Theorem perfect1 15637

Description: Euclid's contribution to the Euclid-Euler theorem. A number of the form 2 ^ ( p - 1 ) x. ( 2 ^ p - 1 ) is a perfect number. (Contributed by Mario Carneiro, 17-May-2016.)

Assertion

Ref	Expression
perfect1	|- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ ( P - 1 ) ) x. ( ( 2 ^ P ) - 1 ) ) ) = ( ( 2 ^ P ) x. ( ( 2 ^ P ) - 1 ) ) )

Proof of Theorem perfect1

Step	Hyp	Ref	Expression
1		mersenne 15636	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. Prime )
2		prmnn 12598	. . . . 5 |- ( P e. Prime -> P e. NN )
3	1, 2	syl 14	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. NN )
4		1sgm2ppw 15634	. . . 4 |- ( P e. NN -> ( 1 sigma ( 2 ^ ( P - 1 ) ) ) = ( ( 2 ^ P ) - 1 ) )
5	3, 4	syl 14	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( 2 ^ ( P - 1 ) ) ) = ( ( 2 ^ P ) - 1 ) )
6		1sgmprm 15633	. . . . 5 |- ( ( ( 2 ^ P ) - 1 ) e. Prime -> ( 1 sigma ( ( 2 ^ P ) - 1 ) ) = ( ( ( 2 ^ P ) - 1 ) + 1 ) )
7	6	adantl 277	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ P ) - 1 ) ) = ( ( ( 2 ^ P ) - 1 ) + 1 ) )
8		2nn 9240	. . . . . . 7 |- 2 e. NN
9	3	nnnn0d 9390	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> P e. NN0 )
10		nnexpcl 10741	. . . . . . 7 |- ( ( 2 e. NN /\ P e. NN0 ) -> ( 2 ^ P ) e. NN )
11	8, 9, 10	sylancr 414	. . . . . 6 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) e. NN )
12	11	nncnd 9092	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) e. CC )
13		ax-1cn 8060	. . . . 5 |- 1 e. CC
14		npcan 8323	. . . . 5 |- ( ( ( 2 ^ P ) e. CC /\ 1 e. CC ) -> ( ( ( 2 ^ P ) - 1 ) + 1 ) = ( 2 ^ P ) )
15	12, 13, 14	sylancl 413	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) + 1 ) = ( 2 ^ P ) )
16	7, 15	eqtrd 2242	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ P ) - 1 ) ) = ( 2 ^ P ) )
17	5, 16	oveq12d 5992	. 2 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 1 sigma ( 2 ^ ( P - 1 ) ) ) x. ( 1 sigma ( ( 2 ^ P ) - 1 ) ) ) = ( ( ( 2 ^ P ) - 1 ) x. ( 2 ^ P ) ) )
18	13	a1i 9	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> 1 e. CC )
19		nnm1nn0 9378	. . . . 5 |- ( P e. NN -> ( P - 1 ) e. NN0 )
20	3, 19	syl 14	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( P - 1 ) e. NN0 )
21		nnexpcl 10741	. . . 4 |- ( ( 2 e. NN /\ ( P - 1 ) e. NN0 ) -> ( 2 ^ ( P - 1 ) ) e. NN )
22	8, 20, 21	sylancr 414	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ ( P - 1 ) ) e. NN )
23		prmnn 12598	. . . 4 |- ( ( ( 2 ^ P ) - 1 ) e. Prime -> ( ( 2 ^ P ) - 1 ) e. NN )
24	23	adantl 277	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. NN )
25	22	nnzd 9536	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ ( P - 1 ) ) e. ZZ )
26		prmz 12599	. . . . . 6 |- ( ( ( 2 ^ P ) - 1 ) e. Prime -> ( ( 2 ^ P ) - 1 ) e. ZZ )
27	26	adantl 277	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. ZZ )
28	25, 27	gcdcomd 12461	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) gcd ( ( 2 ^ P ) - 1 ) ) = ( ( ( 2 ^ P ) - 1 ) gcd ( 2 ^ ( P - 1 ) ) ) )
29		iddvds 12281	. . . . . . . 8 |- ( ( ( 2 ^ P ) - 1 ) e. ZZ -> ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ P ) - 1 ) )
30	27, 29	syl 14	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ P ) - 1 ) )
31		prmuz2 12619	. . . . . . . . . 10 |- ( ( ( 2 ^ P ) - 1 ) e. Prime -> ( ( 2 ^ P ) - 1 ) e. ( ZZ>= ` 2 ) )
32	31	adantl 277	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. ( ZZ>= ` 2 ) )
33		eluz2gt1 9765	. . . . . . . . 9 |- ( ( ( 2 ^ P ) - 1 ) e. ( ZZ>= ` 2 ) -> 1 < ( ( 2 ^ P ) - 1 ) )
34	32, 33	syl 14	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> 1 < ( ( 2 ^ P ) - 1 ) )
35		ndvdsp1 12409	. . . . . . . 8 |- ( ( ( ( 2 ^ P ) - 1 ) e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. NN /\ 1 < ( ( 2 ^ P ) - 1 ) ) -> ( ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ P ) - 1 ) -> -. ( ( 2 ^ P ) - 1 ) || ( ( ( 2 ^ P ) - 1 ) + 1 ) ) )
