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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtnoge3 | Structured version Visualization version GIF version |
Description: Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.) |
Ref | Expression |
---|---|
fmtnoge3 | ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ≥‘3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmtno 42285 | . 2 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) | |
2 | 3z 11745 | . . . 4 ⊢ 3 ∈ ℤ | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 3 ∈ ℤ) |
4 | 2nn0 11644 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℕ0) |
6 | id 22 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
7 | 5, 6 | nn0expcld 13334 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℕ0) |
8 | 5, 7 | nn0expcld 13334 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(2↑𝑁)) ∈ ℕ0) |
9 | peano2nn0 11667 | . . . . 5 ⊢ ((2↑(2↑𝑁)) ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ ℕ0) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ ℕ0) |
11 | 10 | nn0zd 11815 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ ℤ) |
12 | 3m1e2 11493 | . . . . 5 ⊢ (3 − 1) = 2 | |
13 | 2cn 11433 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
14 | exp1 13167 | . . . . . . 7 ⊢ (2 ∈ ℂ → (2↑1) = 2) | |
15 | 13, 14 | ax-mp 5 | . . . . . 6 ⊢ (2↑1) = 2 |
16 | 2re 11432 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
17 | 16 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
18 | 1le2 11574 | . . . . . . . . 9 ⊢ 1 ≤ 2 | |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 ≤ 2) |
20 | 17, 6, 19 | expge1d 13328 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 1 ≤ (2↑𝑁)) |
21 | 1zzd 11743 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℤ) | |
22 | 7 | nn0zd 11815 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℤ) |
23 | 1lt2 11536 | . . . . . . . . 9 ⊢ 1 < 2 | |
24 | 23 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 < 2) |
25 | 17, 21, 22, 24 | leexp2d 13342 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (1 ≤ (2↑𝑁) ↔ (2↑1) ≤ (2↑(2↑𝑁)))) |
26 | 20, 25 | mpbid 224 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑1) ≤ (2↑(2↑𝑁))) |
27 | 15, 26 | syl5eqbrr 4911 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 2 ≤ (2↑(2↑𝑁))) |
28 | 12, 27 | syl5eqbr 4910 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (3 − 1) ≤ (2↑(2↑𝑁))) |
29 | 3re 11438 | . . . . . 6 ⊢ 3 ∈ ℝ | |
30 | 29 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 3 ∈ ℝ) |
31 | 1red 10364 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
32 | 8 | nn0red 11686 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(2↑𝑁)) ∈ ℝ) |
33 | 30, 31, 32 | lesubaddd 10956 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((3 − 1) ≤ (2↑(2↑𝑁)) ↔ 3 ≤ ((2↑(2↑𝑁)) + 1))) |
34 | 28, 33 | mpbid 224 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 3 ≤ ((2↑(2↑𝑁)) + 1)) |
35 | eluz2 11981 | . . 3 ⊢ (((2↑(2↑𝑁)) + 1) ∈ (ℤ≥‘3) ↔ (3 ∈ ℤ ∧ ((2↑(2↑𝑁)) + 1) ∈ ℤ ∧ 3 ≤ ((2↑(2↑𝑁)) + 1))) | |
36 | 3, 11, 34, 35 | syl3anbrc 1447 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ (ℤ≥‘3)) |
37 | 1, 36 | eqeltrd 2906 | 1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ≥‘3)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ‘cfv 6127 (class class class)co 6910 ℂcc 10257 ℝcr 10258 1c1 10260 + caddc 10262 < clt 10398 ≤ cle 10399 − cmin 10592 2c2 11413 3c3 11414 ℕ0cn0 11625 ℤcz 11711 ℤ≥cuz 11975 ↑cexp 13161 FermatNocfmtno 42283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-n0 11626 df-z 11712 df-uz 11976 df-rp 12120 df-seq 13103 df-exp 13162 df-fmtno 42284 |
This theorem is referenced by: fmtnonn 42287 prmdvdsfmtnof 42342 prmdvdsfmtnof1 42343 |
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