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Theorem perfect1 20483
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
StepHypRef Expression
1 mersenne 20482 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  P  e.  Prime )
2 prmnn 12777 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
31, 2syl 15 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  P  e.  NN )
4 1sgm2ppw 20455 . . . 4  |-  ( P  e.  NN  ->  (
1  sigma  ( 2 ^ ( P  -  1 ) ) )  =  ( ( 2 ^ P )  -  1 ) )
53, 4syl 15 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( 2 ^ ( P  - 
1 ) ) )  =  ( ( 2 ^ P )  - 
1 ) )
6 1sgmprm 20454 . . . . 5  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( 1 
sigma  ( ( 2 ^ P )  -  1 ) )  =  ( ( ( 2 ^ P )  -  1 )  +  1 ) )
76adantl 452 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( ( 2 ^ P )  -  1 ) )  =  ( ( ( 2 ^ P )  -  1 )  +  1 ) )
8 2nn 9893 . . . . . . 7  |-  2  e.  NN
93nnnn0d 10034 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  P  e.  NN0 )
10 nnexpcl 11132 . . . . . . 7  |-  ( ( 2  e.  NN  /\  P  e.  NN0 )  -> 
( 2 ^ P
)  e.  NN )
118, 9, 10sylancr 644 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ P
)  e.  NN )
1211nncnd 9778 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ P
)  e.  CC )
13 ax-1cn 8811 . . . . 5  |-  1  e.  CC
14 npcan 9076 . . . . 5  |-  ( ( ( 2 ^ P
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2 ^ P )  - 
1 )  +  1 )  =  ( 2 ^ P ) )
1512, 13, 14sylancl 643 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  +  1 )  =  ( 2 ^ P ) )
167, 15eqtrd 2328 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( ( 2 ^ P )  -  1 ) )  =  ( 2 ^ P ) )
175, 16oveq12d 5892 . 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
) ) )
1813a1i 10 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  1  e.  CC )
19 nnm1nn0 10021 . . . . 5  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
203, 19syl 15 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( P  -  1 )  e.  NN0 )
21 nnexpcl 11132 . . . 4  |-  ( ( 2  e.  NN  /\  ( P  -  1
)  e.  NN0 )  ->  ( 2 ^ ( P  -  1 ) )  e.  NN )
228, 20, 21sylancr 644 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ ( P  -  1 ) )  e.  NN )
23 prmnn 12777 . . . 4  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( ( 2 ^ P )  -  1 )  e.  NN )
2423adantl 452 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  NN )
2522nnzd 10132 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ ( P  -  1 ) )  e.  ZZ )
26 prmz 12778 . . . . . 6  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( ( 2 ^ P )  -  1 )  e.  ZZ )
2726adantl 452 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  ZZ )
28 gcdcom 12715 . . . . 5  |-  ( ( ( 2 ^ ( P  -  1 ) )  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  ZZ )  ->  ( ( 2 ^ ( P  - 
1 ) )  gcd  ( ( 2 ^ P )  -  1 ) )  =  ( ( ( 2 ^ P )  -  1 )  gcd  ( 2 ^ ( P  - 
1 ) ) ) )
2925, 27, 28syl2anc 642 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( P  -  1 ) )  gcd  (
( 2 ^ P
)  -  1 ) )  =  ( ( ( 2 ^ P
)  -  1 )  gcd  ( 2 ^ ( P  -  1 ) ) ) )
30 iddvds 12558 . . . . . . . 8  |-  ( ( ( 2 ^ P
)  -  1 )  e.  ZZ  ->  (
( 2 ^ P
)  -  1 ) 
||  ( ( 2 ^ P )  - 
1 ) )
3127, 30syl 15 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  ||  ( ( 2 ^ P )  -  1 ) )
32 prmuz2 12792 . . . . . . . . . 10  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( ( 2 ^ P )  -  1 )  e.  ( ZZ>= `  2 )
)
3332adantl 452 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  ( ZZ>= ` 
2 ) )
34 eluz2b2 10306 . . . . . . . . . 10  |-  ( ( ( 2 ^ P
)  -  1 )  e.  ( ZZ>= `  2
)  <->  ( ( ( 2 ^ P )  -  1 )  e.  NN  /\  1  < 
( ( 2 ^ P )  -  1 ) ) )
3534simprbi 450 . . . . . . . . 9  |-  ( ( ( 2 ^ P
