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Mirrors > Home > MPE Home > Th. List > Mathboxes > fmtnoodd | Structured version Visualization version GIF version |
Description: Each Fermat number is odd. (Contributed by AV, 26-Jul-2021.) |
Ref | Expression |
---|---|
fmtnoodd | ⊢ (𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 12035 | . . . . . 6 ⊢ 2 ∈ ℕ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℕ) |
3 | id 22 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
4 | 2, 3 | nnexpcld 13949 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℕ) |
5 | nnm1nn0 12263 | . . . . . 6 ⊢ ((2↑𝑁) ∈ ℕ → ((2↑𝑁) − 1) ∈ ℕ0) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) − 1) ∈ ℕ0) |
7 | 2, 6 | nnexpcld 13949 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2↑((2↑𝑁) − 1)) ∈ ℕ) |
8 | 7 | nnzd 12414 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (2↑((2↑𝑁) − 1)) ∈ ℤ) |
9 | oveq2 7277 | . . . . 5 ⊢ (𝑘 = (2↑((2↑𝑁) − 1)) → (2 · 𝑘) = (2 · (2↑((2↑𝑁) − 1)))) | |
10 | 9 | oveq1d 7284 | . . . 4 ⊢ (𝑘 = (2↑((2↑𝑁) − 1)) → ((2 · 𝑘) + 1) = ((2 · (2↑((2↑𝑁) − 1))) + 1)) |
11 | fmtno 44938 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) | |
12 | 10, 11 | eqeqan12rd 2753 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑘 = (2↑((2↑𝑁) − 1))) → (((2 · 𝑘) + 1) = (FermatNo‘𝑁) ↔ ((2 · (2↑((2↑𝑁) − 1))) + 1) = ((2↑(2↑𝑁)) + 1))) |
13 | 2cnd 12040 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) | |
14 | 7 | nncnd 11978 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑((2↑𝑁) − 1)) ∈ ℂ) |
15 | 13, 14 | mulcomd 10985 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2 · (2↑((2↑𝑁) − 1))) = ((2↑((2↑𝑁) − 1)) · 2)) |
16 | expm1t 13800 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ (2↑𝑁) ∈ ℕ) → (2↑(2↑𝑁)) = ((2↑((2↑𝑁) − 1)) · 2)) | |
17 | 13, 4, 16 | syl2anc 584 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(2↑𝑁)) = ((2↑((2↑𝑁) − 1)) · 2)) |
18 | 15, 17 | eqtr4d 2781 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2 · (2↑((2↑𝑁) − 1))) = (2↑(2↑𝑁))) |
19 | 18 | oveq1d 7284 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((2 · (2↑((2↑𝑁) − 1))) + 1) = ((2↑(2↑𝑁)) + 1)) |
20 | 8, 12, 19 | rspcedvd 3564 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (FermatNo‘𝑁)) |
21 | fmtnonn 44940 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℕ) | |
22 | 21 | nnzd 12414 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℤ) |
23 | odd2np1 16039 | . . 3 ⊢ ((FermatNo‘𝑁) ∈ ℤ → (¬ 2 ∥ (FermatNo‘𝑁) ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (FermatNo‘𝑁))) | |
24 | 22, 23 | syl 17 | . 2 ⊢ (𝑁 ∈ ℕ0 → (¬ 2 ∥ (FermatNo‘𝑁) ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (FermatNo‘𝑁))) |
25 | 20, 24 | mpbird 256 | 1 ⊢ (𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 class class class wbr 5075 ‘cfv 6428 (class class class)co 7269 ℂcc 10858 1c1 10861 + caddc 10863 · cmul 10865 − cmin 11194 ℕcn 11962 2c2 12017 ℕ0cn0 12222 ℤcz 12308 ↑cexp 13771 ∥ cdvds 15952 FermatNocfmtno 44936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8487 df-en 8723 df-dom 8724 df-sdom 8725 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-div 11622 df-nn 11963 df-2 12025 df-3 12026 df-n0 12223 df-z 12309 df-uz 12572 df-rp 12720 df-seq 13711 df-exp 13772 df-dvds 15953 df-fmtno 44937 |
This theorem is referenced by: goldbachthlem2 44955 fmtnoprmfac1 44974 fmtnoprmfac2 44976 |
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