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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 43951) 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 11736 | . . . 4 ⊢ 9 ∈ ℕ | |
2 | 1 | elexi 3513 | . . 3 ⊢ 9 ∈ V |
3 | eleq1 2900 | . . . 4 ⊢ (𝑝 = 9 → (𝑝 ∈ (ℤ≥‘3) ↔ 9 ∈ (ℤ≥‘3))) | |
4 | oveq2 7164 | . . . . . . . 8 ⊢ (𝑝 = 9 → (8↑𝑝) = (8↑9)) | |
5 | id 22 | . . . . . . . 8 ⊢ (𝑝 = 9 → 𝑝 = 9) | |
6 | 4, 5 | oveq12d 7174 | . . . . . . 7 ⊢ (𝑝 = 9 → ((8↑𝑝) mod 𝑝) = ((8↑9) mod 9)) |
7 | oveq2 7164 | . . . . . . 7 ⊢ (𝑝 = 9 → (8 mod 𝑝) = (8 mod 9)) | |
8 | 6, 7 | eqeq12d 2837 | . . . . . 6 ⊢ (𝑝 = 9 → (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) ↔ ((8↑9) mod 9) = (8 mod 9))) |
9 | eleq1 2900 | . . . . . 6 ⊢ (𝑝 = 9 → (𝑝 ∈ ℙ ↔ 9 ∈ ℙ)) | |
10 | 8, 9 | imbi12d 347 | . . . . 5 ⊢ (𝑝 = 9 → ((((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ) ↔ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ))) |
11 | 10 | notbid 320 | . . . 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 12016 | . . . . 5 ⊢ 3 ∈ ℤ | |
14 | 1 | nnzi 12007 | . . . . 5 ⊢ 9 ∈ ℤ |
15 | 3re 11718 | . . . . . 6 ⊢ 3 ∈ ℝ | |
16 | 9re 11737 | . . . . . 6 ⊢ 9 ∈ ℝ | |
17 | 3lt9 11842 | . . . . . 6 ⊢ 3 < 9 | |
18 | 15, 16, 17 | ltleii 10763 | . . . . 5 ⊢ 3 ≤ 9 |
19 | eluz2 12250 | . . . . 5 ⊢ (9 ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ 9 ∈ ℤ ∧ 3 ≤ 9)) | |
20 | 13, 14, 18, 19 | mpbir3an 1337 | . . . 4 ⊢ 9 ∈ (ℤ≥‘3) |
21 | 8nn 11733 | . . . . . . 7 ⊢ 8 ∈ ℕ | |
22 | 8nn0 11921 | . . . . . . 7 ⊢ 8 ∈ ℕ0 | |
23 | 0z 11993 | . . . . . . 7 ⊢ 0 ∈ ℤ | |
24 | 1nn0 11914 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
25 | 8exp8mod9 43950 | . . . . . . . 8 ⊢ ((8↑8) mod 9) = 1 | |
26 | 1re 10641 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
27 | nnrp 12401 | . . . . . . . . . 10 ⊢ (9 ∈ ℕ → 9 ∈ ℝ+) | |
28 | 1, 27 | ax-mp 5 | . . . . . . . . 9 ⊢ 9 ∈ ℝ+ |
29 | 0le1 11163 | . . . . . . . . 9 ⊢ 0 ≤ 1 | |
30 | 1lt9 11844 | . . . . . . . . 9 ⊢ 1 < 9 | |
31 | modid 13265 | . . . . . . . . 9 ⊢ (((1 ∈ ℝ ∧ 9 ∈ ℝ+) ∧ (0 ≤ 1 ∧ 1 < 9)) → (1 mod 9) = 1) | |
32 | 26, 28, 29, 30, 31 | mp4an 691 | . . . . . . . 8 ⊢ (1 mod 9) = 1 |
33 | 25, 32 | eqtr4i 2847 | . . . . . . 7 ⊢ ((8↑8) mod 9) = (1 mod 9) |
34 | 8p1e9 11788 | . . . . . . 7 ⊢ (8 + 1) = 9 | |
35 | 8cn 11735 | . . . . . . . . 9 ⊢ 8 ∈ ℂ | |
36 | 35 | addid2i 10828 | . . . . . . . 8 ⊢ (0 + 8) = 8 |
37 | 9cn 11738 | . . . . . . . . . 10 ⊢ 9 ∈ ℂ | |
38 | 37 | mul02i 10829 | . . . . . . . . 9 ⊢ (0 · 9) = 0 |
39 | 38 | oveq1i 7166 | . . . . . . . 8 ⊢ ((0 · 9) + 8) = (0 + 8) |
40 | 35 | mulid2i 10646 | . . . . . . . 8 ⊢ (1 · 8) = 8 |
41 | 36, 39, 40 | 3eqtr4i 2854 | . . . . . . 7 ⊢ ((0 · 9) + 8) = (1 · 8) |
42 | 1, 21, 22, 23, 24, 22, 33, 34, 41 | modxp1i 16406 | . . . . . 6 ⊢ ((8↑9) mod 9) = (8 mod 9) |
43 | 9nprm 16446 | . . . . . 6 ⊢ ¬ 9 ∈ ℙ | |
44 | 42, 43 | pm3.2i 473 | . . . . 5 ⊢ (((8↑9) mod 9) = (8 mod 9) ∧ ¬ 9 ∈ ℙ) |
45 | annim 406 | . . . . 5 ⊢ ((((8↑9) mod 9) = (8 mod 9) ∧ ¬ 9 ∈ ℙ) ↔ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ)) | |
46 | 44, 45 | mpbi 232 | . . . 4 ⊢ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ) |
47 | 20, 46 | pm3.2i 473 | . . 3 ⊢ (9 ∈ (ℤ≥‘3) ∧ ¬ (((8↑9) mod 9) = (8 mod 9) → 9 ∈ ℙ)) |
48 | 2, 12, 47 | ceqsexv2d 3542 | . 2 ⊢ ∃𝑝(𝑝 ∈ (ℤ≥‘3) ∧ ¬ (((8↑𝑝) mod 𝑝) = (8 mod 𝑝) → 𝑝 ∈ ℙ)) |
49 | df-rex 3144 | . 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 233 | 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 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ∃wrex 3139 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 < clt 10675 ≤ cle 10676 ℕcn 11638 3c3 11694 8c8 11699 9c9 11700 ℤcz 11982 ℤ≥cuz 12244 ℝ+crp 12390 mod cmo 13238 ↑cexp 13430 ℙcprime 16015 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-prm 16016 |
This theorem is referenced by: nfermltlrev 43958 |
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