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 11904 | . . . 4 ⊢ 4 ∈ ℕ0 | |
2 | fmtno 43568 | . . . 4 ⊢ (4 ∈ ℕ0 → (FermatNo‘4) = ((2↑(2↑4)) + 1)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (FermatNo‘4) = ((2↑(2↑4)) + 1) |
4 | 2nn 11698 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | 2nn0 11902 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
6 | 5, 1 | nn0expcli 13443 | . . . . . 6 ⊢ (2↑4) ∈ ℕ0 |
7 | nnexpcl 13430 | . . . . . 6 ⊢ ((2 ∈ ℕ ∧ (2↑4) ∈ ℕ0) → (2↑(2↑4)) ∈ ℕ) | |
8 | 4, 6, 7 | mp2an 688 | . . . . 5 ⊢ (2↑(2↑4)) ∈ ℕ |
9 | 2re 11699 | . . . . . 6 ⊢ 2 ∈ ℝ | |
10 | nnexpcl 13430 | . . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 4 ∈ ℕ0) → (2↑4) ∈ ℕ) | |
11 | 4, 1, 10 | mp2an 688 | . . . . . 6 ⊢ (2↑4) ∈ ℕ |
12 | 1lt2 11796 | . . . . . 6 ⊢ 1 < 2 | |
13 | expgt1 13455 | . . . . . 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 12309 | . . . . 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 12289 | . . . 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 2906 | . 2 ⊢ (FermatNo‘4) ∈ (ℤ≥‘2) |
20 | elinel2 4170 | . . . . . . 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 4169 | . . . . . . . 8 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → 𝑝 ∈ (2...(⌊‘(√‘(FermatNo‘4))))) | |
24 | elfzle2 12899 | . . . . . . . 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 43612 | . . . . . 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 43613 | . . . . . 6 ⊢ ¬ ;;193 ∥ (FermatNo‘4) | |
30 | breq1 5060 | . . . . . 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 3145 | . 2 ⊢ ∀𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ¬ 𝑝 ∥ (FermatNo‘4) |
35 | isprm7 16040 | . 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 3135 ∩ cin 3932 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 1c1 10526 + caddc 10528 < clt 10663 ≤ cle 10664 ℕcn 11626 2c2 11680 3c3 11681 4c4 11682 9c9 11687 ℕ0cn0 11885 ;cdc 12086 ℤ≥cuz 12231 ...cfz 12880 ⌊cfl 13148 ↑cexp 13417 √csqrt 14580 ∥ cdvds 15595 ℙcprime 16003 FermatNocfmtno 43566 |
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 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-ioo 12730 df-ico 12732 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-prod 15248 df-dvds 15596 df-gcd 15832 df-prm 16004 df-odz 16090 df-phi 16091 df-pc 16162 df-lgs 25798 df-fmtno 43567 |
This theorem is referenced by: 65537prm 43615 fmtnofz04prm 43616 |
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