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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 39979 | . 2 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) | |
2 | 3z 11287 | . . . 4 ⊢ 3 ∈ ℤ | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 3 ∈ ℤ) |
4 | 2nn0 11186 | . . . . . . 7 ⊢ 2 ∈ ℕ0 | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℕ0) |
6 | id 22 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
7 | 5, 6 | nn0expcld 12893 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℕ0) |
8 | 5, 7 | nn0expcld 12893 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(2↑𝑁)) ∈ ℕ0) |
9 | peano2nn0 11210 | . . . . 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 11356 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ ℤ) |
12 | 3m1e2 11014 | . . . . 5 ⊢ (3 − 1) = 2 | |
13 | 2cn 10968 | . . . . . . 7 ⊢ 2 ∈ ℂ | |
14 | exp1 12728 | . . . . . . 7 ⊢ (2 ∈ ℂ → (2↑1) = 2) | |
15 | 13, 14 | ax-mp 5 | . . . . . 6 ⊢ (2↑1) = 2 |
16 | 2re 10967 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
17 | 16 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℝ) |
18 | 1le2 11118 | . . . . . . . . 9 ⊢ 1 ≤ 2 | |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 ≤ 2) |
20 | 17, 6, 19 | expge1d 12889 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 1 ≤ (2↑𝑁)) |
21 | 1zzd 11285 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℤ) | |
22 | 7 | nn0zd 11356 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℤ) |
23 | 1lt2 11071 | . . . . . . . . 9 ⊢ 1 < 2 | |
24 | 23 | a1i 11 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 1 < 2) |
25 | 17, 21, 22, 24 | leexp2d 12901 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (1 ≤ (2↑𝑁) ↔ (2↑1) ≤ (2↑(2↑𝑁)))) |
26 | 20, 25 | mpbid 221 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑1) ≤ (2↑(2↑𝑁))) |
27 | 15, 26 | syl5eqbrr 4619 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 2 ≤ (2↑(2↑𝑁))) |
28 | 12, 27 | syl5eqbr 4618 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (3 − 1) ≤ (2↑(2↑𝑁))) |
29 | 3re 10971 | . . . . . 6 ⊢ 3 ∈ ℝ | |
30 | 29 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 3 ∈ ℝ) |
31 | 1red 9934 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 1 ∈ ℝ) | |
32 | 8 | nn0red 11229 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(2↑𝑁)) ∈ ℝ) |
33 | 30, 31, 32 | lesubaddd 10503 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((3 − 1) ≤ (2↑(2↑𝑁)) ↔ 3 ≤ ((2↑(2↑𝑁)) + 1))) |
34 | 28, 33 | mpbid 221 | . . 3 ⊢ (𝑁 ∈ ℕ0 → 3 ≤ ((2↑(2↑𝑁)) + 1)) |
35 | eluz2 11569 | . . 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 1239 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑(2↑𝑁)) + 1) ∈ (ℤ≥‘3)) |
37 | 1, 36 | eqeltrd 2688 | 1 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ≥‘3)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 1c1 9816 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 2c2 10947 3c3 10948 ℕ0cn0 11169 ℤcz 11254 ℤ≥cuz 11563 ↑cexp 12722 FermatNocfmtno 39977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-fmtno 39978 |
This theorem is referenced by: fmtnonn 39981 prmdvdsfmtnof 40036 prmdvdsfmtnof1 40037 |
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