)  -  1 )  e.  ( ZZ>= `  2
)  ->  1  <  ( ( 2 ^ P
)  -  1 ) )
3633, 35syl 15 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  1  <  ( ( 2 ^ P )  -  1 ) )
37 ndvdsp1 12624 . . . . . . . 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 ) ) )
3827, 24, 36, 37syl3anc 1182 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  ||  (
( 2 ^ P
)  -  1 )  ->  -.  ( (
2 ^ P )  -  1 )  ||  ( ( ( 2 ^ P )  - 
1 )  +  1 ) ) )
3931, 38mpd 14 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  -.  ( ( 2 ^ P )  - 
1 )  ||  (
( ( 2 ^ P )  -  1 )  +  1 ) )
40 2z 10070 . . . . . . . . 9  |-  2  e.  ZZ
4140a1i 10 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  2  e.  ZZ )
42 dvdsmultr1 12579 . . . . . . . 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 ) ) )
4327, 25, 41, 42syl3anc 1182 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  ||  (
2 ^ ( P  -  1 ) )  ->  ( ( 2 ^ P )  - 
1 )  ||  (
( 2 ^ ( P  -  1 ) )  x.  2 ) ) )
44 2cn 9832 . . . . . . . . . 10  |-  2  e.  CC
45 expm1t 11146 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  P  e.  NN )  ->  ( 2 ^ P
)  =  ( ( 2 ^ ( P  -  1 ) )  x.  2 ) )
4644, 3, 45sylancr 644 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ P
)  =  ( ( 2 ^ ( P  -  1 ) )  x.  2 ) )
4715, 46eqtr2d 2329 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( P  -  1 ) )  x.  2 )  =  ( ( ( 2 ^ P
)  -  1 )  +  1 ) )
4847breq2d 4051 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  ||  (
( 2 ^ ( P  -  1 ) )  x.  2 )  <-> 
( ( 2 ^ P )  -  1 )  ||  ( ( ( 2 ^ P
)  -  1 )  +  1 ) ) )
4943, 48sylibd 205 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  ||  (
2 ^ ( P  -  1 ) )  ->  ( ( 2 ^ P )  - 
1 )  ||  (
( ( 2 ^ P )  -  1 )  +  1 ) ) )
5039, 49mtod 168 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  -.  ( ( 2 ^ P )  - 
1 )  ||  (
2 ^ ( P  -  1 ) ) )
51 simpr 447 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  Prime )
52 coprm 12795 . . . . . 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 ) )
5351, 25, 52syl2anc 642 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( -.  ( ( 2 ^ P )  -  1 )  ||  ( 2 ^ ( P  -  1 ) )  <->  ( ( ( 2 ^ P )  -  1 )  gcd  ( 2 ^ ( P  -  1 ) ) )  =  1 ) )
5450, 53mpbid 201 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  gcd  (
2 ^ ( P  -  1 ) ) )  =  1 )
5529, 54eqtrd 2328 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( P  -  1 ) )  gcd  (
( 2 ^ P
)  -  1 ) )  =  1 )
56 sgmmul 20456 . . 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 ) ) ) )
5718, 22, 24, 55, 56syl13anc 1184 . 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 ) ) ) )
58 subcl 9067 . . . 4  |-  ( ( ( 2 ^ P
)  e.  CC  /\  1  e.  CC )  ->  ( ( 2 ^ P )  -  1 )  e.  CC )
5912, 13, 58sylancl 643 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  CC )
6012, 59mulcomd 8872 . 2  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  x.  (
( 2 ^ P
)  -  1 ) )  =  ( ( ( 2 ^ P
)  -  1 )  x.  ( 2 ^ P ) ) )
6117, 57, 603eqtr4d 2338 1  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( ( 2 ^ ( P  -  1 ) )  x.  ( ( 2 ^ P )  - 
1 ) ) )  =  ( ( 2 ^ P )  x.  ( ( 2 ^ P )  -  1 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053   NNcn 9762   2c2 9811   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ^cexp 11120    || cdivides 12547    gcd cgcd 12701   Primecprime 12774    sigma csgm 20349
This theorem is referenced by:  perfect  20486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ioc 10677  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365  df-sin 12367  df-cos 12368  df-pi 12370  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233  df-log 19930  df-cxp 19931  df-sgm 20355
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