MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  perfect Unicode version

Theorem perfect 20397
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 734 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  2  ||  N )
2 2prm 12701 . . . . . . . 8  |-  2  e.  Prime
3 simpll 733 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  NN )
4 pcelnn 12849 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  (
( 2  pCnt  N
)  e.  NN  <->  2  ||  N ) )
52, 3, 4sylancr 647 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2  pCnt  N )  e.  NN  <->  2  ||  N
) )
61, 5mpbird 225 . . . . . 6  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  NN )
76nnzd 10048 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  ZZ )
87peano2zd 10052 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2  pCnt  N )  +  1 )  e.  ZZ )
9 pcdvds 12843 . . . . . . . . 9  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  (
2 ^ ( 2 
pCnt  N ) )  ||  N )
102, 3, 9sylancr 647 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  ||  N )
11 2nn 9809 . . . . . . . . . 10  |-  2  e.  NN
126nnnn0d 9950 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  NN0 )
13 nnexpcl 11047 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  ( 2  pCnt  N
)  e.  NN0 )  ->  ( 2 ^ (
2  pCnt  N )
)  e.  NN )
1411, 12, 13sylancr 647 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  e.  NN )
15 nndivdivides 12464 . . . . . . . . 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 645 . . . . . . . 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 203 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  e.  NN )
18 pcndvds2 12847 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  -.  2  ||  ( N  / 
( 2 ^ (
2  pCnt  N )
) ) )
192, 3, 18sylancr 647 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  -.  2  ||  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) )
20 simpr 449 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 1 
sigma  N )  =  ( 2  x.  N ) )
21 nncn 9687 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
2221ad2antrr 709 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  CC )
2314nncnd 9695 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  e.  CC )
2414nnne0d 9723 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  =/=  0 )
2522, 23, 24divcan2d 9471 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( 2 
pCnt  N ) )  x.  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) )  =  N )
2625oveq2d 5773 . . . . . . . 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 5773 . . . . . . . 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 2298 . . . . . . 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 20396 . . . . . 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 451 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  =  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) )
3129simpld 447 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  e. 
Prime )
3230, 31eqeltrrd 2331 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( ( 2  pCnt  N )  +  1 ) )  -  1 )  e. 
Prime )
336nncnd 9695 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  CC )
34 ax-1cn 8728 . . . . . . . . 9  |-  1  e.  CC
35 pncan 8990 . . . . . . . . 9  |-  ( ( ( 2  pCnt  N
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2 
pCnt  N )  +  1 )  -  1 )  =  ( 2  pCnt 
N ) )
3633, 34, 35sylancl 646 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
( 2  pCnt  N
)  +  1 )  -  1 )  =  ( 2  pCnt  N
) )
3736eqcomd 2261 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  =  ( ( ( 2  pCnt 
N )  +  1 )  -  1 ) )
3837oveq2d 5773 . . . . . 6  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
3938, 30oveq12d 5775 . . . . 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 2290 . . . 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 5765 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ p )  =  ( 2 ^ (
( 2  pCnt  N
)  +  1 ) ) )
4241oveq1d 5772 . . . . . . 7  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
2 ^ p )  -  1 )  =  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) )
4342eleq1d 2322 . . . . . 6  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
( 2 ^ p
)  -  1 )  e.  Prime  <->  ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 )  e.  Prime )
)
44 oveq1 5764 . . . . . . . . 9  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( p  -  1 )  =  ( ( ( 2 
pCnt  N )  +  1 )  -  1 ) )
4544oveq2d 5773 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ ( p  - 
1 ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
4645, 42oveq12d 5775 . . . . . . 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 2267 . . . . . 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 694 . . . . 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 ) ) ) ) )
4948rcla4ev 2835 . . . 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 1185 . . 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 425 . 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 20394 . . . . . 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 9749 . . . . . . . . 9  |-  2  e.  CC
