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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 12439 | . . . 4 ⊢ 4 ∈ ℕ0 | |
2 | fmtno 45795 | . . . 4 ⊢ (4 ∈ ℕ0 → (FermatNo‘4) = ((2↑(2↑4)) + 1)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (FermatNo‘4) = ((2↑(2↑4)) + 1) |
4 | 2nn 12233 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | 2nn0 12437 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
6 | 5, 1 | nn0expcli 14001 | . . . . . 6 ⊢ (2↑4) ∈ ℕ0 |
7 | nnexpcl 13987 | . . . . . 6 ⊢ ((2 ∈ ℕ ∧ (2↑4) ∈ ℕ0) → (2↑(2↑4)) ∈ ℕ) | |
8 | 4, 6, 7 | mp2an 691 | . . . . 5 ⊢ (2↑(2↑4)) ∈ ℕ |
9 | 2re 12234 | . . . . . 6 ⊢ 2 ∈ ℝ | |
10 | nnexpcl 13987 | . . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 4 ∈ ℕ0) → (2↑4) ∈ ℕ) | |
11 | 4, 1, 10 | mp2an 691 | . . . . . 6 ⊢ (2↑4) ∈ ℕ |
12 | 1lt2 12331 | . . . . . 6 ⊢ 1 < 2 | |
13 | expgt1 14013 | . . . . . 6 ⊢ ((2 ∈ ℝ ∧ (2↑4) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(2↑4))) | |
14 | 9, 11, 12, 13 | mp3an 1462 | . . . . 5 ⊢ 1 < (2↑(2↑4)) |
15 | eluz2b2 12853 | . . . . 5 ⊢ ((2↑(2↑4)) ∈ (ℤ≥‘2) ↔ ((2↑(2↑4)) ∈ ℕ ∧ 1 < (2↑(2↑4)))) | |
16 | 8, 14, 15 | mpbir2an 710 | . . . 4 ⊢ (2↑(2↑4)) ∈ (ℤ≥‘2) |
17 | peano2uz 12833 | . . . 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 4161 | . . . . . . 7 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → 𝑝 ∈ ℙ) | |
21 | 20 | adantr 482 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ∈ ℙ) |
22 | simpr 486 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ∥ (FermatNo‘4)) | |
23 | elinel1 4160 | . . . . . . . 8 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → 𝑝 ∈ (2...(⌊‘(√‘(FermatNo‘4))))) | |
24 | elfzle2 13452 | . . . . . . . 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 482 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ≤ (⌊‘(√‘(FermatNo‘4)))) |
27 | fmtno4prmfac193 45839 | . . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑝 ∥ (FermatNo‘4) ∧ 𝑝 ≤ (⌊‘(√‘(FermatNo‘4)))) → 𝑝 = ;;193) | |
28 | 21, 22, 26, 27 | syl3anc 1372 | . . . . 5 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 = ;;193) |
29 | fmtno4nprmfac193 45840 | . . . . . 6 ⊢ ¬ ;;193 ∥ (FermatNo‘4) | |
30 | breq1 5113 | . . . . . 6 ⊢ (𝑝 = ;;193 → (𝑝 ∥ (FermatNo‘4) ↔ ;;193 ∥ (FermatNo‘4))) | |
31 | 29, 30 | mtbiri 327 | . . . . 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 798 | . . 3 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → ¬ 𝑝 ∥ (FermatNo‘4)) |
34 | 33 | rgen 3067 | . 2 ⊢ ∀𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ¬ 𝑝 ∥ (FermatNo‘4) |
35 | isprm7 16591 | . 2 ⊢ ((FermatNo‘4) ∈ ℙ ↔ ((FermatNo‘4) ∈ (ℤ≥‘2) ∧ ∀𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ¬ 𝑝 ∥ (FermatNo‘4))) | |
36 | 19, 34, 35 | mpbir2an 710 | 1 ⊢ (FermatNo‘4) ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∩ cin 3914 class class class wbr 5110 ‘cfv 6501 (class class class)co 7362 ℝcr 11057 1c1 11059 + caddc 11061 < clt 11196 ≤ cle 11197 ℕcn 12160 2c2 12215 3c3 12216 4c4 12217 9c9 12222 ℕ0cn0 12420 ;cdc 12625 ℤ≥cuz 12770 ...cfz 13431 ⌊cfl 13702 ↑cexp 13974 √csqrt 15125 ∥ cdvds 16143 ℙcprime 16554 FermatNocfmtno 45793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-xnn0 12493 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-ioo 13275 df-ico 13277 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-fac 14181 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-prod 15796 df-dvds 16144 df-gcd 16382 df-prm 16555 df-odz 16644 df-phi 16645 df-pc 16716 df-lgs 26659 df-fmtno 45794 |
This theorem is referenced by: 65537prm 45842 fmtnofz04prm 45843 |
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