)) | |
5 | id 22 | . . . . . . . 8 ⊢ (𝑝 = 9 → 𝑝 = 9) | |
6 | 4, 5 | oveq12d 7034 | . . . . . . 7 ⊢ (𝑝 = 9 → ((8↑𝑝) mod 𝑝) = ((8↑9) mod 9)) |
7 | oveq2 7024 | . . . . . . 7 ⊢ (𝑝 = 9 → (8 mod 𝑝) = (8 mod 9)) | |
8 | 6, 7 | eqeq12d 2810 | . . . . . 6 ⊢ (𝑝 = 9 → (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) ↔ ((8↑9) mod 9) = (8 mod 9))) |
9 | eleq1 2870 | . . . . . 6 ⊢ (𝑝 = 9 → (𝑝 ∈ ℙ ↔ 9 ∈ ℙ)) | |
10 | 8, 9 | imbi12d 346 | . . . . 5 ⊢ (𝑝 = 9 → ((((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ) ↔ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ))) |
11 | 10 | notbid 319 | . . . 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 630 | . . 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 11864 | . . . . 5 ⊢ 3 ∈ ℤ | |
14 | 1 | nnzi 11855 | . . . . 5 ⊢ 9 ∈ ℤ |
15 | 3re 11565 | . . . . . 6 ⊢ 3 ∈ ℝ | |
16 | 9re 11584 | . . . . . 6 ⊢ 9 ∈ ℝ | |
17 | 3lt9 11689 | . . . . . 6 ⊢ 3 < 9 | |
18 | 15, 16, 17 | ltleii 10610 | . . . . 5 ⊢ 3 ≤ 9 |
19 | eluz2 12099 | . . . . 5 ⊢ (9 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 9 ∈ ℤ ∧ 3 ≤ 9)) | |
20 | 13, 14, 18, 19 | mpbir3an 1334 | . . . 4 ⊢ 9 ∈ (ℤ≥‘3) |
21 | 8nn 11580 | . . . . . . 7 ⊢ 8 ∈ ℕ | |
22 | 8nn0 11768 | . . . . . . 7 ⊢ 8 ∈ ℕ0 | |
23 | 0z 11840 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
24 | 1nn0 11761 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
25 | 8exp8mod9 43383 | . . . . . . . 8 ⊢ ((8↑8) mod 9) = 1 | |
26 | 1re 10487 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
27 | nnrp 12250 | . . . . . . . . . 10 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
28 | 1, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ 9 ∈ ℝ+ |
29 | 0le1 11011 | . . . . . . . . 9 ⊢ 0 ≤ 1 | |
30 | 1lt9 11691 | . . . . . . . . 9 ⊢ 1 < 9 | |
31 | modid 13114 | . . . . . . . . 9 ⊢ (((1 ∈ ℝ ∧ 9 ∈ ℝ+) ∧ (0 ≤ 1 ∧ 1 < 9)) → (1 mod 9) = 1) | |
32 | 26, 28, 29, 30, 31 | mp4an 689 | . . . . . . . 8 ⊢ (1 mod 9) = 1 |
33 | 25, 32 | eqtr4i 2822 | . . . . . . 7 ⊢ ((8↑8) mod 9) = (1 mod 9) |
34 | 8p1e9 11635 | . . . . . . 7 ⊢ (8 + 1) = 9 | |
35 | 8cn 11582 | . . . . . . . . 9 ⊢ 8 ∈ ℂ | |
36 | 35 | addid2i 10675 | . . . . . . . 8 ⊢ (0 + 8) = 8 |
37 | 9cn 11585 | . . . . . . . . . 10 ⊢ 9 ∈ ℂ | |
38 | 37 | mul02i 10676 | . . . . . . . . 9 ⊢ (0 · 9) = 0 |
39 | 38 | oveq1i 7026 | . . . . . . . 8 ⊢ ((0 · 9) + 8) = (0 + 8) |
40 | 35 | mulid2i 10492 | . . . . . . . 8 ⊢ (1 · 8) = 8 |
41 | 36, 39, 40 | 3eqtr4i 2829 | . . . . . . 7 ⊢ ((0 · 9) + 8) = (1 · 8) |
42 | 1, 21, 22, 23, 24, 22, 33, 34, 41 | modxp1i 16235 | . . . . . 6 ⊢ ((8↑9) mod 9) = (8 mod 9) |
43 | 9nprm 16275 | . . . . . 6 ⊢ ¬ 9 ∈ ℙ | |
44 | 42, 43 | pm3.2i 471 | . . . . 5 ⊢ (((8↑9) mod 9) = (8 mod 9) ∧ ¬ 9 ∈ ℙ) |
45 | annim 404 | . . . . 5 ⊢ ((((8↑9) mod 9) = (8 mod 9) ∧ ¬ 9 ∈ ℙ) ↔ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ)) | |
46 | 44, 45 | mpbi 231 | . . . 4 ⊢ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ) |
47 | 20, 46 | pm3.2i 471 | . . 3 ⊢ (9 ∈ (ℤ≥‘3) ∧ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ)) |
48 | 2, 12, 47 | ceqsexv2d 3485 | . 2 ⊢ ∃𝑝(𝑝 ∈ (ℤ≥‘3) ∧ ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ)) |
49 | df-rex 3111 | . 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 232 | 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 396 = wceq 1522 ∃wex 1761 ∈ wcel 2081 ∃wrex 3106 class class class wbr 4962 ‘cfv 6225 (class class class)co 7016 ℝcr 10382 0cc0 10383 1c1 10384 + caddc 10386 · cmul 10388 < clt 10521 ≤ cle 10522 ℕcn 11486 3c3 11541 8c8 11546 9c9 11547 ℤcz 11829 ℤ≥cuz 12093 ℝ+crp 12239 mod cmo 13087 ↑cexp 13279 ℙcprime 15844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-rp 12240 df-fl 13012 df-mod 13088 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-dvds 15441 df-prm 15845 |
This theorem is referenced by: nfermltlrev 43391 |
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