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 11917 | . . . 4 ⊢ 4 ∈ ℕ0 | |
2 | fmtno 43711 | . . . 4 ⊢ (4 ∈ ℕ0 → (FermatNo‘4) = ((2↑(2↑4)) + 1)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ (FermatNo‘4) = ((2↑(2↑4)) + 1) |
4 | 2nn 11711 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | 2nn0 11915 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
6 | 5, 1 | nn0expcli 13456 | . . . . . 6 ⊢ (2↑4) ∈ ℕ0 |
7 | nnexpcl 13443 | . . . . . 6 ⊢ ((2 ∈ ℕ ∧ (2↑4) ∈ ℕ0) → (2↑(2↑4)) ∈ ℕ) | |
8 | 4, 6, 7 | mp2an 690 | . . . . 5 ⊢ (2↑(2↑4)) ∈ ℕ |
9 | 2re 11712 | . . . . . 6 ⊢ 2 ∈ ℝ | |
10 | nnexpcl 13443 | . . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 4 ∈ ℕ0) → (2↑4) ∈ ℕ) | |
11 | 4, 1, 10 | mp2an 690 | . . . . . 6 ⊢ (2↑4) ∈ ℕ |
12 | 1lt2 11809 | . . . . . 6 ⊢ 1 < 2 | |
13 | expgt1 13468 | . . . . . 6 ⊢ ((2 ∈ ℝ ∧ (2↑4) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(2↑4))) | |
14 | 9, 11, 12, 13 | mp3an 1457 | . . . . 5 ⊢ 1 < (2↑(2↑4)) |
15 | eluz2b2 12322 | . . . . 5 ⊢ ((2↑(2↑4)) ∈ (ℤ≥‘2) ↔ ((2↑(2↑4)) ∈ ℕ ∧ 1 < (2↑(2↑4)))) | |
16 | 8, 14, 15 | mpbir2an 709 | . . . 4 ⊢ (2↑(2↑4)) ∈ (ℤ≥‘2) |
17 | peano2uz 12302 | . . . 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 4173 | . . . . . . 7 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → 𝑝 ∈ ℙ) | |
21 | 20 | adantr 483 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ∈ ℙ) |
22 | simpr 487 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ∥ (FermatNo‘4)) | |
23 | elinel1 4172 | . . . . . . . 8 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → 𝑝 ∈ (2...(⌊‘(√‘(FermatNo‘4))))) | |
24 | elfzle2 12912 | . . . . . . . 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 483 | . . . . . 6 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 ≤ (⌊‘(√‘(FermatNo‘4)))) |
27 | fmtno4prmfac193 43755 | . . . . . 6 ⊢ ((𝑝 ∈ ℙ ∧ 𝑝 ∥ (FermatNo‘4) ∧ 𝑝 ≤ (⌊‘(√‘(FermatNo‘4)))) → 𝑝 = ;;193) | |
28 | 21, 22, 26, 27 | syl3anc 1367 | . . . . 5 ⊢ ((𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ∧ 𝑝 ∥ (FermatNo‘4)) → 𝑝 = ;;193) |
29 | fmtno4nprmfac193 43756 | . . . . . 6 ⊢ ¬ ;;193 ∥ (FermatNo‘4) | |
30 | breq1 5069 | . . . . . 6 ⊢ (𝑝 = ;;193 → (𝑝 ∥ (FermatNo‘4) ↔ ;;193 ∥ (FermatNo‘4))) | |
31 | 29, 30 | mtbiri 329 | . . . . 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 797 | . . 3 ⊢ (𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) → ¬ 𝑝 ∥ (FermatNo‘4)) |
34 | 33 | rgen 3148 | . 2 ⊢ ∀𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ¬ 𝑝 ∥ (FermatNo‘4) |
35 | isprm7 16052 | . 2 ⊢ ((FermatNo‘4) ∈ ℙ ↔ ((FermatNo‘4) ∈ (ℤ≥‘2) ∧ ∀𝑝 ∈ ((2...(⌊‘(√‘(FermatNo‘4)))) ∩ ℙ) ¬ 𝑝 ∥ (FermatNo‘4))) | |
36 | 19, 34, 35 | mpbir2an 709 | 1 ⊢ (FermatNo‘4) ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∩ cin 3935 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 1c1 10538 + caddc 10540 < clt 10675 ≤ cle 10676 ℕcn 11638 2c2 11693 3c3 11694 4c4 11695 9c9 11700 ℕ0cn0 11898 ;cdc 12099 ℤ≥cuz 12244 ...cfz 12893 ⌊cfl 13161 ↑cexp 13430 √csqrt 14592 ∥ cdvds 15607 ℙcprime 16015 FermatNocfmtno 43709 |
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-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 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-fal 1550 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-int 4877 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-se 5515 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-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 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-xnn0 11969 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-ioo 12743 df-ico 12745 df-fz 12894 df-fzo 13035 df-fl 13163 df-mod 13239 df-seq 13371 df-exp 13431 df-fac 13635 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-prod 15260 df-dvds 15608 df-gcd 15844 df-prm 16016 df-odz 16102 df-phi 16103 df-pc 16174 df-lgs 25871 df-fmtno 43710 |
This theorem is referenced by: 65537prm 43758 fmtnofz04prm 43759 |
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