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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 341fppr2 | Structured version Visualization version GIF version | ||
| Description: 341 is the (smallest) Poulet number (Fermat pseudoprime to the base 2). (Contributed by AV, 3-Jun-2023.) |
| Ref | Expression |
|---|---|
| 341fppr2 | ⊢ ;;341 ∈ ( FPPr ‘2) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 4z 12651 | . . 3 ⊢ 4 ∈ ℤ | |
| 2 | 3nn0 12544 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
| 3 | 4nn0 12545 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
| 4 | 2, 3 | deccl 12748 | . . . . 5 ⊢ ;34 ∈ ℕ0 |
| 5 | 1nn 12277 | . . . . 5 ⊢ 1 ∈ ℕ | |
| 6 | 4, 5 | decnncl 12753 | . . . 4 ⊢ ;;341 ∈ ℕ |
| 7 | 6 | nnzi 12641 | . . 3 ⊢ ;;341 ∈ ℤ |
| 8 | 4nn 12349 | . . . . 5 ⊢ 4 ∈ ℕ | |
| 9 | 2, 8 | decnncl 12753 | . . . 4 ⊢ ;34 ∈ ℕ |
| 10 | 1nn0 12542 | . . . 4 ⊢ 1 ∈ ℕ0 | |
| 11 | 4re 12350 | . . . . 5 ⊢ 4 ∈ ℝ | |
| 12 | 9re 12365 | . . . . 5 ⊢ 9 ∈ ℝ | |
| 13 | 4lt9 12469 | . . . . 5 ⊢ 4 < 9 | |
| 14 | 11, 12, 13 | ltleii 11384 | . . . 4 ⊢ 4 ≤ 9 |
| 15 | 9, 10, 3, 14 | declei 12769 | . . 3 ⊢ 4 ≤ ;;341 |
| 16 | eluz2 12884 | . . 3 ⊢ (;;341 ∈ (ℤ≥‘4) ↔ (4 ∈ ℤ ∧ ;;341 ∈ ℤ ∧ 4 ≤ ;;341)) | |
| 17 | 1, 7, 15, 16 | mpbir3an 1342 | . 2 ⊢ ;;341 ∈ (ℤ≥‘4) |
| 18 | 2z 12649 | . . . . 5 ⊢ 2 ∈ ℤ | |
| 19 | 10, 5 | decnncl 12753 | . . . . . 6 ⊢ ;11 ∈ ℕ |
| 20 | 19 | nnzi 12641 | . . . . 5 ⊢ ;11 ∈ ℤ |
| 21 | 2nn0 12543 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
| 22 | 2re 12340 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
| 23 | 2lt9 12471 | . . . . . . 7 ⊢ 2 < 9 | |
| 24 | 22, 12, 23 | ltleii 11384 | . . . . . 6 ⊢ 2 ≤ 9 |
| 25 | 5, 10, 21, 24 | declei 12769 | . . . . 5 ⊢ 2 ≤ ;11 |
| 26 | eluz2 12884 | . . . . 5 ⊢ (;11 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;11 ∈ ℤ ∧ 2 ≤ ;11)) | |
| 27 | 18, 20, 25, 26 | mpbir3an 1342 | . . . 4 ⊢ ;11 ∈ (ℤ≥‘2) |
| 28 | 2, 5 | decnncl 12753 | . . . . . 6 ⊢ ;31 ∈ ℕ |
| 29 | 28 | nnzi 12641 | . . . . 5 ⊢ ;31 ∈ ℤ |
| 30 | 3nn 12345 | . . . . . 6 ⊢ 3 ∈ ℕ | |
| 31 | 30, 10, 21, 24 | declei 12769 | . . . . 5 ⊢ 2 ≤ ;31 |
| 32 | eluz2 12884 | . . . . 5 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31)) | |
| 33 | 18, 29, 31, 32 | mpbir3an 1342 | . . . 4 ⊢ ;31 ∈ (ℤ≥‘2) |
| 34 | nprm 16725 | . . . 4 ⊢ ((;11 ∈ (ℤ≥‘2) ∧ ;31 ∈ (ℤ≥‘2)) → ¬ (;11 · ;31) ∈ ℙ) | |
| 35 | 27, 33, 34 | mp2an 692 | . . 3 ⊢ ¬ (;11 · ;31) ∈ ℙ |
| 36 | df-nel 3047 | . . . 4 ⊢ (;;341 ∉ ℙ ↔ ¬ ;;341 ∈ ℙ) | |
| 37 | 11t31e341 47719 | . . . . . 6 ⊢ (;11 · ;31) = ;;341 | |
| 38 | 37 | eqcomi 2746 | . . . . 5 ⊢ ;;341 = (;11 · ;31) |
| 39 | 38 | eleq1i 2832 | . . . 4 ⊢ (;;341 ∈ ℙ ↔ (;11 · ;31) ∈ ℙ) |
| 40 | 36, 39 | xchbinx 334 | . . 3 ⊢ (;;341 ∉ ℙ ↔ ¬ (;11 · ;31) ∈ ℙ) |
| 41 | 35, 40 | mpbir 231 | . 2 ⊢ ;;341 ∉ ℙ |
| 42 | eqid 2737 | . . . . . 6 ⊢ ;;341 = ;;341 | |
| 43 | eqid 2737 | . . . . . 6 ⊢ (;34 + 1) = (;34 + 1) | |
| 44 | 1m1e0 12338 | . . . . . 6 ⊢ (1 − 1) = 0 | |
| 45 | 4, 10, 10, 42, 43, 44 | decsubi 12796 | . . . . 5 ⊢ (;;341 − 1) = ;;340 |
| 46 | 45 | oveq2i 7442 | . . . 4 ⊢ (2↑(;;341 − 1)) = (2↑;;340) |
| 47 | 46 | oveq1i 7441 | . . 3 ⊢ ((2↑(;;341 − 1)) mod ;;341) = ((2↑;;340) mod ;;341) |
| 48 | 2exp340mod341 47720 | . . 3 ⊢ ((2↑;;340) mod ;;341) = 1 | |
| 49 | 47, 48 | eqtri 2765 | . 2 ⊢ ((2↑(;;341 − 1)) mod ;;341) = 1 |
| 50 | 2nn 12339 | . . 3 ⊢ 2 ∈ ℕ | |
| 51 | fpprel 47715 | . . 3 ⊢ (2 ∈ ℕ → (;;341 ∈ ( FPPr ‘2) ↔ (;;341 ∈ (ℤ≥‘4) ∧ ;;341 ∉ ℙ ∧ ((2↑(;;341 − 1)) mod ;;341) = 1))) | |
| 52 | 50, 51 | ax-mp 5 | . 2 ⊢ (;;341 ∈ ( FPPr ‘2) ↔ (;;341 ∈ (ℤ≥‘4) ∧ ;;341 ∉ ℙ ∧ ((2↑(;;341 − 1)) mod ;;341) = 1)) |
| 53 | 17, 41, 49, 52 | mpbir3an 1342 | 1 ⊢ ;;341 ∈ ( FPPr ‘2) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∉ wnel 3046 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 ≤ cle 11296 − cmin 11492 ℕcn 12266 2c2 12321 3c3 12322 4c4 12323 9c9 12328 ℤcz 12613 ;cdc 12733 ℤ≥cuz 12878 mod cmo 13909 ↑cexp 14102 ℙcprime 16708 FPPr cfppr 47711 |
| 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 df-fppr 47712 |
| This theorem is referenced by: nfermltl2rev 47730 |
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