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Theorem perfect1 20881
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 20880 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  P  e.  Prime )
2 prmnn 13011 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
31, 2syl 16 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  P  e.  NN )
4 1sgm2ppw 20853 . . . 4  |-  ( P  e.  NN  ->  (
1  sigma  ( 2 ^ ( P  -  1 ) ) )  =  ( ( 2 ^ P )  -  1 ) )
53, 4syl 16 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( 2 ^ ( P  - 
1 ) ) )  =  ( ( 2 ^ P )  - 
1 ) )
6 1sgmprm 20852 . . . . 5  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( 1 
sigma  ( ( 2 ^ P )  -  1 ) )  =  ( ( ( 2 ^ P )  -  1 )  +  1 ) )
76adantl 453 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( ( 2 ^ P )  -  1 ) )  =  ( ( ( 2 ^ P )  -  1 )  +  1 ) )
8 2nn 10067 . . . . . . 7  |-  2  e.  NN
93nnnn0d 10208 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  P  e.  NN0 )
10 nnexpcl 11323 . . . . . . 7  |-  ( ( 2  e.  NN  /\  P  e.  NN0 )  -> 
( 2 ^ P
)  e.  NN )
118, 9, 10sylancr 645 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ P
)  e.  NN )
1211nncnd 9950 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ P
)  e.  CC )
13 ax-1cn 8983 . . . . 5  |-  1  e.  CC
14 npcan 9248 . . . . 5  |-  ( ( ( 2 ^ P
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2 ^ P )  - 
1 )  +  1 )  =  ( 2 ^ P ) )
1512, 13, 14sylancl 644 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  +  1 )  =  ( 2 ^ P ) )
167, 15eqtrd 2421 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( ( 2 ^ P )  -  1 ) )  =  ( 2 ^ P ) )
175, 16oveq12d 6040 . 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 11 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  1  e.  CC )
19 nnm1nn0 10195 . . . . 5  |-  ( P  e.  NN  ->  ( P  -  1 )  e.  NN0 )
203, 19syl 16 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( P  -  1 )  e.  NN0 )
21 nnexpcl 11323 . . . 4  |-  ( ( 2  e.  NN  /\  ( P  -  1
)  e.  NN0 )  ->  ( 2 ^ ( P  -  1 ) )  e.  NN )
228, 20, 21sylancr 645 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ ( P  -  1 ) )  e.  NN )
23 prmnn 13011 . . . 4  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( ( 2 ^ P )  -  1 )  e.  NN )
2423adantl 453 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  NN )
2522nnzd 10308 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ ( P  -  1 ) )  e.  ZZ )
26 prmz 13012 . . . . . 6  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( ( 2 ^ P )  -  1 )  e.  ZZ )
2726adantl 453 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  ZZ )
28 gcdcom 12949 . . . . 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 643 . . . 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 12792 . . . . . . . 8  |-  ( ( ( 2 ^ P
)  -  1 )  e.  ZZ  ->  (
( 2 ^ P
)  -  1 ) 
||  ( ( 2 ^ P )  - 
1 ) )
3127, 30syl 16 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  ||  ( ( 2 ^ P )  -  1 ) )
32 prmuz2 13026 . . . . . . . . . 10  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( ( 2 ^ P )  -  1 )  e.  ( ZZ>= `  2 )
)
3332adantl 453 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  ( ZZ>= ` 
2 ) )
34 eluz2b2 10482 . . . . . . . . . 10  |-  ( ( ( 2 ^ P
)  -  1 )  e.  ( ZZ>= `  2
)  <->  ( ( ( 2 ^ P )  -  1 )  e.  NN  /\  1  < 
( ( 2 ^ P )  -  1 ) ) )
3534simprbi 451 . . . . . . . . 9  |-  ( ( ( 2 ^ P
)  -  1 )  e.  ( ZZ>= `  2
)  ->  1  <  ( ( 2 ^ P
)  -  1 ) )
3633, 35syl 16 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  1  <  ( ( 2 ^ P )  -  1 ) )
37 ndvdsp1 12858 . . . . . . . 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 1184 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  ||  (
( 2 ^ P
)  -  1 )  ->  -.  ( (
2 ^ P )  -  1 )  ||  ( ( ( 2 ^ P )  - 
1 )  +  1 ) ) )
3931, 38mpd 15 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  -.  ( ( 2 ^ P )  - 
1 )  ||  (
( ( 2 ^ P )  -  1 )  +  1 ) )
40 2z 10246 . . . . . . . . 9  |-  2  e.  ZZ
4140a1i 11 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  2  e.  ZZ )
