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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 10583 | . . 3 ⊢ 1 ∈ ℂ | |
2 | 2prm 16024 | . . 3 ⊢ 2 ∈ ℙ | |
3 | nnm1nn0 11926 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
4 | sgmppw 25700 | . . 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 1456 | . 2 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((2↑𝑐1)↑𝑘)) |
6 | 2cn 11700 | . . . . . 6 ⊢ 2 ∈ ℂ | |
7 | cxp1 25181 | . . . . . 6 ⊢ (2 ∈ ℂ → (2↑𝑐1) = 2) | |
8 | 6, 7 | mp1i 13 | . . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (2↑𝑐1) = 2) |
9 | 8 | oveq1d 7160 | . . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → ((2↑𝑐1)↑𝑘) = (2↑𝑘)) |
10 | 9 | sumeq2i 15044 | . . 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 11833 | . . . . . 6 ⊢ 1 ≠ 2 | |
13 | 12 | necomi 3067 | . . . . 5 ⊢ 2 ≠ 1 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → 2 ≠ 1) |
15 | nnnn0 11892 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
16 | 11, 14, 15 | geoser 15210 | . . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2))) |
17 | 10, 16 | syl5eq 2865 | . 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))((2↑𝑐1)↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2))) |
18 | 2nn 11698 | . . . . . . 7 ⊢ 2 ∈ ℕ | |
19 | nnexpcl 13430 | . . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℕ) | |
20 | 18, 15, 19 | sylancr 587 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ) |
21 | 20 | nncnd 11642 | . . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ) |
22 | subcl 10873 | . . . . 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 10594 | . . . . 5 ⊢ 1 ≠ 0 | |
26 | 25 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → 1 ≠ 0) |
27 | 23, 24, 26 | div2negd 11419 | . . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1)) |
28 | negsubdi2 10933 | . . . . 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 10861 | . . . . . 6 ⊢ -1 = (0 − 1) | |
31 | 0cn 10621 | . . . . . . 7 ⊢ 0 ∈ ℂ | |
32 | pnpcan 10913 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + 0) − (1 + 1)) = (0 − 1)) | |
33 | 1, 31, 1, 32 | mp3an 1452 | . . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (0 − 1) |
34 | 1p0e1 11749 | . . . . . . 7 ⊢ (1 + 0) = 1 | |
35 | 1p1e2 11750 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
36 | 34, 35 | oveq12i 7157 | . . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (1 − 2) |
37 | 30, 33, 36 | 3eqtr2i 2847 | . . . . 5 ⊢ -1 = (1 − 2) |
38 | 37 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → -1 = (1 − 2)) |
39 | 29, 38 | oveq12d 7163 | . . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2))) |
40 | 23 | div1d 11396 | . . 3 ⊢ (𝑁 ∈ ℕ → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1)) |
41 | 27, 39, 40 | 3eqtr3d 2861 | . 2 ⊢ (𝑁 ∈ ℕ → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1)) |
42 | 5, 17, 41 | 3eqtrd 2857 | 1 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 − cmin 10858 -cneg 10859 / cdiv 11285 ℕcn 11626 2c2 11680 ℕ0cn0 11885 ...cfz 12880 ↑cexp 13417 Σcsu 15030 ℙcprime 16003 ↑𝑐ccxp 25066 σ csgm 25600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-fac 13622 df-bc 13651 df-hash 13679 df-shft 14414 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-limsup 14816 df-clim 14833 df-rlim 14834 df-sum 15031 df-ef 15409 df-sin 15411 df-cos 15412 df-pi 15414 df-dvds 15596 df-gcd 15832 df-prm 16004 df-pc 16162 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-lp 21672 df-perf 21673 df-cn 21763 df-cnp 21764 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-limc 24391 df-dv 24392 df-log 25067 df-cxp 25068 df-sgm 25606 |
This theorem is referenced by: perfect1 25731 perfectlem1 25732 perfectlem2 25733 perfectALTVlem1 43763 perfectALTVlem2 43764 |
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