Mathbox for Alexander van der Vekens |
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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 12017 | . . 3 ⊢ 4 ∈ ℤ | |
2 | 3nn0 11916 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
3 | 4nn0 11917 | . . . . . 6 ⊢ 4 ∈ ℕ0 | |
4 | 2, 3 | deccl 12114 | . . . . 5 ⊢ ;34 ∈ ℕ0 |
5 | 1nn 11649 | . . . . 5 ⊢ 1 ∈ ℕ | |
6 | 4, 5 | decnncl 12119 | . . . 4 ⊢ ;;341 ∈ ℕ |
7 | 6 | nnzi 12007 | . . 3 ⊢ ;;341 ∈ ℤ |
8 | 4nn 11721 | . . . . 5 ⊢ 4 ∈ ℕ | |
9 | 2, 8 | decnncl 12119 | . . . 4 ⊢ ;34 ∈ ℕ |
10 | 1nn0 11914 | . . . 4 ⊢ 1 ∈ ℕ0 | |
11 | 4re 11722 | . . . . 5 ⊢ 4 ∈ ℝ | |
12 | 9re 11737 | . . . . 5 ⊢ 9 ∈ ℝ | |
13 | 4lt9 11841 | . . . . 5 ⊢ 4 < 9 | |
14 | 11, 12, 13 | ltleii 10763 | . . . 4 ⊢ 4 ≤ 9 |
15 | 9, 10, 3, 14 | declei 12135 | . . 3 ⊢ 4 ≤ ;;341 |
16 | eluz2 12250 | . . 3 ⊢ (;;341 ∈ (ℤ≥‘4) ↔ (4 ∈ ℤ ∧ ;;341 ∈ ℤ ∧ 4 ≤ ;;341)) | |
17 | 1, 7, 15, 16 | mpbir3an 1337 | . 2 ⊢ ;;341 ∈ (ℤ≥‘4) |
18 | 2z 12015 | . . . . 5 ⊢ 2 ∈ ℤ | |
19 | 10, 5 | decnncl 12119 | . . . . . 6 ⊢ ;11 ∈ ℕ |
20 | 19 | nnzi 12007 | . . . . 5 ⊢ ;11 ∈ ℤ |
21 | 2nn0 11915 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
22 | 2re 11712 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
23 | 2lt9 11843 | . . . . . . 7 ⊢ 2 < 9 | |
24 | 22, 12, 23 | ltleii 10763 | . . . . . 6 ⊢ 2 ≤ 9 |
25 | 5, 10, 21, 24 | declei 12135 | . . . . 5 ⊢ 2 ≤ ;11 |
26 | eluz2 12250 | . . . . 5 ⊢ (;11 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;11 ∈ ℤ ∧ 2 ≤ ;11)) | |
27 | 18, 20, 25, 26 | mpbir3an 1337 | . . . 4 ⊢ ;11 ∈ (ℤ≥‘2) |
28 | 2, 5 | decnncl 12119 | . . . . . 6 ⊢ ;31 ∈ ℕ |
29 | 28 | nnzi 12007 | . . . . 5 ⊢ ;31 ∈ ℤ |
30 | 3nn 11717 | . . . . . 6 ⊢ 3 ∈ ℕ | |
31 | 30, 10, 21, 24 | declei 12135 | . . . . 5 ⊢ 2 ≤ ;31 |
32 | eluz2 12250 | . . . . 5 ⊢ (;31 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ ;31 ∈ ℤ ∧ 2 ≤ ;31)) | |
33 | 18, 29, 31, 32 | mpbir3an 1337 | . . . 4 ⊢ ;31 ∈ (ℤ≥‘2) |
34 | nprm 16032 | . . . 4 ⊢ ((;11 ∈ (ℤ≥‘2) ∧ ;31 ∈ (ℤ≥‘2)) → ¬ (;11 · ;31) ∈ ℙ) | |
35 | 27, 33, 34 | mp2an 690 | . . 3 ⊢ ¬ (;11 · ;31) ∈ ℙ |
36 | df-nel 3124 | . . . 4 ⊢ (;;341 ∉ ℙ ↔ ¬ ;;341 ∈ ℙ) | |
37 | 11t31e341 43946 | . . . . . 6 ⊢ (;11 · ;31) = ;;341 | |
38 | 37 | eqcomi 2830 | . . . . 5 ⊢ ;;341 = (;11 · ;31) |
39 | 38 | eleq1i 2903 | . . . 4 ⊢ (;;341 ∈ ℙ ↔ (;11 · ;31) ∈ ℙ) |
40 | 36, 39 | xchbinx 336 | . . 3 ⊢ (;;341 ∉ ℙ ↔ ¬ (;11 · ;31) ∈ ℙ) |
41 | 35, 40 | mpbir 233 | . 2 ⊢ ;;341 ∉ ℙ |
42 | eqid 2821 | . . . . . 6 ⊢ ;;341 = ;;341 | |
43 | eqid 2821 | . . . . . 6 ⊢ (;34 + 1) = (;34 + 1) | |
44 | 1m1e0 11710 | . . . . . 6 ⊢ (1 − 1) = 0 | |
45 | 4, 10, 10, 42, 43, 44 | decsubi 12162 | . . . . 5 ⊢ (;;341 − 1) = ;;340 |
46 | 45 | oveq2i 7167 | . . . 4 ⊢ (2↑(;;341 − 1)) = (2↑;;340) |
47 | 46 | oveq1i 7166 | . . 3 ⊢ ((2↑(;;341 − 1)) mod ;;341) = ((2↑;;340) mod ;;341) |
48 | 2exp340mod341 43947 | . . 3 ⊢ ((2↑;;340) mod ;;341) = 1 | |
49 | 47, 48 | eqtri 2844 | . 2 ⊢ ((2↑(;;341 − 1)) mod ;;341) = 1 |
50 | 2nn 11711 | . . 3 ⊢ 2 ∈ ℕ | |
51 | fpprel 43942 | . . 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 1337 | 1 ⊢ ;;341 ∈ ( FPPr ‘2) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∉ wnel 3123 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ≤ cle 10676 − cmin 10870 ℕcn 11638 2c2 11693 3c3 11694 4c4 11695 9c9 11700 ℤcz 11982 ;cdc 12099 ℤ≥cuz 12244 mod cmo 13238 ↑cexp 13430 ℙcprime 16015 FPPr cfppr 43938 |
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 df-fppr 43939 |
This theorem is referenced by: nfermltl2rev 43957 |
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