1sgm2ppw - Metamath Proof Explorer

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Theorem 1sgm2ppw 27244

Description: The sum of the divisors of 2↑(𝑁 − 1). (Contributed by Mario Carneiro, 17-May-2016.)

**Assertion**
Ref	Expression
1sgm2ppw	⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1))

Proof of Theorem 1sgm2ppw Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1cn 11213	. . 3 ⊢ 1 ∈ ℂ
2		2prm 16729	. . 3 ⊢ 2 ∈ ℙ
3		nnm1nn0 12567	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
4		sgmppw 27241	. . 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 1467	. 2 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘))
6		2cn 12341	. . . . . 6 ⊢ 2 ∈ ℂ
7		cxp1 26713	. . . . . 6 ⊢ (2 ∈ ℂ → (2↑_𝑐1) = 2)
8	6, 7	mp1i 13	. . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (2↑_𝑐1) = 2)
9	8	oveq1d 7446	. . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → ((2↑_𝑐1)↑𝑘) = (2↑𝑘))
10	9	sumeq2i 15734	. . 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 12474	. . . . . 6 ⊢ 1 ≠ 2
13	12	necomi 2995	. . . . 5 ⊢ 2 ≠ 1
14	13	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ → 2 ≠ 1)
15		nnnn0 12533	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
16	11, 14, 15	geoser 15903	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2)))
17	10, 16	eqtrid 2789	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2)))
18		2nn 12339	. . . . . . 7 ⊢ 2 ∈ ℕ
19		nnexpcl 14115	. . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℕ)
20	18, 15, 19	sylancr 587	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ)
21	20	nncnd 12282	. . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
22		subcl 11507	. . . . 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 11224	. . . . 5 ⊢ 1 ≠ 0
26	25	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ → 1 ≠ 0)
27	23, 24, 26	div2negd 12058	. . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1))
28		negsubdi2 11568	. . . . 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 11495	. . . . . 6 ⊢ -1 = (0 − 1)
31		0cn 11253	. . . . . . 7 ⊢ 0 ∈ ℂ
32		pnpcan 11548	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + 0) − (1 + 1)) = (0 − 1))
33	1, 31, 1, 32	mp3an 1463	. . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (0 − 1)
34		1p0e1 12390	. . . . . . 7 ⊢ (1 + 0) = 1
35		1p1e2 12391	. . . . . . 7 ⊢ (1 + 1) = 2
36	34, 35	oveq12i 7443	. . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (1 − 2)
37	30, 33, 36	3eqtr2i 2771	. . . . 5 ⊢ -1 = (1 − 2)
38	37	a1i 11	. . . 4 ⊢ (𝑁 ∈ ℕ → -1 = (1 − 2))
39	29, 38	oveq12d 7449	. . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2)))
40	23	div1d 12035	. . 3 ⊢ (𝑁 ∈ ℕ → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1))
41	27, 39, 40	3eqtr3d 2785	. 2 ⊢ (𝑁 ∈ ℕ → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1))
42	5, 17, 41	3eqtrd 2781	1 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 − cmin 11492 -cneg 11493 / cdiv 11920 ℕcn 12266 2c2 12321 ℕ₀cn0 12526 ...cfz 13547 ↑cexp 14102 Σcsu 15722 ℙcprime 16708 ↑_𝑐ccxp 26597 σ csgm 27139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-fac 14313 df-bc 14342 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 df-sin 16105 df-cos 16106 df-pi 16108 df-dvds 16291 df-gcd 16532 df-prm 16709 df-pc 16875 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-limc 25901 df-dv 25902 df-log 26598 df-cxp 26599 df-sgm 27145

This theorem is referenced by: perfect1 27272 perfectlem1 27273 perfectlem2 27274 perfectALTVlem1 47708 perfectALTVlem2 47709

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