54 mersenne 20393 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  Prime )
55 prmnn 12689 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
5654, 55syl 17 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  NN )
57 expm1t 11061 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  p  e.  NN )  ->  ( 2 ^ p
)  =  ( ( 2 ^ ( p  -  1 ) )  x.  2 ) )
5853, 56, 57sylancr 647 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( ( 2 ^ ( p  -  1 ) )  x.  2 ) )
59 nnm1nn0 9937 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (
p  -  1 )  e.  NN0 )
6056, 59syl 17 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( p  -  1 )  e.  NN0 )
61 expcl 11052 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  ( p  -  1
)  e.  NN0 )  ->  ( 2 ^ (
p  -  1 ) )  e.  CC )
6253, 60, 61sylancr 647 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ (
p  -  1 ) )  e.  CC )
63 mulcom 8756 . . . . . . . . 9  |-  ( ( ( 2 ^ (
p  -  1 ) )  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^ ( p  -  1 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6462, 53, 63sylancl 646 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( p  -  1 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6558, 64eqtrd 2288 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6665oveq1d 5772 . . . . . 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 12 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  2  e.  CC )
68 prmnn 12689 . . . . . . . . 9  |-  ( ( ( 2 ^ p
)  -  1 )  e.  Prime  ->  ( ( 2 ^ p )  -  1 )  e.  NN )
6968adantl 454 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  NN )
7069nncnd 9695 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  CC )
7167, 62, 70mulassd 8791 . . . . . 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 2292 . . . . 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 5765 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
1  sigma  N )  =  ( 1  sigma  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )
74 oveq2 5765 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
2  x.  N )  =  ( 2  x.  ( ( 2 ^ ( p  -  1 ) )  x.  (
( 2 ^ p
)  -  1 ) ) ) )
7573, 74eqeq12d 2270 . . . . 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 215 . . . 4  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( N  =  ( ( 2 ^ (
p  -  1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  ( 1  sigma  N )  =  ( 2  x.  N ) ) )
7776impr 605 . . 3  |-  ( ( p  e.  ZZ  /\  ( ( ( 2 ^ p )  - 
1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )  ->  ( 1 
sigma  N )  =  ( 2  x.  N ) )
7877rexlimiva 2633 . 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 196 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 5    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2517   class class class wbr 3963  (class class class)co 5757   CCcc 8668   1c1 8671    + caddc 8673    x. cmul 8675    - cmin 8970    / cdiv 9356   NNcn 9679   2c2 9728   NN0cn0 9897   ZZcz 9956   ^cexp 11035    || cdivides 12458   Primecprime 12685    pCnt cpc 12816    sigma csgm 20260
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748  ax-addf 8749  ax-mulf 8750
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-iin 3849  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-of 5977  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-er 6593  df-map 6707  df-pm 6708  df-ixp 6751  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-fi 7098  df-sup 7127  df-oi 7158  df-card 7505  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-4 9739  df-5 9740  df-6 9741  df-7 9742  df-8 9743  df-9 9744  df-10 9745  df-n0 9898  df-z 9957  df-dec 10057  df-uz 10163  df-q 10249  df-rp 10287  df-xneg 10384  df-xadd 10385  df-xmul 10386  df-ioo 10591  df-ioc 10592  df-ico 10593  df-icc 10594  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-shft 11492  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-limsup 11875  df-clim 11892  df-rlim 11893  df-sum 12089  df-ef 12276  df-sin 12278  df-cos 12279  df-pi 12281  df-divides 12459  df-gcd 12613  df-prime 12686  df-pc 12817  df-struct 13077  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-mulr 13149  df-starv 13150  df-sca 13151  df-vsca 13152  df-tset 13154  df-ple 13155  df-ds 13157  df-hom 13159  df-cco 13160  df-rest 13254  df-topn 13255  df-topgen 13271  df-pt 13272  df-prds 13275  df-xrs 13330  df-0g 13331  df-gsum 13332  df-qtop 13337  df-imas 13338  df-xps 13340  df-mre 13415  df-mrc 13416  df-acs 13418  df-mnd 14294  df-submnd 14343  df-mulg 14419  df-cntz 14720  df-cmn 15018  df-xmet 16300  df-met 16301  df-bl 16302  df-mopn 16303  df-cnfld 16305  df-top 16563  df-bases 16565  df-topon 16566  df-topsp 16567  df-cld 16683  df-ntr 16684  df-cls 16685  df-nei 16762  df-lp 16795  df-perf 16796  df-cn 16884  df-cnp 16885  df-haus 16970  df-tx 17184  df-hmeo 17373  df-fbas 17447  df-fg 17448  df-fil 17468  df-fm 17560  df-flim 17561  df-flf 17562  df-xms 17812  df-ms 17813  df-tms 17814  df-cncf 18309  df-limc 19143  df-dv 19144  df-log 19841  df-cxp 19842  df-sgm 20266
  Copyright terms: Public domain W3C validator