Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 11704 | . . . . . 6 ⊢ 2 ∈ ℕ | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℕ) |
3 | id 22 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0) | |
4 | 2, 3 | nnexpcld 13600 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℕ) |
5 | nnm1nn0 11932 | . . . . . 6 ⊢ ((2↑𝑁) ∈ ℕ → ((2↑𝑁) − 1) ∈ ℕ0) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) − 1) ∈ ℕ0) |
7 | 2, 6 | nnexpcld 13600 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2↑((2↑𝑁) − 1)) ∈ ℕ) |
8 | 7 | nnzd 12080 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (2↑((2↑𝑁) − 1)) ∈ ℤ) |
9 | oveq2 7158 | . . . . 5 ⊢ (𝑘 = (2↑((2↑𝑁) − 1)) → (2 · 𝑘) = (2 · (2↑((2↑𝑁) − 1)))) | |
10 | 9 | oveq1d 7165 | . . . 4 ⊢ (𝑘 = (2↑((2↑𝑁) − 1)) → ((2 · 𝑘) + 1) = ((2 · (2↑((2↑𝑁) − 1))) + 1)) |
11 | fmtno 43684 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1)) | |
12 | 10, 11 | eqeqan12rd 2840 | . . 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 11709 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) | |
14 | 7 | nncnd 11648 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2↑((2↑𝑁) − 1)) ∈ ℂ) |
15 | 13, 14 | mulcomd 10656 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2 · (2↑((2↑𝑁) − 1))) = ((2↑((2↑𝑁) − 1)) · 2)) |
16 | expm1t 13451 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ (2↑𝑁) ∈ ℕ) → (2↑(2↑𝑁)) = ((2↑((2↑𝑁) − 1)) · 2)) | |
17 | 13, 4, 16 | syl2anc 586 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(2↑𝑁)) = ((2↑((2↑𝑁) − 1)) · 2)) |
18 | 15, 17 | eqtr4d 2859 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2 · (2↑((2↑𝑁) − 1))) = (2↑(2↑𝑁))) |
19 | 18 | oveq1d 7165 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((2 · (2↑((2↑𝑁) − 1))) + 1) = ((2↑(2↑𝑁)) + 1)) |
20 | 8, 12, 19 | rspcedvd 3626 | . 2 ⊢ (𝑁 ∈ ℕ0 → ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = (FermatNo‘𝑁)) |
21 | fmtnonn 43686 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℕ) | |
22 | 21 | nnzd 12080 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℤ) |
23 | odd2np1 15684 | . . 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 259 | 1 ⊢ (𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 1c1 10532 + caddc 10534 · cmul 10536 − cmin 10864 ℕcn 11632 2c2 11686 ℕ0cn0 11891 ℤcz 11975 ↑cexp 13423 ∥ cdvds 15601 FermatNocfmtno 43682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13364 df-exp 13424 df-dvds 15602 df-fmtno 43683 |
This theorem is referenced by: goldbachthlem2 43701 fmtnoprmfac1 43720 fmtnoprmfac2 43722 |
Copyright terms: Public domain | W3C validator |