1sgm2ppw - Intuitionistic Logic Explorer

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Theorem 1sgm2ppw 15725

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 8125	. . 3 ⊢ 1 ∈ ℂ
2		2prm 12704	. . 3 ⊢ 2 ∈ ℙ
3		nnm1nn0 9443	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
4		sgmppw 15722	. . 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 1377	. 2 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘))
6		2rp 9893	. . . . . 6 ⊢ 2 ∈ ℝ⁺
7		rpcxp1 15629	. . . . . 6 ⊢ (2 ∈ ℝ⁺ → (2↑_𝑐1) = 2)
8	6, 7	mp1i 10	. . . . 5 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → (2↑_𝑐1) = 2)
9	8	oveq1d 6033	. . . 4 ⊢ (𝑘 ∈ (0...(𝑁 − 1)) → ((2↑_𝑐1)↑𝑘) = (2↑𝑘))
10	9	sumeq2i 11929	. . 3 ⊢ Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘) = Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘)
11		2cn 9214	. . . . 5 ⊢ 2 ∈ ℂ
12	11	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ)
13		1ap2 9351	. . . . . 6 ⊢ 1 # 2
14		apsym 8786	. . . . . . 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 9409	. . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
19	12, 17, 18	geoserap 12073	. . 3 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))(2↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2)))
20	10, 19	eqtrid 2276	. 2 ⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...(𝑁 − 1))((2↑_𝑐1)↑𝑘) = ((1 − (2↑𝑁)) / (1 − 2)))
21		2nn 9305	. . . . . . 7 ⊢ 2 ∈ ℕ
22		nnexpcl 10815	. . . . . . 7 ⊢ ((2 ∈ ℕ ∧ 𝑁 ∈ ℕ₀) → (2↑𝑁) ∈ ℕ)
23	21, 18, 22	sylancr 414	. . . . . 6 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℕ)
24	23	nncnd 9157	. . . . 5 ⊢ (𝑁 ∈ ℕ → (2↑𝑁) ∈ ℂ)
25		subcl 8378	. . . . 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 8770	. . . . 5 ⊢ 1 # 0
29	28	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → 1 # 0)
30	26, 27, 29	div2negapd 8985	. . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = (((2↑𝑁) − 1) / 1))
31		negsubdi2 8438	. . . . 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 8353	. . . . . 6 ⊢ -1 = (0 − 1)
34		0cn 8171	. . . . . . 7 ⊢ 0 ∈ ℂ
35		pnpcan 8418	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0 ∈ ℂ ∧ 1 ∈ ℂ) → ((1 + 0) − (1 + 1)) = (0 − 1))
36	1, 34, 1, 35	mp3an 1373	. . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (0 − 1)
37		1p0e1 9259	. . . . . . 7 ⊢ (1 + 0) = 1
38		1p1e2 9260	. . . . . . 7 ⊢ (1 + 1) = 2
39	37, 38	oveq12i 6030	. . . . . 6 ⊢ ((1 + 0) − (1 + 1)) = (1 − 2)
40	33, 36, 39	3eqtr2i 2258	. . . . 5 ⊢ -1 = (1 − 2)
41	40	a1i 9	. . . 4 ⊢ (𝑁 ∈ ℕ → -1 = (1 − 2))
42	32, 41	oveq12d 6036	. . 3 ⊢ (𝑁 ∈ ℕ → (-((2↑𝑁) − 1) / -1) = ((1 − (2↑𝑁)) / (1 − 2)))
43	26	div1d 8960	. . 3 ⊢ (𝑁 ∈ ℕ → (((2↑𝑁) − 1) / 1) = ((2↑𝑁) − 1))
44	30, 42, 43	3eqtr3d 2272	. 2 ⊢ (𝑁 ∈ ℕ → ((1 − (2↑𝑁)) / (1 − 2)) = ((2↑𝑁) − 1))
45	5, 20, 44	3eqtrd 2268	1 ⊢ (𝑁 ∈ ℕ → (1 σ (2↑(𝑁 − 1))) = ((2↑𝑁) − 1))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 = wceq 1397 ∈ wcel 2202 class class class wbr 4088 (class class class)co 6018 ℂcc 8030 0cc0 8032 1c1 8033 + caddc 8035 − cmin 8350 -cneg 8351 # cap 8761 / cdiv 8852 ℕcn 9143 2c2 9194 ℕ₀cn0 9402 ℝ⁺crp 9888 ...cfz 10243 ↑cexp 10801 Σcsu 11918 ℙcprime 12684 ↑_𝑐ccxp 15587 σ csgm 15711

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152 ax-pre-suploc 8153 ax-addf 8154 ax-mulf 8155

This theorem depends on definitions: df-bi 117 df-stab 838 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-disj 4065 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-isom 5335 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-of 6235 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-frec 6557 df-1o 6582 df-2o 6583 df-oadd 6586 df-er 6702 df-map 6819 df-pm 6820 df-en 6910 df-dom 6911 df-fin 6912 df-sup 7183 df-inf 7184 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-xnn0 9466 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-xneg 10007 df-xadd 10008 df-ioo 10127 df-ico 10129 df-icc 10130 df-fz 10244 df-fzo 10378 df-fl 10531 df-mod 10586 df-seqfrec 10711 df-exp 10802 df-fac 10989 df-bc 11011 df-ihash 11039 df-shft 11380 df-cj 11407 df-re 11408 df-im 11409 df-rsqrt 11563 df-abs 11564 df-clim 11844 df-sumdc 11919 df-ef 12214 df-e 12215 df-dvds 12354 df-gcd 12530 df-prm 12685 df-pc 12863 df-rest 13329 df-topgen 13348 df-psmet 14563 df-xmet 14564 df-met 14565 df-bl 14566 df-mopn 14567 df-top 14728 df-topon 14741 df-bases 14773 df-ntr 14826 df-cn 14918 df-cnp 14919 df-tx 14983 df-cncf 15301 df-limced 15386 df-dvap 15387 df-relog 15588 df-rpcxp 15589 df-sgm 15712

This theorem is referenced by: perfect1 15728 perfectlem1 15729 perfectlem2 15730

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