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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 47662) 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 12362 | . . . 4 ⊢ 9 ∈ ℕ | |
2 | 1 | elexi 3501 | . . 3 ⊢ 9 ∈ V |
3 | eleq1 2827 | . . . 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 2751 | . . . . . 6 ⊢ (𝑝 = 9 → (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) ↔ ((8↑9) mod 9) = (8 mod 9))) |
9 | eleq1 2827 | . . . . . 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 12648 | . . . . 5 ⊢ 3 ∈ ℤ | |
14 | 1 | nnzi 12639 | . . . . 5 ⊢ 9 ∈ ℤ |
15 | 3re 12344 | . . . . . 6 ⊢ 3 ∈ ℝ | |
16 | 9re 12363 | . . . . . 6 ⊢ 9 ∈ ℝ | |
17 | 3lt9 12468 | . . . . . 6 ⊢ 3 < 9 | |
18 | 15, 16, 17 | ltleii 11382 | . . . . 5 ⊢ 3 ≤ 9 |
19 | eluz2 12882 | . . . . 5 ⊢ (9 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 9 ∈ ℤ ∧ 3 ≤ 9)) | |
20 | 13, 14, 18, 19 | mpbir3an 1340 | . . . 4 ⊢ 9 ∈ (ℤ≥‘3) |
21 | 8nn 12359 | . . . . . . 7 ⊢ 8 ∈ ℕ | |
22 | 8nn0 12547 | . . . . . . 7 ⊢ 8 ∈ ℕ0 | |
23 | 0z 12622 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
24 | 1nn0 12540 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
25 | 8exp8mod9 47661 | . . . . . . . 8 ⊢ ((8↑8) mod 9) = 1 | |
26 | 1re 11259 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
27 | nnrp 13044 | . . . . . . . . . 10 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
28 | 1, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ 9 ∈ ℝ+ |
29 | 0le1 11784 | . . . . . . . . 9 ⊢ 0 ≤ 1 | |
30 | 1lt9 12470 | . . . . . . . . 9 ⊢ 1 < 9 | |
31 | modid 13933 | . . . . . . . . 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 2766 | . . . . . . 7 ⊢ ((8↑8) mod 9) = (1 mod 9) |
34 | 8p1e9 12414 | . . . . . . 7 ⊢ (8 + 1) = 9 | |
35 | 8cn 12361 | . . . . . . . . 9 ⊢ 8 ∈ ℂ | |
36 | 35 | addlidi 11447 | . . . . . . . 8 ⊢ (0 + 8) = 8 |
37 | 9cn 12364 | . . . . . . . . . 10 ⊢ 9 ∈ ℂ | |
38 | 37 | mul02i 11448 | . . . . . . . . 9 ⊢ (0 · 9) = 0 |
39 | 38 | oveq1i 7441 | . . . . . . . 8 ⊢ ((0 · 9) + 8) = (0 + 8) |
40 | 35 | mullidi 11264 | . . . . . . . 8 ⊢ (1 · 8) = 8 |
41 | 36, 39, 40 | 3eqtr4i 2773 | . . . . . . 7 ⊢ ((0 · 9) + 8) = (1 · 8) |
42 | 1, 21, 22, 23, 24, 22, 33, 34, 41 | modxp1i 17104 | . . . . . 6 ⊢ ((8↑9) mod 9) = (8 mod 9) |
43 | 9nprm 17147 | . . . . . 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 3069 | . 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 1537 ∃wex 1776 ∈ wcel 2106 ∃wrex 3068 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 ≤ cle 11294 ℕcn 12264 3c3 12320 8c8 12325 9c9 12326 ℤcz 12611 ℤ≥cuz 12876 ℝ+crp 13032 mod cmo 13906 ↑cexp 14099 ℙcprime 16705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-prm 16706 |
This theorem is referenced by: nfermltlrev 47669 |
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