Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtno4prm | Structured version Visualization version GIF version |
Description: The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.) |
Ref | Expression |
---|---|
fmtno4prm | ⊢ (FermatNo‘4) ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn0 12109 | . . . 4 ⊢ 4 ∈ ℕ0 | |
2 | fmtno 44654 | . . . 4 ⊢ (4 ∈ ℕ0 → (FermatNo‘4) = ((2↑(2↑4)) + 1)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (FermatNo‘4) = ((2↑(2↑4)) + 1) |
4 | 2nn 11903 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | 2nn0 12107 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
6 | 5, 1 | nn0expcli 13661 | . . . . . 6 ⊢ (2↑4) ∈ ℕ0 |
7 | nnexpcl 13648 | . . . . . 6 ⊢ ((2 ∈ ℕ ∧ (2↑4) ∈ ℕ0) → (2↑(2↑4)) ∈ ℕ) | |
8 | 4, 6, 7 | mp2an 692 | . . . . 5 ⊢ (2↑(2↑4)) ∈ ℕ |
9 | 2re 11904 | . . . . . 6 ⊢ 2 ∈ ℝ | |
10 | nnexpcl 13648 | . . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 4 ∈ ℕ0) → (2↑4) ∈ ℕ) | |
11 | 4, 1, 10 | mp2an 692 | . . . . . 6 ⊢ (2↑4) ∈ ℕ |
12 | 1lt2 12001 | . . . . . 6 ⊢ 1 < 2 | |
13 | expgt1 13673 | . . . . . 6 ⊢ ((2 ∈ ℝ ∧ (2↑4) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(2↑4))) | |
14 | 9, 11, 12, 13 | mp3an 1463 | . . . . 5 ⊢ 1 < (2↑(2↑4)) |
15 | eluz2b2 12517 | . . . . 5 ⊢ ((2↑(2↑4)) ∈ (ℤ≥‘2) ↔ ((2↑(2↑4)) ∈ ℕ ∧ 1 < (2↑(2↑4)))) | |
16 | 8, 14, 15 | mpbir2an 711 | . . . 4 ⊢ (2↑(2↑4)) ∈ (ℤ≥‘2) |
17 | peano2uz 12497 | . . . 4 ⊢ ((2↑(2↑4)) ∈ (ℤ≥‘2) → ((2↑(2↑4)) + 1) ∈ (ℤ≥‘2)) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ ((2↑(2↑4)) + 1) ∈ (ℤ≥‘2) |
19 | 3, 18 | eqeltri 2834 | . 2 ⊢ (FermatNo‘4) ∈ (ℤ≥‘2) |
20 | elinel2 4110 | . . . . . . 7 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → 𝑝 ∈ ℙ) | |
21 | 20 | adantr 484 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ∈ ℙ) |
22 | simpr 488 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ∥ (FermatNo‘4)) | |
23 | elinel1 4109 | . . . . . . . 8 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → 𝑝 ∈ (2...(⌊‘(√‘(FermatNo‘4))))) | |
24 | elfzle2 13116 | . . . . . . . 8 ⊢ (𝑝 ∈ (2...(⌊‘(√‘(FermatNo‘4)))) → 𝑝 ≤ (⌊‘(√‘(FermatNo‘4)))) | |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → 𝑝 ≤ (⌊‘(√‘(FermatNo‘4)))) |
26 | 25 | adantr 484 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ≤ (⌊‘(√‘(FermatNo‘4)))) |
27 | fmtno4prmfac193 44698 | . . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑝 ∥ (FermatNo‘4) ∧ 𝑝 ≤ (⌊‘(√‘(FermatNo‘4)))) → 𝑝 = ;;193) | |
28 | 21, 22, 26, 27 | syl3anc 1373 | . . . . 5 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 = ;;193) |
29 | fmtno4nprmfac193 44699 | . . . . . 6 ⊢ ¬ ;;193 ∥ (FermatNo‘4) | |
30 | breq1 5056 | . . . . . 6 ⊢ (𝑝 = ;;193 → (𝑝 ∥ (FermatNo‘4) ↔ ;;193 ∥ (FermatNo‘4))) | |
31 | 29, 30 | mtbiri 330 | . . . . 5 ⊢ (𝑝 = ;;193 → ¬ 𝑝 ∥ (FermatNo‘4)) |
32 | 28, 31 | syl 17 | . . . 4 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → ¬ 𝑝 ∥ (FermatNo‘4)) |
33 | 32 | pm2.01da 799 | . . 3 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → ¬ 𝑝 ∥ (FermatNo‘4)) |
34 | 33 | rgen 3071 | . 2 ⊢ ∀𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ¬ 𝑝 ∥ (FermatNo‘4) |
35 | isprm7 16265 | . 2 ⊢ ((FermatNo‘4) ∈ ℙ ↔ ((FermatNo‘4) ∈ (ℤ≥‘2) ∧ ∀𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ¬ 𝑝 ∥ (FermatNo‘4))) | |
36 | 19, 34, 35 | mpbir2an 711 | 1 ⊢ (FermatNo‘4) ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∩ cin 3865 class class class wbr 5053 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 1c1 10730 + caddc 10732 < clt 10867 ≤ cle 10868 ℕcn 11830 2c2 11885 3c3 11886 4c4 11887 9c9 11892 ℕ0cn0 12090 ;cdc 12293 ℤ≥cuz 12438 ...cfz 13095 ⌊cfl 13365 ↑cexp 13635 √csqrt 14796 ∥ cdvds 15815 ℙcprime 16228 FermatNocfmtno 44652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-2o 8203 df-oadd 8206 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-oi 9126 df-dju 9517 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-xnn0 12163 df-z 12177 df-dec 12294 df-uz 12439 df-q 12545 df-rp 12587 df-ioo 12939 df-ico 12941 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-seq 13575 df-exp 13636 df-fac 13840 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-prod 15468 df-dvds 15816 df-gcd 16054 df-prm 16229 df-odz 16318 df-phi 16319 df-pc 16390 df-lgs 26176 df-fmtno 44653 |
This theorem is referenced by: 65537prm 44701 fmtnofz04prm 44702 |
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