1sgm2ppw - Intuitionistic Logic Explorer

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Theorem 1sgm2ppw 15975

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 8236	. . 3 ⊢ 1 ∈ ℂ
2		2prm 12849	. . 3 ⊢ 2 ∈ ℙ
3		nnm1nn0 9554	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
4		sgmppw 15972	. . 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 1378	. 2 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘))
6		2rp 10009	. . . . . 6 ⊢ 2 ∈ ℝ⁺
7		rpcxp1 15876	. . . . . 6 ⊢ (2 ∈ ℝ⁺ → (2↑_𝑐1) = 2)
8	6, 7	mp1i 10	. . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (2↑_𝑐1) = 2)
9	8	oveq1d 6073	. . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → ((2↑_𝑐1)↑𝑘) = (2↑𝑘))
10	9	sumeq2i 12074	. . 3 ⊢ Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘)
11		2cn 9325	. . . . 5 ⊢ 2 ∈ ℂ
12	11	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ)
13		1ap2 9462	. . . . . 6 ⊢ 1 # 2
14		apsym 8897	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 2 ∈ ℂ) → (1 # 2 ↔ 2 # 1))
15	1, 11, 14	mp2an 426	. . . . . 6 ⊢ (1 # 2 ↔ 2 # 1)
16	13, 15	mpbi 145	. . . . 5 ⊢ 2 # 1
17	16	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → 2 # 1)
18		nnnn0 9520	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
19	12, 17, 18	geoserap 12218	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2)))
20	10, 19	eqtrid 2279	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2)))
21		2nn 9416	. . . . . . 7 ⊢ 2 ∈ ℕ
22		nnexpcl 10938	. . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℕ)
23	21, 18, 22	sylancr 414	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ)
24	23	nncnd 9268	. . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
25		subcl 8488	. . . . 5 ⊢ (((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → ((2↑𝑁) − 1) ∈ ℂ)
26	24, 1, 25	sylancl 413	. . . 4 ⊢ (𝑁 ∈ ℕ → ((2↑𝑁) − 1) ∈ ℂ)
27	1	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → 1 ∈ ℂ)
28		1ap0 8881	. . . . 5 ⊢ 1 # 0
29	28	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → 1 # 0)
30	26, 27, 29	div2negapd 9096	. . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1))
31		negsubdi2 8548	. . . . 5 ⊢ (((2↑𝑁) ∈ ℂ ∧ 1 ∈ ℂ) → -((2↑𝑁) − 1) = (1 − (2↑𝑁)))
32	24, 1, 31	sylancl 413	. . . 4 ⊢ (𝑁 ∈ ℕ → -((2↑𝑁) − 1) = (1 − (2↑𝑁)))
33		df-neg 8463	. . . . . 6 ⊢ -1 = (0 − 1)
34		0cn 8282	. . . . . . 7 ⊢ 0 ∈ ℂ
35		pnpcan 8528	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + 0) − (1 + 1)) = (0 − 1))
36	1, 34, 1, 35	mp3an 1374	. . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (0 − 1)
37		1p0e1 9370	. . . . . . 7 ⊢ (1 + 0) = 1
38		1p1e2 9371	. . . . . . 7 ⊢ (1 + 1) = 2
39	37, 38	oveq12i 6070	. . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (1 − 2)
40	33, 36, 39	3eqtr2i 2261	. . . . 5 ⊢ -1 = (1 − 2)
41	40	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → -1 = (1 − 2))
42	32, 41	oveq12d 6076	. . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2)))
43	26	div1d 9071	. . 3 ⊢ (𝑁 ∈ ℕ → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1))
44	30, 42, 43	3eqtr3d 2275	. 2 ⊢ (𝑁 ∈ ℕ → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1))
45	5, 20, 44	3eqtrd 2271	1 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1398 ∈ wcel 2205 class class class wbr 4114 (class class class)co 6058 ℂcc 8141 0cc0 8143 1c1 8144 + caddc 8146 − cmin 8460 -cneg 8461 # cap 8872 / cdiv 8963 ℕcn 9254 2c2 9305 ℕ₀cn0 9513 ℝ⁺crp 10004 ...cfz 10361 ↑cexp 10924 Σcsu 12063 ℙcprime 12829 ↑_𝑐ccxp 15834 σ csgm 15961

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261 ax-arch 8262 ax-caucvg 8263 ax-pre-suploc 8264 ax-addf 8265 ax-mulf 8266

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-disj 4091 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-isom 5366 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-of 6275 df-1st 6347 df-2nd 6348 df-recs 6549 df-irdg 6614 df-frec 6635 df-1o 6660 df-2o 6661 df-oadd 6664 df-er 6780 df-map 6897 df-pm 6898 df-en 6989 df-dom 6990 df-fin 6991 df-sup 7288 df-inf 7289 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-n0 9514 df-xnn0 9581 df-z 9595 df-uz 9872 df-q 9970 df-rp 10005 df-xneg 10124 df-xadd 10125 df-ioo 10244 df-ico 10246 df-icc 10247 df-fz 10362 df-fzo 10499 df-fl 10654 df-mod 10709 df-seqfrec 10834 df-exp 10925 df-fac 11113 df-bc 11135 df-ihash 11164 df-shft 11525 df-cj 11552 df-re 11553 df-im 11554 df-rsqrt 11708 df-abs 11709 df-clim 11989 df-sumdc 12064 df-ef 12359 df-e 12360 df-dvds 12499 df-gcd 12675 df-prm 12830 df-pc 13008 df-rest 13538 df-topgen 13557 df-psmet 14803 df-xmet 14804 df-met 14805 df-bl 14806 df-mopn 14807 df-top 14975 df-topon 14988 df-bases 15020 df-ntr 15073 df-cn 15165 df-cnp 15166 df-tx 15230 df-cncf 15548 df-limced 15633 df-dvap 15634 df-relog 15835 df-rpcxp 15836 df-sgm 15962

This theorem is referenced by: perfect1 15978 perfectlem1 15979 perfectlem2 15980

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