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Theorem perfect 20693
Description: The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer  N is a perfect number (that is, its divisor sum is  2 N) if and only if it is of the form  2 ^ ( p  - 
1 )  x.  (
2 ^ p  - 
1 ), where  2 ^ p  -  1 is prime (a Mersenne prime). (It follows from this that  p is also prime.) (Contributed by Mario Carneiro, 17-May-2016.)
Assertion
Ref Expression
perfect  |-  ( ( N  e.  NN  /\  2  ||  N )  -> 
( ( 1  sigma  N )  =  ( 2  x.  N )  <->  E. p  e.  ZZ  ( ( ( 2 ^ p )  - 
1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) ) )
Distinct variable group:    N, p

Proof of Theorem perfect
StepHypRef Expression
1 simplr 731 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  2  ||  N )
2 2prm 12982 . . . . . . . 8  |-  2  e.  Prime
3 simpll 730 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  NN )
4 pcelnn 13130 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  (
( 2  pCnt  N
)  e.  NN  <->  2  ||  N ) )
52, 3, 4sylancr 644 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2  pCnt  N )  e.  NN  <->  2  ||  N
) )
61, 5mpbird 223 . . . . . 6  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  NN )
76nnzd 10267 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  ZZ )
87peano2zd 10271 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2  pCnt  N )  +  1 )  e.  ZZ )
9 pcdvds 13124 . . . . . . . . 9  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  (
2 ^ ( 2 
pCnt  N ) )  ||  N )
102, 3, 9sylancr 644 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  ||  N )
11 2nn 10026 . . . . . . . . . 10  |-  2  e.  NN
126nnnn0d 10167 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  NN0 )
13 nnexpcl 11281 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  ( 2  pCnt  N
)  e.  NN0 )  ->  ( 2 ^ (
2  pCnt  N )
)  e.  NN )
1411, 12, 13sylancr 644 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  e.  NN )
15 nndivdvds 12745 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( 2 ^ (
2  pCnt  N )
)  e.  NN )  ->  ( ( 2 ^ ( 2  pCnt 
N ) )  ||  N 
<->  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) )  e.  NN ) )
163, 14, 15syl2anc 642 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( 2 
pCnt  N ) )  ||  N 
<->  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) )  e.  NN ) )
1710, 16mpbid 201 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  e.  NN )
18 pcndvds2 13128 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  -.  2  ||  ( N  / 
( 2 ^ (
2  pCnt  N )
) ) )
192, 3, 18sylancr 644 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  -.  2  ||  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) )
20 simpr 447 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 1 
sigma  N )  =  ( 2  x.  N ) )
21 nncn 9901 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
2221ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  CC )
2314nncnd 9909 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  e.  CC )
2414nnne0d 9937 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  =/=  0 )
2522, 23, 24divcan2d 9685 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( 2 
pCnt  N ) )  x.  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) )  =  N )
2625oveq2d 5997 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 1 
sigma  ( ( 2 ^ ( 2  pCnt  N
) )  x.  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) ) ) )  =  ( 1 
sigma  N ) )
2725oveq2d 5997 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2  x.  ( ( 2 ^ ( 2  pCnt 
N ) )  x.  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) ) )  =  ( 2  x.  N ) )
2820, 26, 273eqtr4d 2408 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 1 
sigma  ( ( 2 ^ ( 2  pCnt  N
) )  x.  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) ) ) )  =  ( 2  x.  ( ( 2 ^ ( 2  pCnt 
N ) )  x.  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) ) ) )
296, 17, 19, 28perfectlem2 20692 . . . . . 6  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  e. 
Prime  /\  ( N  / 
( 2 ^ (
2  pCnt  N )
) )  =  ( ( 2 ^ (
( 2  pCnt  N
)  +  1 ) )  -  1 ) ) )
3029simprd 449 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  =  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) )
3129simpld 445 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  e. 
Prime )
3230, 31eqeltrrd 2441 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( ( 2  pCnt  N )  +  1 ) )  -  1 )  e. 
