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 11905 | . . . 4 ⊢ 4 ∈ ℕ0 | |
2 | fmtno 43538 | . . . 4 ⊢ (4 ∈ ℕ0 → (FermatNo‘4) = ((2↑(2↑4)) + 1)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (FermatNo‘4) = ((2↑(2↑4)) + 1) |
4 | 2nn 11699 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | 2nn0 11903 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
6 | 5, 1 | nn0expcli 13445 | . . . . . 6 ⊢ (2↑4) ∈ ℕ0 |
7 | nnexpcl 13432 | . . . . . 6 ⊢ ((2 ∈ ℕ ∧ (2↑4) ∈ ℕ0) → (2↑(2↑4)) ∈ ℕ) | |
8 | 4, 6, 7 | mp2an 688 | . . . . 5 ⊢ (2↑(2↑4)) ∈ ℕ |
9 | 2re 11700 | . . . . . 6 ⊢ 2 ∈ ℝ | |
10 | nnexpcl 13432 | . . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 4 ∈ ℕ0) → (2↑4) ∈ ℕ) | |
11 | 4, 1, 10 | mp2an 688 | . . . . . 6 ⊢ (2↑4) ∈ ℕ |
12 | 1lt2 11797 | . . . . . 6 ⊢ 1 < 2 | |
13 | expgt1 13457 | . . . . . 6 ⊢ ((2 ∈ ℝ ∧ (2↑4) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(2↑4))) | |
14 | 9, 11, 12, 13 | mp3an 1452 | . . . . 5 ⊢ 1 < (2↑(2↑4)) |
15 | eluz2b2 12310 | . . . . 5 ⊢ ((2↑(2↑4)) ∈ (ℤ≥‘2) ↔ ((2↑(2↑4)) ∈ ℕ ∧ 1 < (2↑(2↑4)))) | |
16 | 8, 14, 15 | mpbir2an 707 | . . . 4 ⊢ (2↑(2↑4)) ∈ (ℤ≥‘2) |
17 | peano2uz 12290 | . . . 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 2909 | . 2 ⊢ (FermatNo‘4) ∈ (ℤ≥‘2) |
20 | elinel2 4172 | . . . . . . 7 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → 𝑝 ∈ ℙ) | |
21 | 20 | adantr 481 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ∈ ℙ) |
22 | simpr 485 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ∥ (FermatNo‘4)) | |
23 | elinel1 4171 | . . . . . . . 8 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → 𝑝 ∈ (2...(⌊‘(√‘(FermatNo‘4))))) | |
24 | elfzle2 12901 | . . . . . . . 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 481 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ≤ (⌊‘(√‘(FermatNo‘4)))) |
27 | fmtno4prmfac193 43582 | . . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑝 ∥ (FermatNo‘4) ∧ 𝑝 ≤ (⌊‘(√‘(FermatNo‘4)))) → 𝑝 = ;;193) | |
28 | 21, 22, 26, 27 | syl3anc 1363 | . . . . 5 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 = ;;193) |
29 | fmtno4nprmfac193 43583 | . . . . . 6 ⊢ ¬ ;;193 ∥ (FermatNo‘4) | |
30 | breq1 5061 | . . . . . 6 ⊢ (𝑝 = ;;193 → (𝑝 ∥ (FermatNo‘4) ↔ ;;193 ∥ (FermatNo‘4))) | |
31 | 29, 30 | mtbiri 328 | . . . . 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 795 | . . 3 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → ¬ 𝑝 ∥ (FermatNo‘4)) |
34 | 33 | rgen 3148 | . 2 ⊢ ∀𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ¬ 𝑝 ∥ (FermatNo‘4) |
35 | isprm7 16042 | . 2 ⊢ ((FermatNo‘4) ∈ ℙ ↔ ((FermatNo‘4) ∈ (ℤ≥‘2) ∧ ∀𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ¬ 𝑝 ∥ (FermatNo‘4))) | |
36 | 19, 34, 35 | mpbir2an 707 | 1 ⊢ (FermatNo‘4) ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3138 ∩ cin 3934 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 ℝcr 10525 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 ℕcn 11627 2c2 11681 3c3 11682 4c4 11683 9c9 11688 ℕ0cn0 11886 ;cdc 12087 ℤ≥cuz 12232 ...cfz 12882 ⌊cfl 13150 ↑cexp 13419 √csqrt 14582 ∥ cdvds 15597 ℙcprime 16005 FermatNocfmtno 43536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-inf 8896 df-oi 8963 df-dju 9319 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-xnn0 11957 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-ioo 12732 df-ico 12734 df-fz 12883 df-fzo 13024 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-fac 13624 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-clim 14835 df-prod 15250 df-dvds 15598 df-gcd 15834 df-prm 16006 df-odz 16092 df-phi 16093 df-pc 16164 df-lgs 25799 df-fmtno 43537 |
This theorem is referenced by: 65537prm 43585 fmtnofz04prm 43586 |
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