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Mirrors > Home > MPE Home > Th. List > 1sgm2ppw | Structured version Visualization version GIF version |
Description: The sum of the divisors of 2↑(𝑁 − 1). (Contributed by Mario Carneiro, 17-May-2016.) |
Ref | Expression |
---|---|
1sgm2ppw | ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 11150 | . . 3 ⊢ 1 ∈ ℂ | |
2 | 2prm 16611 | . . 3 ⊢ 2 ∈ ℙ | |
3 | nnm1nn0 12495 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
4 | sgmppw 26627 | . . 3 ⊢ ((1 ∈ ℂ ∧ 2 ∈ ℙ ∧ (𝑁 − 1) ∈ ℕ0) → (1 σ (2↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((2↑𝑐1)↑𝑘)) | |
5 | 1, 2, 3, 4 | mp3an12i 1465 | . 2 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((2↑𝑐1)↑𝑘)) |
6 | 2cn 12269 | . . . . . 6 ⊢ 2 ∈ ℂ | |
7 | cxp1 26108 | . . . . . 6 ⊢ (2 ∈ ℂ → (2↑𝑐1) = 2) | |
8 | 6, 7 | mp1i 13 | . . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (2↑𝑐1) = 2) |
9 | 8 | oveq1d 7408 | . . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → ((2↑𝑐1)↑𝑘) = (2↑𝑘)) |
10 | 9 | sumeq2i 15627 | . . 3 ⊢ Σ𝑘 ∈ (0...(𝑁 − 1))((2↑𝑐1)↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) |
11 | 6 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) |
12 | 1ne2 12402 | . . . . . 6 ⊢ 1 ≠ 2 | |
13 | 12 | necomi 2994 | . . . . 5 ⊢ 2 ≠ 1 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → 2 ≠ 1) |
15 | nnnn0 12461 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
16 | 11, 14, 15 | geoser 15795 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2))) |
17 | 10, 16 | eqtrid 2783 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))((2↑𝑐1)↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2))) |
18 | 2nn 12267 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
19 | nnexpcl 14022 | . . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℕ) | |
20 | 18, 15, 19 | sylancr 587 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ) |
21 | 20 | nncnd 12210 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ) |
22 | subcl 11441 | . . . . 5 ⊢ (((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → ((2↑𝑁) − 1) ∈ ℂ) | |
23 | 21, 1, 22 | sylancl 586 | . . . 4 ⊢ (𝑁 ∈ ℕ → ((2↑𝑁) − 1) ∈ ℂ) |
24 | 1 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ) |
25 | ax-1ne0 11161 | . . . . 5 ⊢ 1 ≠ 0 | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ≠ 0) |
27 | 23, 24, 26 | div2negd 11987 | . . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1)) |
28 | negsubdi2 11501 | . . . . 5 ⊢ (((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → -((2↑𝑁) − 1) = (1 − (2↑𝑁))) | |
29 | 21, 1, 28 | sylancl 586 | . . . 4 ⊢ (𝑁 ∈ ℕ → -((2↑𝑁) − 1) = (1 − (2↑𝑁))) |
30 | df-neg 11429 | . . . . . 6 ⊢ -1 = (0 − 1) | |
31 | 0cn 11188 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
32 | pnpcan 11481 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + 0) − (1 + 1)) = (0 − 1)) | |
33 | 1, 31, 1, 32 | mp3an 1461 | . . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (0 − 1) |
34 | 1p0e1 12318 | . . . . . . 7 ⊢ (1 + 0) = 1 | |
35 | 1p1e2 12319 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
36 | 34, 35 | oveq12i 7405 | . . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (1 − 2) |
37 | 30, 33, 36 | 3eqtr2i 2765 | . . . . 5 ⊢ -1 = (1 − 2) |
38 | 37 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → -1 = (1 − 2)) |
39 | 29, 38 | oveq12d 7411 | . . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2))) |
40 | 23 | div1d 11964 | . . 3 ⊢ (𝑁 ∈ ℕ → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1)) |
41 | 27, 39, 40 | 3eqtr3d 2779 | . 2 ⊢ (𝑁 ∈ ℕ → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1)) |
42 | 5, 17, 41 | 3eqtrd 2775 | 1 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 (class class class)co 7393 ℂcc 11090 0cc0 11092 1c1 11093 + caddc 11095 − cmin 11426 -cneg 11427 / cdiv 11853 ℕcn 12194 2c2 12249 ℕ0cn0 12454 ...cfz 13466 ↑cexp 14009 Σcsu 15614 ℙcprime 16590 ↑𝑐ccxp 25993 σ csgm 26527 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-er 8686 df-map 8805 df-pm 8806 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13467 df-fzo 13610 df-fl 13739 df-mod 13817 df-seq 13949 df-exp 14010 df-fac 14216 df-bc 14245 df-hash 14273 df-shft 14996 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-limsup 15397 df-clim 15414 df-rlim 15415 df-sum 15615 df-ef 15993 df-sin 15995 df-cos 15996 df-pi 15998 df-dvds 16180 df-gcd 16418 df-prm 16591 df-pc 16752 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17350 df-topn 17351 df-0g 17369 df-gsum 17370 df-topgen 17371 df-pt 17372 df-prds 17375 df-xrs 17430 df-qtop 17435 df-imas 17436 df-xps 17438 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-mulg 18923 df-cntz 19147 df-cmn 19614 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-fbas 20875 df-fg 20876 df-cnfld 20879 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-cld 22452 df-ntr 22453 df-cls 22454 df-nei 22531 df-lp 22569 df-perf 22570 df-cn 22660 df-cnp 22661 df-haus 22748 df-tx 22995 df-hmeo 23188 df-fil 23279 df-fm 23371 df-flim 23372 df-flf 23373 df-xms 23755 df-ms 23756 df-tms 23757 df-cncf 24323 df-limc 25312 df-dv 25313 df-log 25994 df-cxp 25995 df-sgm 26533 |
This theorem is referenced by: perfect1 26658 perfectlem1 26659 perfectlem2 26660 perfectALTVlem1 46161 perfectALTVlem2 46162 |
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