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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 12336 | . . . . . 6 ⊢ 2 ∈ ℕ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℕ) |
3 | id 22 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
4 | 2, 3 | nnexpcld 14280 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℕ) |
5 | nnm1nn0 12564 | . . . . . 6 ⊢ ((2↑𝑁) ∈ ℕ → ((2↑𝑁) − 1) ∈ ℕ0) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) − 1) ∈ ℕ0) |
7 | 2, 6 | nnexpcld 14280 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2↑((2↑𝑁) − 1)) ∈ ℕ) |
8 | 7 | nnzd 12637 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (2↑((2↑𝑁) − 1)) ∈ ℤ) |
9 | oveq2 7438 | . . . . 5 ⊢ (𝑘 = (2↑((2↑𝑁) − 1)) → (2 · 𝑘) = (2 · (2↑((2↑𝑁) − 1)))) | |
10 | 9 | oveq1d 7445 | . . . 4 ⊢ (𝑘 = (2↑((2↑𝑁) − 1)) → ((2 · 𝑘) + 1) = ((2 · (2↑((2↑𝑁) − 1))) + 1)) |
11 | fmtno 47453 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) | |
12 | 10, 11 | eqeqan12rd 2749 | . . 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 12341 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) | |
14 | 7 | nncnd 12279 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑((2↑𝑁) − 1)) ∈ ℂ) |
15 | 13, 14 | mulcomd 11279 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2 · (2↑((2↑𝑁) − 1))) = ((2↑((2↑𝑁) − 1)) · 2)) |
16 | expm1t 14127 | . . . . . 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 2777 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2 · (2↑((2↑𝑁) − 1))) = (2↑(2↑𝑁))) |
19 | 18 | oveq1d 7445 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((2 · (2↑((2↑𝑁) − 1))) + 1) = ((2↑(2↑𝑁)) + 1)) |
20 | 8, 12, 19 | rspcedvd 3623 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (FermatNo‘𝑁)) |
21 | fmtnonn 47455 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℕ) | |
22 | 21 | nnzd 12637 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℤ) |
23 | odd2np1 16374 | . . 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 257 | 1 ⊢ (𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 class class class wbr 5147 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 1c1 11153 + caddc 11155 · cmul 11157 − cmin 11489 ℕcn 12263 2c2 12318 ℕ0cn0 12523 ℤcz 12610 ↑cexp 14098 ∥ cdvds 16286 FermatNocfmtno 47451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-seq 14039 df-exp 14099 df-dvds 16287 df-fmtno 47452 |
This theorem is referenced by: goldbachthlem2 47470 fmtnoprmfac1 47489 fmtnoprmfac2 47491 |
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