42 dvdsmultr1 12813 . . . . . . . 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 1184 . . . . . . 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 10004 . . . . . . . . . 10  |-  2  e.  CC
45 expm1t 11337 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  P  e.  NN )  ->  ( 2 ^ P
)  =  ( ( 2 ^ ( P  -  1 ) )  x.  2 ) )
4644, 3, 45sylancr 645 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ P
)  =  ( ( 2 ^ ( P  -  1 ) )  x.  2 ) )
4715, 46eqtr2d 2422 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( P  -  1 ) )  x.  2 )  =  ( ( ( 2 ^ P
)  -  1 )  +  1 ) )
4847breq2d 4167 . . . . . . 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 206 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  ||  (
2 ^ ( P  -  1 ) )  ->  ( ( 2 ^ P )  - 
1 )  ||  (
( ( 2 ^ P )  -  1 )  +  1 ) ) )
5039, 49mtod 170 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  -.  ( ( 2 ^ P )  - 
1 )  ||  (
2 ^ ( P  -  1 ) ) )
51 simpr 448 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  Prime )
52 coprm 13029 . . . . . 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 643 . . . . 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 202 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  gcd  (
2 ^ ( P  -  1 ) ) )  =  1 )
5529, 54eqtrd 2421 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( P  -  1 ) )  gcd  (
( 2 ^ P
)  -  1 ) )  =  1 )
56 sgmmul 20854 . . 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 1186 . 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 9239 . . . 4  |-  ( ( ( 2 ^ P
)  e.  CC  /\  1  e.  CC )  ->  ( ( 2 ^ P )  -  1 )  e.  CC )
5912, 13, 58sylancl 644 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  CC )
6012, 59mulcomd 9044 . 2  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  x.  (
( 2 ^ P
)  -  1 ) )  =  ( ( ( 2 ^ P
)  -  1 )  x.  ( 2 ^ P ) ) )
6117, 57, 603eqtr4d 2431 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   class class class wbr 4155   ` cfv 5396  (class class class)co 6022   CCcc 8923   1c1 8926    + caddc 8928    x. cmul 8930    < clt 9055    - cmin 9225   NNcn 9934   2c2 9983   NN0cn0 10155   ZZcz 10216   ZZ>=cuz 10422   ^cexp 11311    || cdivides 12781    gcd cgcd 12935   Primecprime 13008    sigma csgm 20747
This theorem is referenced by:  perfect  20884
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002  ax-pre-sup 9003  ax-addf 9004  ax-mulf 9005
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-fi 7353  df-sup 7383  df-oi 7414  df-card 7761  df-cda 7983  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-dec 10317  df-uz 10423  df-q 10509  df-rp 10547  df-xneg 10644  df-xadd 10645  df-xmul 10646  df-ioo 10854  df-ioc 10855  df-ico 10856  df-icc 10857  df-fz 10978  df-fzo 11068  df-fl 11131  df-mod 11180  df-seq 11253  df-exp 11312  df-fac 11496  df-bc 11523  df-hash 11548  df-shft 11811  df-cj 11833  df-re 11834  df-im 11835  df-sqr 11969  df-abs 11970  df-limsup 12194  df-clim 12211  df-rlim 12212  df-sum 12409  df-ef 12599  df-sin 12601  df-cos 12602  df-pi 12604  df-dvds 12782  df-gcd 12936  df-prm 13009  df-pc 13140  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-starv 13473  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-ds 13480  df-unif 13481  df-hom 13482  df-cco 13483  df-rest 13579  df-topn 13580  df-topgen 13596  df-pt 13597  df-prds 13600  df-xrs 13655  df-0g 13656  df-gsum 13657  df-qtop 13662  df-imas 13663  df-xps 13665  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-submnd 14668  df-mulg 14744  df-cntz 15045  df-cmn 15343  df-xmet 16621  df-met 16622  df-bl 16623  df-mopn 16624  df-fbas 16625  df-fg 16626  df-cnfld 16629  df-top 16888  df-bases 16890  df-topon 16891  df-topsp 16892  df-cld 17008  df-ntr 17009  df-cls 17010  df-nei 17087  df-lp 17125  df-perf 17126  df-cn 17215  df-cnp 17216  df-haus 17303  df-tx 17517  df-hmeo 17710  df-fil 17801  df-fm 17893  df-flim 17894  df-flf 17895  df-xms 18261  df-ms 18262  df-tms 18263  df-cncf 18781  df-limc 19622  df-dv 19623  df-log 20323  df-cxp 20324  df-sgm 20753
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