Prime )
336nncnd 9909 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  CC )
34 ax-1cn 8942 . . . . . . . . 9  |-  1  e.  CC
35 pncan 9204 . . . . . . . . 9  |-  ( ( ( 2  pCnt  N
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2 
pCnt  N )  +  1 )  -  1 )  =  ( 2  pCnt 
N ) )
3633, 34, 35sylancl 643 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
( 2  pCnt  N
)  +  1 )  -  1 )  =  ( 2  pCnt  N
) )
3736eqcomd 2371 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  =  ( ( ( 2  pCnt 
N )  +  1 )  -  1 ) )
3837oveq2d 5997 . . . . . 6  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
3938, 30oveq12d 5999 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( 2 
pCnt  N ) )  x.  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) )  =  ( ( 2 ^ ( ( ( 2  pCnt  N
)  +  1 )  -  1 ) )  x.  ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 ) ) )
4025, 39eqtr3d 2400 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  =  ( ( 2 ^ ( ( ( 2 
pCnt  N )  +  1 )  -  1 ) )  x.  ( ( 2 ^ ( ( 2  pCnt  N )  +  1 ) )  -  1 ) ) )
41 oveq2 5989 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ p )  =  ( 2 ^ (
( 2  pCnt  N
)  +  1 ) ) )
4241oveq1d 5996 . . . . . . 7  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
2 ^ p )  -  1 )  =  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) )
4342eleq1d 2432 . . . . . 6  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
( 2 ^ p
)  -  1 )  e.  Prime  <->  ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 )  e.  Prime )
)
44 oveq1 5988 . . . . . . . . 9  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( p  -  1 )  =  ( ( ( 2 
pCnt  N )  +  1 )  -  1 ) )
4544oveq2d 5997 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ ( p  - 
1 ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
4645, 42oveq12d 5999 . . . . . . 7  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) )  =  ( ( 2 ^ ( ( ( 2 
pCnt  N )  +  1 )  -  1 ) )  x.  ( ( 2 ^ ( ( 2  pCnt  N )  +  1 ) )  -  1 ) ) )
4746eqeq2d 2377 . . . . . 6  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  <->  N  =  ( ( 2 ^ ( ( ( 2 
pCnt  N )  +  1 )  -  1 ) )  x.  ( ( 2 ^ ( ( 2  pCnt  N )  +  1 ) )  -  1 ) ) ) )
4843, 47anbi12d 691 . . . . 5  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
( ( 2 ^ p )  -  1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) ) )  <->  ( (
( 2 ^ (
( 2  pCnt  N
)  +  1 ) )  -  1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( ( ( 2 
pCnt  N )  +  1 )  -  1 ) )  x.  ( ( 2 ^ ( ( 2  pCnt  N )  +  1 ) )  -  1 ) ) ) ) )
4948rspcev 2969 . . . 4  |-  ( ( ( ( 2  pCnt 
N )  +  1 )  e.  ZZ  /\  ( ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( ( ( 2  pCnt  N )  +  1 )  - 
1 ) )  x.  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) ) ) )  ->  E. p  e.  ZZ  ( ( ( 2 ^ p )  - 
1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )
508, 32, 40, 49syl12anc 1181 . . 3  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  E. p  e.  ZZ  ( ( ( 2 ^ p )  -  1 )  e. 
Prime  /\  N  =  ( ( 2 ^ (
p  -  1 ) )  x.  ( ( 2 ^ p )  -  1 ) ) ) )
5150ex 423 . 2  |-  ( ( N  e.  NN  /\  2  ||  N )  -> 
( ( 1  sigma  N )  =  ( 2  x.  N )  ->  E. p  e.  ZZ  ( ( ( 2 ^ p )  - 
1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) ) )
52 perfect1 20690 . . . . . 6  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) )  =  ( ( 2 ^ p )  x.  ( ( 2 ^ p )  -  1 ) ) )
53 2cn 9963 . . . . . . . . 9  |-  2  e.  CC
54 mersenne 20689 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  Prime )
55 prmnn 12969 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
5654, 55syl 15 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  NN )
57 expm1t 11295 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  p  e.  NN )  ->  ( 2 ^ p
)  =  ( ( 2 ^ ( p  -  1 ) )  x.  2 ) )
5853, 56, 57sylancr 644 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( ( 2 ^ ( p  -  1 ) )  x.  2 ) )
59 nnm1nn0 10154 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (
p  -  1 )  e.  NN0 )
6056, 59syl 15 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( p  -  1 )  e.  NN0 )
61 expcl 11286 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  ( p  -  1
)  e.  NN0 )  ->  ( 2 ^ (
p  -  1 ) )  e.  CC )
6253, 60, 61sylancr 644 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ (
p  -  1 ) )  e.  CC )
63 mulcom 8970 . . . . . . . . 9  |-  ( ( ( 2 ^ (
