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Theorem perfect1 22510
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 22509 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  P  e.  Prime )
2 prmnn 13762 . . . . 5  |-  ( P  e.  Prime  ->  P  e.  NN )
31, 2syl 16 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  P  e.  NN )
4 1sgm2ppw 22482 . . . 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 22481 . . . . 5  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( 1 
sigma  ( ( 2 ^ P )  -  1 ) )  =  ( ( ( 2 ^ P )  -  1 )  +  1 ) )
76adantl 463 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( ( 2 ^ P )  -  1 ) )  =  ( ( ( 2 ^ P )  -  1 )  +  1 ) )
8 2nn 10475 . . . . . . 7  |-  2  e.  NN
93nnnn0d 10632 . . . . . . 7  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  P  e.  NN0 )
10 nnexpcl 11874 . . . . . . 7  |-  ( ( 2  e.  NN  /\  P  e.  NN0 )  -> 
( 2 ^ P
)  e.  NN )
118, 9, 10sylancr 658 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ P
)  e.  NN )
1211nncnd 10334 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ P
)  e.  CC )
13 ax-1cn 9336 . . . . 5  |-  1  e.  CC
14 npcan 9615 . . . . 5  |-  ( ( ( 2 ^ P
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2 ^ P )  - 
1 )  +  1 )  =  ( 2 ^ P ) )
1512, 13, 14sylancl 657 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  +  1 )  =  ( 2 ^ P ) )
167, 15eqtrd 2473 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( ( 2 ^ P )  -  1 ) )  =  ( 2 ^ P ) )
175, 16oveq12d 6108 . 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 10617 . . . . 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 11874 . . . 4  |-  ( ( 2  e.  NN  /\  ( P  -  1
)  e.  NN0 )  ->  ( 2 ^ ( P  -  1 ) )  e.  NN )
228, 20, 21sylancr 658 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ ( P  -  1 ) )  e.  NN )
23 prmnn 13762 . . . 4  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( ( 2 ^ P )  -  1 )  e.  NN )
2423adantl 463 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  NN )
2522nnzd 10742 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ ( P  -  1 ) )  e.  ZZ )
26 prmz 13763 . . . . . 6  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( ( 2 ^ P )  -  1 )  e.  ZZ )
2726adantl 463 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  ZZ )
28 gcdcom 13700 . . . . 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 656 . . . 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 13542 . . . . . . . 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 13777 . . . . . . . . . 10  |-  ( ( ( 2 ^ P
)  -  1 )  e.  Prime  ->  ( ( 2 ^ P )  -  1 )  e.  ( ZZ>= `  2 )
)
3332adantl 463 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  ( ZZ>= ` 
2 ) )
34 eluz2b2 10923 . . . . . . . . . 10  |-  ( ( ( 2 ^ P
)  -  1 )  e.  ( ZZ>= `  2
)  <->  ( ( ( 2 ^ P )  -  1 )  e.  NN  /\  1  < 
( ( 2 ^ P )  -  1 ) ) )
3534simprbi 461 . . . . . . . . 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 13609 . . . . . . . 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 1213 . . . . . . 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 10674 . . . . . . . . 9  |-  2  e.  ZZ
4140a1i 11 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  2  e.  ZZ )
42 dvdsmultr1 13563 . . . . . . . 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 1213 . . . . . . 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 10388 . . . . . . . . . 10  |-  2  e.  CC
45 expm1t 11888 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  P  e.  NN )  ->  ( 2 ^ P
)  =  ( ( 2 ^ ( P  -  1 ) )  x.  2 ) )
4644, 3, 45sylancr 658 . . . . . . . . 9  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( 2 ^ P
)  =  ( ( 2 ^ ( P  -  1 ) )  x.  2 ) )
4715, 46eqtr2d 2474 . . . . . . . 8  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( P  -  1 ) )  x.  2 )  =  ( ( ( 2 ^ P
)  -  1 )  +  1 ) )
4847breq2d 4301 . . . . . . 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 214 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  ||  (
2 ^ ( P  -  1 ) )  ->  ( ( 2 ^ P )  - 
1 )  ||  (
( ( 2 ^ P )  -  1 )  +  1 ) ) )
5039, 49mtod 177 . . . . 5  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  -.  ( ( 2 ^ P )  - 
1 )  ||  (
2 ^ ( P  -  1 ) ) )
51 simpr 458 . . . . . 6  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  Prime )
52 coprm 13782 . . . . . 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 656 . . . . 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 210 . . . 4  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( ( 2 ^ P )  - 
1 )  gcd  (
2 ^ ( P  -  1 ) ) )  =  1 )
5529, 54eqtrd 2473 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( P  -  1 ) )  gcd  (
( 2 ^ P
)  -  1 ) )  =  1 )
56 sgmmul 22483 . . 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 1215 . 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 9605 . . . 4  |-  ( ( ( 2 ^ P
)  e.  CC  /\  1  e.  CC )  ->  ( ( 2 ^ P )  -  1 )  e.  CC )
5912, 13, 58sylancl 657 . . 3  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  -  1 )  e.  CC )
6012, 59mulcomd 9403 . 2  |-  ( ( P  e.  ZZ  /\  ( ( 2 ^ P )  -  1 )  e.  Prime )  ->  ( ( 2 ^ P )  x.  (
( 2 ^ P
)  -  1 ) )  =  ( ( ( 2 ^ P
)  -  1 )  x.  ( 2 ^ P ) ) )
6117, 57, 603eqtr4d 2483 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1364    e. wcel 1761   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   1c1 9279    + caddc 9281    x. cmul 9283    < clt 9414    - cmin 9591   NNcn 10318   2c2 10367   NN0cn0 10575   ZZcz 10642   ZZ>=cuz 10857   ^cexp 11861    || cdivides 13531    gcd cgcd 13686   Primecprime 13759    sigma csgm 22376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-pi 13354  df-dvds 13532  df-gcd 13687  df-prm 13760  df-pc 13900  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cn 18731  df-cnp 18732  df-haus 18819  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-xms 19795  df-ms 19796  df-tms 19797  df-cncf 20354  df-limc 21241  df-dv 21242  df-log 21951  df-cxp 21952  df-sgm 22382
This theorem is referenced by:  perfect  22513
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