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Theorem perfect 20466
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 12770 . . . . . . . 8  |-  2  e.  Prime
3 simpll 730 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  NN )
4 pcelnn 12918 . . . . . . . 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 10112 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  ZZ )
87peano2zd 10116 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2  pCnt  N )  +  1 )  e.  ZZ )
9 pcdvds 12912 . . . . . . . . 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 9873 . . . . . . . . . 10  |-  2  e.  NN
126nnnn0d 10014 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  NN0 )
13 nnexpcl 11112 . . . . . . . . . 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 12533 . . . . . . . . 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 12916 . . . . . . . 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 9750 . . . . . . . . . . 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 9758 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  e.  CC )
2414nnne0d 9786 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  =/=  0 )
2522, 23, 24divcan2d 9534 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( 2 
pCnt  N ) )  x.  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) )  =  N )
2625oveq2d 5836 . . . . . . . 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 5836 . . . . . . . 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 2326 . . . . . . 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 20465 . . . . . 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 2359 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( ( 2  pCnt  N )  +  1 ) )  -  1 )  e. 
Prime )
336nncnd 9758 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  CC )
34 ax-1cn 8791 . . . . . . . . 9  |-  1  e.  CC
35 pncan 9053 . . . . . . . . 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 2289 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  =  ( ( ( 2  pCnt 
N )  +  1 )  -  1 ) )
3837oveq2d 5836 . . . . . 6  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
3938, 30oveq12d 5838 . . . . 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 2318 . . . 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 5828 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ p )  =  ( 2 ^ (
( 2  pCnt  N
)  +  1 ) ) )
4241oveq1d 5835 . . . . . . 7  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
2 ^ p )  -  1 )  =  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) )
4342eleq1d 2350 . . . . . 6  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
( 2 ^ p
)  -  1 )  e.  Prime  <->  ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 )  e.  Prime )
)
44 oveq1 5827 . . . . . . . . 9  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( p  -  1 )  =  ( ( ( 2 
pCnt  N )  +  1 )  -  1 ) )
4544oveq2d 5836 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ ( p  - 
1 ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
4645, 42oveq12d 5838 . . . . . . 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 2295 . . . . . 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 2885 . . . 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 1180 . . 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 20463 . . . . . 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 9812 . . . . . . . . 9  |-  2  e.  CC
54 mersenne 20462 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  Prime )
55 prmnn 12757 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
5654, 55syl 15 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  NN )
57 expm1t 11126 . . . . . . . . 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 10001 . . . . . . . . . . 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 11117 . . . . . . . . . 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 8819 . . . . . . . . 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 2316 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6665oveq1d 5835 . . . . . 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 12757 . . . . . . . . 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 9758 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  CC )
7167, 62, 70mulassd 8854 . . . . . 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 2320 . . . . 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 5828 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
1  sigma  N )  =  ( 1  sigma  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )
74 oveq2 5828 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
2  x.  N )  =  ( 2  x.  ( ( 2 ^ ( p  -  1 ) )  x.  (
( 2 ^ p
)  -  1 ) ) ) )
7573, 74eqeq12d 2298 . . . . 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 2663 . 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 1623    e. wcel 1685   E.wrex 2545   class class class wbr 4024  (class class class)co 5820   CCcc 8731   1c1 8734    + caddc 8736    x. cmul 8738    - cmin 9033    / cdiv 9419   NNcn 9742   2c2 9791   NN0cn0 9961   ZZcz 10020   ^cexp 11100    || cdivides 12527   Primecprime 12754    pCnt cpc 12885    sigma csgm 20329
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6656  df-map 6770  df-pm 6771  df-ixp 6814  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-fi 7161  df-sup 7190  df-oi 7221  df-card 7568  df-cda 7790  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-7 9805  df-8 9806  df-9 9807  df-10 9808  df-n0 9962  df-z 10021  df-dec 10121  df-uz 10227  df-q 10313  df-rp 10351  df-xneg 10448  df-xadd 10449  df-xmul 10450  df-ioo 10656  df-ioc 10657  df-ico 10658  df-icc 10659  df-fz 10779  df-fzo 10867  df-fl 10921  df-mod 10970  df-seq 11043  df-exp 11101  df-fac 11285  df-bc 11312  df-hash 11334  df-shft 11558  df-cj 11580  df-re 11581  df-im 11582  df-sqr 11716  df-abs 11717  df-limsup 11941  df-clim 11958  df-rlim 11959  df-sum 12155  df-ef 12345  df-sin 12347  df-cos 12348  df-pi 12350  df-dvds 12528  df-gcd 12682  df-prm 12755  df-pc 12886  df-struct 13146  df-ndx 13147  df-slot 13148  df-base 13149  df-sets 13150  df-ress 13151  df-plusg 13217  df-mulr 13218  df-starv 13219  df-sca 13220  df-vsca 13221  df-tset 13223  df-ple 13224  df-ds 13226  df-hom 13228  df-cco 13229  df-rest 13323  df-topn 13324  df-topgen 13340  df-pt 13341  df-prds 13344  df-xrs 13399  df-0g 13400  df-gsum 13401  df-qtop 13406  df-imas 13407  df-xps 13409  df-mre 13484  df-mrc 13485  df-acs 13487  df-mnd 14363  df-submnd 14412  df-mulg 14488  df-cntz 14789  df-cmn 15087  df-xmet 16369  df-met 16370  df-bl 16371  df-mopn 16372  df-cnfld 16374  df-top 16632  df-bases 16634  df-topon 16635  df-topsp 16636  df-cld 16752  df-ntr 16753  df-cls 16754  df-nei 16831  df-lp 16864  df-perf 16865  df-cn 16953  df-cnp 16954  df-haus 17039  df-tx 17253  df-hmeo 17442  df-fbas 17516  df-fg 17517  df-fil 17537  df-fm 17629  df-flim 17630  df-flf 17631  df-xms 17881  df-ms 17882  df-tms 17883  df-cncf 18378  df-limc 19212  df-dv 19213  df-log 19910  df-cxp 19911  df-sgm 20335
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