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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 43568 | . 2 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) | |
2 | 3z 12003 | . . . 4 ⊢ 3 ∈ ℤ | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 3 ∈ ℤ) |
4 | 2nn0 11902 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℕ0) |
6 | id 22 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
7 | 5, 6 | nn0expcld 13595 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℕ0) |
8 | 5, 7 | nn0expcld 13595 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(2↑𝑁)) ∈ ℕ0) |
9 | peano2nn0 11925 | . . . . 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 12073 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ ℤ) |
12 | 3m1e2 11753 | . . . . 5 ⊢ (3 − 1) = 2 | |
13 | 2cn 11700 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
14 | exp1 13423 | . . . . . . 7 ⊢ (2 ∈ ℂ → (2↑1) = 2) | |
15 | 13, 14 | ax-mp 5 | . . . . . 6 ⊢ (2↑1) = 2 |
16 | 2re 11699 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
17 | 16 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
18 | 1le2 11834 | . . . . . . . . 9 ⊢ 1 ≤ 2 | |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 ≤ 2) |
20 | 17, 6, 19 | expge1d 13517 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 1 ≤ (2↑𝑁)) |
21 | 1zzd 12001 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℤ) | |
22 | 7 | nn0zd 12073 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℤ) |
23 | 1lt2 11796 | . . . . . . . . 9 ⊢ 1 < 2 | |
24 | 23 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 < 2) |
25 | 17, 21, 22, 24 | leexp2d 13603 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (1 ≤ (2↑𝑁) ↔ (2↑1) ≤ (2↑(2↑𝑁)))) |
26 | 20, 25 | mpbid 233 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑1) ≤ (2↑(2↑𝑁))) |
27 | 15, 26 | eqbrtrrid 5093 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 2 ≤ (2↑(2↑𝑁))) |
28 | 12, 27 | eqbrtrid 5092 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (3 − 1) ≤ (2↑(2↑𝑁))) |
29 | 3re 11705 | . . . . . 6 ⊢ 3 ∈ ℝ | |
30 | 29 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 3 ∈ ℝ) |
31 | 1red 10630 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
32 | 8 | nn0red 11944 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(2↑𝑁)) ∈ ℝ) |
33 | 30, 31, 32 | lesubaddd 11225 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((3 − 1) ≤ (2↑(2↑𝑁)) ↔ 3 ≤ ((2↑(2↑𝑁)) + 1))) |
34 | 28, 33 | mpbid 233 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 3 ≤ ((2↑(2↑𝑁)) + 1)) |
35 | eluz2 12237 | . . 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 1335 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ (ℤ≥‘3)) |
37 | 1, 36 | eqeltrd 2910 | 1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ≥‘3)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 1c1 10526 + caddc 10528 < clt 10663 ≤ cle 10664 − cmin 10858 2c2 11680 3c3 11681 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 ↑cexp 13417 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-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 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-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-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-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 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-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-fmtno 43567 |
This theorem is referenced by: fmtnonn 43570 prmdvdsfmtnof 43625 prmdvdsfmtnof1 43626 |
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