36	27, 24, 34, 35	syl3anc 1252	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ P ) - 1 ) -> -. ( ( 2 ^ P ) - 1 ) || ( ( ( 2 ^ P ) - 1 ) + 1 ) ) )
37	30, 36	mpd 13	. . . . . 6 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> -. ( ( 2 ^ P ) - 1 ) || ( ( ( 2 ^ P ) - 1 ) + 1 ) )
38		2z 9442	. . . . . . . . 9 |- 2 e. ZZ
39	38	a1i 9	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> 2 e. ZZ )
40		dvdsmultr1 12308	. . . . . . . 8 |- ( ( ( ( 2 ^ P ) - 1 ) e. ZZ /\ ( 2 ^ ( P - 1 ) ) e. ZZ /\ 2 e. ZZ ) -> ( ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) -> ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ ( P - 1 ) ) x. 2 ) ) )
41	27, 25, 39, 40	syl3anc 1252	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) -> ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ ( P - 1 ) ) x. 2 ) ) )
42		2cn 9149	. . . . . . . . . 10 |- 2 e. CC
43		expm1t 10756	. . . . . . . . . 10 |- ( ( 2 e. CC /\ P e. NN ) -> ( 2 ^ P ) = ( ( 2 ^ ( P - 1 ) ) x. 2 ) )
44	42, 3, 43	sylancr 414	. . . . . . . . 9 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 2 ^ P ) = ( ( 2 ^ ( P - 1 ) ) x. 2 ) )
45	15, 44	eqtr2d 2243	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) x. 2 ) = ( ( ( 2 ^ P ) - 1 ) + 1 ) )
46	45	breq2d 4074	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) || ( ( 2 ^ ( P - 1 ) ) x. 2 ) <-> ( ( 2 ^ P ) - 1 ) || ( ( ( 2 ^ P ) - 1 ) + 1 ) ) )
47	41, 46	sylibd 149	. . . . . 6 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) -> ( ( 2 ^ P ) - 1 ) || ( ( ( 2 ^ P ) - 1 ) + 1 ) ) )
48	37, 47	mtod 667	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> -. ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) )
49		simpr 110	. . . . . 6 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. Prime )
50		coprm 12632	. . . . . 6 |- ( ( ( ( 2 ^ P ) - 1 ) e. Prime /\ ( 2 ^ ( P - 1 ) ) e. ZZ ) -> ( -. ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) <-> ( ( ( 2 ^ P ) - 1 ) gcd ( 2 ^ ( P - 1 ) ) ) = 1 ) )
51	49, 25, 50	syl2anc 411	. . . . 5 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( -. ( ( 2 ^ P ) - 1 ) || ( 2 ^ ( P - 1 ) ) <-> ( ( ( 2 ^ P ) - 1 ) gcd ( 2 ^ ( P - 1 ) ) ) = 1 ) )
52	48, 51	mpbid 147	. . . 4 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( ( 2 ^ P ) - 1 ) gcd ( 2 ^ ( P - 1 ) ) ) = 1 )
53	28, 52	eqtrd 2242	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ ( P - 1 ) ) gcd ( ( 2 ^ P ) - 1 ) ) = 1 )
54		sgmmul 15635	. . 3 |- ( ( 1 e. CC /\ ( ( 2 ^ ( P - 1 ) ) e. NN /\ ( ( 2 ^ P ) - 1 ) e. NN /\ ( ( 2 ^ ( P - 1 ) ) gcd ( ( 2 ^ P ) - 1 ) ) = 1 ) ) -> ( 1 sigma ( ( 2 ^ ( P - 1 ) ) x. ( ( 2 ^ P ) - 1 ) ) ) = ( ( 1 sigma ( 2 ^ ( P - 1 ) ) ) x. ( 1 sigma ( ( 2 ^ P ) - 1 ) ) ) )
55	18, 22, 24, 53, 54	syl13anc 1254	. 2 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ ( P - 1 ) ) x. ( ( 2 ^ P ) - 1 ) ) ) = ( ( 1 sigma ( 2 ^ ( P - 1 ) ) ) x. ( 1 sigma ( ( 2 ^ P ) - 1 ) ) ) )
56		subcl 8313	. . . 4 |- ( ( ( 2 ^ P ) e. CC /\ 1 e. CC ) -> ( ( 2 ^ P ) - 1 ) e. CC )
57	12, 13, 56	sylancl 413	. . 3 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) - 1 ) e. CC )
58	12, 57	mulcomd 8136	. 2 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( ( 2 ^ P ) x. ( ( 2 ^ P ) - 1 ) ) = ( ( ( 2 ^ P ) - 1 ) x. ( 2 ^ P ) ) )
59	17, 55, 58	3eqtr4d 2252	1 |- ( ( P e. ZZ /\ ( ( 2 ^ P ) - 1 ) e. Prime ) -> ( 1 sigma ( ( 2 ^ ( P - 1 ) ) x. ( ( 2 ^ P ) - 1 ) ) ) = ( ( 2 ^ P ) x. ( ( 2 ^ P ) - 1 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1375   e. wcel 2180   class class class wbr 4062   ` cfv 5294  (class class class)co 5974   CCcc 7965   1c1 7968   + caddc 7970   x. cmul 7972   < clt 8149   - cmin 8285   NNcn 9078   2c2 9129   NN0cn0 9337   ZZcz 9414   ZZ>=cuz 9690   ^cexp 10727   || cdvds 12264   gcd cgcd 12440   Primecprime 12595   sigma csgm 15620