p  -  1 ) )  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^ ( p  -  1 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6462, 53, 63sylancl 643 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( p  -  1 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6558, 64eqtrd 2398 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6665oveq1d 5996 . . . . . 6  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  x.  (
( 2 ^ p
)  -  1 ) )  =  ( ( 2  x.  ( 2 ^ ( p  - 
1 ) ) )  x.  ( ( 2 ^ p )  - 
1 ) ) )
6753a1i 10 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  2  e.  CC )
68 prmnn 12969 . . . . . . . . 9  |-  ( ( ( 2 ^ p
)  -  1 )  e.  Prime  ->  ( ( 2 ^ p )  -  1 )  e.  NN )
6968adantl 452 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  NN )
7069nncnd 9909 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  CC )
7167, 62, 70mulassd 9005 . . . . . 6  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2  x.  ( 2 ^ (
p  -  1 ) ) )  x.  (
( 2 ^ p
)  -  1 ) )  =  ( 2  x.  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) ) ) )
7252, 66, 713eqtrd 2402 . . . . 5  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) )  =  ( 2  x.  ( ( 2 ^ ( p  -  1 ) )  x.  (
( 2 ^ p
)  -  1 ) ) ) )
73 oveq2 5989 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
1  sigma  N )  =  ( 1  sigma  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )
74 oveq2 5989 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
2  x.  N )  =  ( 2  x.  ( ( 2 ^ ( p  -  1 ) )  x.  (
( 2 ^ p
)  -  1 ) ) ) )
7573, 74eqeq12d 2380 . . . . 5  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
( 1  sigma  N )  =  ( 2  x.  N )  <->  ( 1 
sigma  ( ( 2 ^ ( p  -  1 ) )  x.  (
( 2 ^ p
)  -  1 ) ) )  =  ( 2  x.  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) ) )
7672, 75syl5ibrcom 213 . . . 4  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( N  =  ( ( 2 ^ (
p  -  1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  ( 1  sigma  N )  =  ( 2  x.  N ) ) )
7776impr 602 . . 3  |-  ( ( p  e.  ZZ  /\  ( ( ( 2 ^ p )  - 
1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )  ->  ( 1 
sigma  N )  =  ( 2  x.  N ) )
7877rexlimiva 2747 . 2  |-  ( E. p  e.  ZZ  (
( ( 2 ^ p )  -  1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) ) )  -> 
( 1  sigma  N )  =  ( 2  x.  N ) )
7951, 78impbid1 194 1  |-  ( ( N  e.  NN  /\  2  ||  N )  -> 
( ( 1  sigma  N )  =  ( 2  x.  N )  <->  E. p  e.  ZZ  ( ( ( 2 ^ p )  - 
1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   E.wrex 2629   class class class wbr 4125  (class class class)co 5981   CCcc 8882   1c1 8885    + caddc 8887    x. cmul 8889    - cmin 9184    / cdiv 9570   NNcn 9893   2c2 9942   NN0cn0 10114   ZZcz 10175   ^cexp 11269    || cdivides 12739   Primecprime 12966    pCnt cpc 13097    sigma csgm 20556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-se 4456  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-isom 5367  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-map 6917  df-pm 6918  df-ixp 6961  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-fi 7312  df-sup 7341  df-oi 7372  df-card 7719  df-cda 7941  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-q 10468  df-rp 10506  df-xneg 10603  df-xadd 10604  df-xmul 10605  df-ioo 10813  df-ioc 10814  df-ico 10815  df-icc 10816  df-fz 10936  df-fzo 11026  df-fl 11089  df-mod 11138  df-seq 11211  df-exp 11270  df-fac 11454  df-bc 11481  df-hash 11506  df-shft 11769  df-cj 11791  df-re 11792  df-im 11793  df-sqr 11927  df-abs 11928  df-limsup 12152  df-clim 12169  df-rlim 12170  df-sum 12367  df-ef 12557  df-sin 12559  df-cos 12560  df-pi 12562  df-dvds 12740  df-gcd 12894  df-prm 12967  df-pc 13098  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-hom 13440  df-cco 13441  df-rest 13537  df-topn 13538  df-topgen 13554  df-pt 13555  df-prds 13558  df-xrs 13613  df-0g 13614  df-gsum 13615  df-qtop 13620  df-imas 13621  df-xps 13623  df-mre 13698  df-mrc 13699  df-acs 13701  df-mnd 14577  df-submnd 14626  df-mulg 14702  df-cntz 15003  df-cmn 15301  df-xmet 16586  df-met 16587  df-bl 16588  df-mopn 16589  df-fbas 16590  df-fg 16591  df-cnfld 16594  df-top 16853  df-bases 16855  df-topon 16856  df-topsp 16857  df-cld 16973  df-ntr 16974  df-cls 16975  df-nei 17052  df-lp 17085  df-perf 17086  df-cn 17174  df-cnp 17175  df-haus 17260  df-tx 17474  df-hmeo 17663  df-fil 17754  df-fm 17846  df-flim 17847  df-flf 17848  df-xms 18098  df-ms 18099  df-tms 18100  df-cncf 18596  df-limc 19431  df-dv 19432  df-log 20132  df-cxp 20133  df-sgm 20562
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