1sgm2ppw - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > 1sgm2ppw		GIF version

Theorem 1sgm2ppw 15690

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 8108	. . 3 ⊢ 1 ∈ ℂ
2		2prm 12670	. . 3 ⊢ 2 ∈ ℙ
3		nnm1nn0 9426	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
4		sgmppw 15687	. . 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 1375	. 2 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘))
6		2rp 9871	. . . . . 6 ⊢ 2 ∈ ℝ⁺
7		rpcxp1 15594	. . . . . 6 ⊢ (2 ∈ ℝ⁺ → (2↑_𝑐1) = 2)
8	6, 7	mp1i 10	. . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (2↑_𝑐1) = 2)
9	8	oveq1d 6025	. . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → ((2↑_𝑐1)↑𝑘) = (2↑𝑘))
10	9	sumeq2i 11896	. . 3 ⊢ Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘)
11		2cn 9197	. . . . 5 ⊢ 2 ∈ ℂ
12	11	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ)
13		1ap2 9334	. . . . . 6 ⊢ 1 # 2
14		apsym 8769	. . . . . . 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 9392	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
19	12, 17, 18	geoserap 12039	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2)))
20	10, 19	eqtrid 2274	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2)))
21		2nn 9288	. . . . . . 7 ⊢ 2 ∈ ℕ
22		nnexpcl 10791	. . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℕ)
23	21, 18, 22	sylancr 414	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ)
24	23	nncnd 9140	. . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
25		subcl 8361	. . . . 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 8753	. . . . 5 ⊢ 1 # 0
29	28	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → 1 # 0)
30	26, 27, 29	div2negapd 8968	. . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1))
31		negsubdi2 8421	. . . . 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 8336	. . . . . 6 ⊢ -1 = (0 − 1)
34		0cn 8154	. . . . . . 7 ⊢ 0 ∈ ℂ
35		pnpcan 8401	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + 0) − (1 + 1)) = (0 − 1))
36	1, 34, 1, 35	mp3an 1371	. . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (0 − 1)
37		1p0e1 9242	. . . . . . 7 ⊢ (1 + 0) = 1
38		1p1e2 9243	. . . . . . 7 ⊢ (1 + 1) = 2
39	37, 38	oveq12i 6022	. . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (1 − 2)
40	33, 36, 39	3eqtr2i 2256	. . . . 5 ⊢ -1 = (1 − 2)
41	40	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → -1 = (1 − 2))
42	32, 41	oveq12d 6028	. . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2)))
43	26	div1d 8943	. . 3 ⊢ (𝑁 ∈ ℕ → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1))
44	30, 42, 43	3eqtr3d 2270	. 2 ⊢ (𝑁 ∈ ℕ → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1))
45	5, 20, 44	3eqtrd 2266	1 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1395 ∈ wcel 2200 class class class wbr 4083 (class class class)co 6010 ℂcc 8013 0cc0 8015 1c1 8016 + caddc 8018 − cmin 8333 -cneg 8334 # cap 8744 / cdiv 8835 ℕcn 9126 2c2 9177 ℕ₀cn0 9385 ℝ⁺crp 9866 ...cfz 10221 ↑cexp 10777 Σcsu 11885 ℙcprime 12650 ↑_𝑐ccxp 15552 σ csgm 15676

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4259 ax-pr 4294 ax-un 4525 ax-setind 4630 ax-iinf 4681 ax-cnex 8106 ax-resscn 8107 ax-1cn 8108 ax-1re 8109 ax-icn 8110 ax-addcl 8111 ax-addrcl 8112 ax-mulcl 8113 ax-mulrcl 8114 ax-addcom 8115 ax-mulcom 8116 ax-addass 8117 ax-mulass 8118 ax-distr 8119 ax-i2m1 8120 ax-0lt1 8121 ax-1rid 8122 ax-0id 8123 ax-rnegex 8124 ax-precex 8125 ax-cnre 8126 ax-pre-ltirr 8127 ax-pre-ltwlin 8128 ax-pre-lttrn 8129 ax-pre-apti 8130 ax-pre-ltadd 8131 ax-pre-mulgt0 8132 ax-pre-mulext 8133 ax-arch 8134 ax-caucvg 8135 ax-pre-suploc 8136 ax-addf 8137 ax-mulf 8138

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-disj 4060 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4385 df-po 4388 df-iso 4389 df-iord 4458 df-on 4460 df-ilim 4461 df-suc 4463 df-iom 4684 df-xp 4726 df-rel 4727 df-cnv 4728 df-co 4729 df-dm 4730 df-rn 4731 df-res 4732 df-ima 4733 df-iota 5281 df-fun 5323 df-fn 5324 df-f 5325 df-f1 5326 df-fo 5327 df-f1o 5328 df-fv 5329 df-isom 5330 df-riota 5963 df-ov 6013 df-oprab 6014 df-mpo 6015 df-of 6227 df-1st 6295 df-2nd 6296 df-recs 6462 df-irdg 6527 df-frec 6548 df-1o 6573 df-2o 6574 df-oadd 6577 df-er 6693 df-map 6810 df-pm 6811 df-en 6901 df-dom 6902 df-fin 6903 df-sup 7167 df-inf 7168 df-pnf 8199 df-mnf 8200 df-xr 8201 df-ltxr 8202 df-le 8203 df-sub 8335 df-neg 8336 df-reap 8738 df-ap 8745 df-div 8836 df-inn 9127 df-2 9185 df-3 9186 df-4 9187 df-n0 9386 df-xnn0 9449 df-z 9463 df-uz 9739 df-q 9832 df-rp 9867 df-xneg 9985 df-xadd 9986 df-ioo 10105 df-ico 10107 df-icc 10108 df-fz 10222 df-fzo 10356 df-fl 10507 df-mod 10562 df-seqfrec 10687 df-exp 10778 df-fac 10965 df-bc 10987 df-ihash 11015 df-shft 11347 df-cj 11374 df-re 11375 df-im 11376 df-rsqrt 11530 df-abs 11531 df-clim 11811 df-sumdc 11886 df-ef 12180 df-e 12181 df-dvds 12320 df-gcd 12496 df-prm 12651 df-pc 12829 df-rest 13295 df-topgen 13314 df-psmet 14528 df-xmet 14529 df-met 14530 df-bl 14531 df-mopn 14532 df-top 14693 df-topon 14706 df-bases 14738 df-ntr 14791 df-cn 14883 df-cnp 14884 df-tx 14948 df-cncf 15266 df-limced 15351 df-dvap 15352 df-relog 15553 df-rpcxp 15554 df-sgm 15677

This theorem is referenced by: perfect1 15693 perfectlem1 15694 perfectlem2 15695

W3C validator