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-nul 4189  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-iinf 4657  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-mulrcl 8066  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-0lt1 8073  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-precex 8077  ax-cnre 8078  ax-pre-ltirr 8079  ax-pre-ltwlin 8080  ax-pre-lttrn 8081  ax-pre-apti 8082  ax-pre-ltadd 8083  ax-pre-mulgt0 8084  ax-pre-mulext 8085  ax-arch 8086  ax-caucvg 8087  ax-pre-suploc 8088  ax-addf 8089  ax-mulf 8090

This theorem depends on definitions:  df-bi 117  df-stab 835  df-dc 839  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-reu 2495  df-rmo 2496  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-iun 3946  df-disj 4039  df-br 4063  df-opab 4125  df-mpt 4126  df-tr 4162  df-id 4361  df-po 4364  df-iso 4365  df-iord 4434  df-on 4436  df-ilim 4437  df-suc 4439  df-iom 4660  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-isom 5303  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-of 6188  df-1st 6256  df-2nd 6257  df-recs 6421  df-irdg 6486  df-frec 6507  df-1o 6532  df-2o 6533  df-oadd 6536  df-er 6650  df-map 6767  df-pm 6768  df-en 6858  df-dom 6859  df-fin 6860  df-sup 7119  df-inf 7120  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154  df-le 8155  df-sub 8287  df-neg 8288  df-reap 8690  df-ap 8697  df-div 8788  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-n0 9338  df-xnn0 9401  df-z 9415  df-uz 9691  df-q 9783  df-rp 9818  df-xneg 9936  df-xadd 9937  df-ioo 10056  df-ico 10058  df-icc 10059  df-fz 10173  df-fzo 10307  df-fl 10457  df-mod 10512  df-seqfrec 10637  df-exp 10728  df-fac 10915  df-bc 10937  df-ihash 10965  df-shft 11292  df-cj 11319  df-re 11320  df-im 11321  df-rsqrt 11475  df-abs 11476  df-clim 11756  df-sumdc 11831  df-ef 12125  df-e 12126  df-dvds 12265  df-gcd 12441  df-prm 12596  df-pc 12774  df-rest 13240  df-topgen 13259  df-psmet 14472  df-xmet 14473  df-met 14474  df-bl 14475  df-mopn 14476  df-top 14637  df-topon 14650  df-bases 14682  df-ntr 14735  df-cn 14827  df-cnp 14828  df-tx 14892  df-cncf 15210  df-limced 15295  df-dvap 15296  df-relog 15497  df-rpcxp 15498  df-sgm 15621

This theorem is referenced by:  perfect  15640