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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 44046 | . 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 13603 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℕ0) |
8 | 5, 7 | nn0expcld 13603 | . . . . 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 13431 | . . . . . . 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 13525 | . . . . . . 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 13611 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (1 ≤ (2↑𝑁) ↔ (2↑1) ≤ (2↑(2↑𝑁)))) |
26 | 20, 25 | mpbid 235 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑1) ≤ (2↑(2↑𝑁))) |
27 | 15, 26 | eqbrtrrid 5066 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 2 ≤ (2↑(2↑𝑁))) |
28 | 12, 27 | eqbrtrid 5065 | . . . 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 10631 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
32 | 8 | nn0red 11944 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(2↑𝑁)) ∈ ℝ) |
33 | 30, 31, 32 | lesubaddd 11226 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((3 − 1) ≤ (2↑(2↑𝑁)) ↔ 3 ≤ ((2↑(2↑𝑁)) + 1))) |
34 | 28, 33 | mpbid 235 | . . 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 1340 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ (ℤ≥‘3)) |
37 | 1, 36 | eqeltrd 2890 | 1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ≥‘3)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 − cmin 10859 2c2 11680 3c3 11681 ℕ0cn0 11885 ℤcz 11969 ℤ≥cuz 12231 ↑cexp 13425 FermatNocfmtno 44044 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-fmtno 44045 |
This theorem is referenced by: fmtnonn 44048 prmdvdsfmtnof 44103 prmdvdsfmtnof1 44104 |
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