1sgm2ppw - Intuitionistic Logic Explorer

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Theorem 1sgm2ppw 15317

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 7991	. . 3 ⊢ 1 ∈ ℂ
2		2prm 12322	. . 3 ⊢ 2 ∈ ℙ
3		nnm1nn0 9309	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
4		sgmppw 15314	. . 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 1352	. 2 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘))
6		2rp 9752	. . . . . 6 ⊢ 2 ∈ ℝ⁺
7		rpcxp1 15221	. . . . . 6 ⊢ (2 ∈ ℝ⁺ → (2↑_𝑐1) = 2)
8	6, 7	mp1i 10	. . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (2↑_𝑐1) = 2)
9	8	oveq1d 5940	. . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → ((2↑_𝑐1)↑𝑘) = (2↑𝑘))
10	9	sumeq2i 11548	. . 3 ⊢ Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘)
11		2cn 9080	. . . . 5 ⊢ 2 ∈ ℂ
12	11	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ)
13		1ap2 9217	. . . . . 6 ⊢ 1 # 2
14		apsym 8652	. . . . . . 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 9275	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
19	12, 17, 18	geoserap 11691	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2)))
20	10, 19	eqtrid 2241	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2)))
21		2nn 9171	. . . . . . 7 ⊢ 2 ∈ ℕ
22		nnexpcl 10663	. . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℕ)
23	21, 18, 22	sylancr 414	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ)
24	23	nncnd 9023	. . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
25		subcl 8244	. . . . 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 8636	. . . . 5 ⊢ 1 # 0
29	28	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → 1 # 0)
30	26, 27, 29	div2negapd 8851	. . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1))
31		negsubdi2 8304	. . . . 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 8219	. . . . . 6 ⊢ -1 = (0 − 1)
34		0cn 8037	. . . . . . 7 ⊢ 0 ∈ ℂ
35		pnpcan 8284	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + 0) − (1 + 1)) = (0 − 1))
36	1, 34, 1, 35	mp3an 1348	. . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (0 − 1)
37		1p0e1 9125	. . . . . . 7 ⊢ (1 + 0) = 1
38		1p1e2 9126	. . . . . . 7 ⊢ (1 + 1) = 2
39	37, 38	oveq12i 5937	. . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (1 − 2)
40	33, 36, 39	3eqtr2i 2223	. . . . 5 ⊢ -1 = (1 − 2)
41	40	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → -1 = (1 − 2))
42	32, 41	oveq12d 5943	. . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2)))
43	26	div1d 8826	. . 3 ⊢ (𝑁 ∈ ℕ → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1))
44	30, 42, 43	3eqtr3d 2237	. 2 ⊢ (𝑁 ∈ ℕ → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1))
45	5, 20, 44	3eqtrd 2233	1 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1364 ∈ wcel 2167 class class class wbr 4034 (class class class)co 5925 ℂcc 7896 0cc0 7898 1c1 7899 + caddc 7901 − cmin 8216 -cneg 8217 # cap 8627 / cdiv 8718 ℕcn 9009 2c2 9060 ℕ₀cn0 9268 ℝ⁺crp 9747 ...cfz 10102 ↑cexp 10649 Σcsu 11537 ℙcprime 12302 ↑_𝑐ccxp 15179 σ csgm 15303

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 ax-pre-mulext 8016 ax-arch 8017 ax-caucvg 8018 ax-pre-suploc 8019 ax-addf 8020 ax-mulf 8021

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-disj 4012 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-isom 5268 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-of 6139 df-1st 6207 df-2nd 6208 df-recs 6372 df-irdg 6437 df-frec 6458 df-1o 6483 df-2o 6484 df-oadd 6487 df-er 6601 df-map 6718 df-pm 6719 df-en 6809 df-dom 6810 df-fin 6811 df-sup 7059 df-inf 7060 df-pnf 8082 df-mnf 8083 df-xr 8084 df-ltxr 8085 df-le 8086 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 df-div 8719 df-inn 9010 df-2 9068 df-3 9069 df-4 9070 df-n0 9269 df-xnn0 9332 df-z 9346 df-uz 9621 df-q 9713 df-rp 9748 df-xneg 9866 df-xadd 9867 df-ioo 9986 df-ico 9988 df-icc 9989 df-fz 10103 df-fzo 10237 df-fl 10379 df-mod 10434 df-seqfrec 10559 df-exp 10650 df-fac 10837 df-bc 10859 df-ihash 10887 df-shft 10999 df-cj 11026 df-re 11027 df-im 11028 df-rsqrt 11182 df-abs 11183 df-clim 11463 df-sumdc 11538 df-ef 11832 df-e 11833 df-dvds 11972 df-gcd 12148 df-prm 12303 df-pc 12481 df-rest 12945 df-topgen 12964 df-psmet 14177 df-xmet 14178 df-met 14179 df-bl 14180 df-mopn 14181 df-top 14320 df-topon 14333 df-bases 14365 df-ntr 14418 df-cn 14510 df-cnp 14511 df-tx 14575 df-cncf 14893 df-limced 14978 df-dvap 14979 df-relog 15180 df-rpcxp 15181 df-sgm 15304

This theorem is referenced by: perfect1 15320 perfectlem1 15321 perfectlem2 15322

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