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Theorem perfect 22455
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 747 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  2  ||  N )
2 2prm 13762 . . . . . . . 8  |-  2  e.  Prime
3 simpll 746 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  NN )
4 pcelnn 13919 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  (
( 2  pCnt  N
)  e.  NN  <->  2  ||  N ) )
52, 3, 4sylancr 656 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2  pCnt  N )  e.  NN  <->  2  ||  N
) )
61, 5mpbird 232 . . . . . 6  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  NN )
76nnzd 10734 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  ZZ )
87peano2zd 10738 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2  pCnt  N )  +  1 )  e.  ZZ )
9 pcdvds 13913 . . . . . . . . 9  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  (
2 ^ ( 2 
pCnt  N ) )  ||  N )
102, 3, 9sylancr 656 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  ||  N )
11 2nn 10467 . . . . . . . . . 10  |-  2  e.  NN
126nnnn0d 10624 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  NN0 )
13 nnexpcl 11862 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  ( 2  pCnt  N
)  e.  NN0 )  ->  ( 2 ^ (
2  pCnt  N )
)  e.  NN )
1411, 12, 13sylancr 656 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  e.  NN )
15 nndivdvds 13524 . . . . . . . . 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 654 . . . . . . . 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 210 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  e.  NN )
18 pcndvds2 13917 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  -.  2  ||  ( N  / 
( 2 ^ (
2  pCnt  N )
) ) )
192, 3, 18sylancr 656 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  -.  2  ||  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) )
20 simpr 458 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 1 
sigma  N )  =  ( 2  x.  N ) )
21 nncn 10318 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
2221ad2antrr 718 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  CC )
2314nncnd 10326 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  e.  CC )
2414nnne0d 10354 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  =/=  0 )
2522, 23, 24divcan2d 10097 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( 2 
pCnt  N ) )  x.  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) )  =  N )
2625oveq2d 6096 . . . . . . . 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 6096 . . . . . . . 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 2475 . . . . . . 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 22454 . . . . . 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 460 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  =  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) )
3129simpld 456 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  e. 
Prime )
3230, 31eqeltrrd 2508 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( ( 2  pCnt  N )  +  1 ) )  -  1 )  e. 
Prime )
336nncnd 10326 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  CC )
34 ax-1cn 9328 . . . . . . . . 9  |-  1  e.  CC
35 pncan 9604 . . . . . . . . 9  |-  ( ( ( 2  pCnt  N
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2 
pCnt  N )  +  1 )  -  1 )  =  ( 2  pCnt 
N ) )
3633, 34, 35sylancl 655 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
( 2  pCnt  N
)  +  1 )  -  1 )  =  ( 2  pCnt  N
) )
3736eqcomd 2438 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  =  ( ( ( 2  pCnt 
N )  +  1 )  -  1 ) )
3837oveq2d 6096 . . . . . 6  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
3938, 30oveq12d 6098 . . . . 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 2467 . . . 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 6088 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ p )  =  ( 2 ^ (
( 2  pCnt  N
)  +  1 ) ) )
4241oveq1d 6095 . . . . . . 7  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
2 ^ p )  -  1 )  =  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) )
4342eleq1d 2499 . . . . . 6  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
( 2 ^ p
)  -  1 )  e.  Prime  <->  ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 )  e.  Prime )
)
44 oveq1 6087 . . . . . . . . 9  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( p  -  1 )  =  ( ( ( 2 
pCnt  N )  +  1 )  -  1 ) )
4544oveq2d 6096 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ ( p  - 
1 ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
4645, 42oveq12d 6098 . . . . . . 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 2444 . . . . . 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 703 . . . . 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 3062 . . . 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 1209 . . 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 434 . 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 22452 . . . . . 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 10380 . . . . . . . . 9  |-  2  e.  CC
54 mersenne 22451 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  Prime )
55 prmnn 13749 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
5654, 55syl 16 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  NN )
57 expm1t 11876 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  p  e.  NN )  ->  ( 2 ^ p
)  =  ( ( 2 ^ ( p  -  1 ) )  x.  2 ) )
5853, 56, 57sylancr 656 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( ( 2 ^ ( p  -  1 ) )  x.  2 ) )
59 nnm1nn0 10609 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (
p  -  1 )  e.  NN0 )
6056, 59syl 16 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( p  -  1 )  e.  NN0 )
61 expcl 11867 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  ( p  -  1
)  e.  NN0 )  ->  ( 2 ^ (
p  -  1 ) )  e.  CC )
6253, 60, 61sylancr 656 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ (
p  -  1 ) )  e.  CC )
63 mulcom 9356 . . . . . . . . 9  |-  ( ( ( 2 ^ (
p  -  1 ) )  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^ ( p  -  1 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6462, 53, 63sylancl 655 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( p  -  1 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6558, 64eqtrd 2465 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6665oveq1d 6095 . . . . . 6  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  x.  (
( 2 ^ p
)  -  1 ) )  =  ( ( 2  x.  ( 2 ^ ( p  - 
1 ) ) )  x.  ( ( 2 ^ p )  - 
1 ) ) )
67 2cnd 10382 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  2  e.  CC )
68 prmnn 13749 . . . . . . . . 9  |-  ( ( ( 2 ^ p
)  -  1 )  e.  Prime  ->  ( ( 2 ^ p )  -  1 )  e.  NN )
6968adantl 463 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  NN )
7069nncnd 10326 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  CC )
7167, 62, 70mulassd 9397 . . . . . 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 2469 . . . . 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 6088 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
1  sigma  N )  =  ( 1  sigma  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )
74 oveq2 6088 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
2  x.  N )  =  ( 2  x.  ( ( 2 ^ ( p  -  1 ) )  x.  (
( 2 ^ p
)  -  1 ) ) ) )
7573, 74eqeq12d 2447 . . . . 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 222 . . . 4  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( N  =  ( ( 2 ^ (
p  -  1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  ( 1  sigma  N )  =  ( 2  x.  N ) ) )
7776impr 614 . . 3  |-  ( ( p  e.  ZZ  /\  ( ( ( 2 ^ p )  - 
1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )  ->  ( 1 
sigma  N )  =  ( 2  x.  N ) )
7877rexlimiva 2826 . 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 203 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362    e. wcel 1755   E.wrex 2706   class class class wbr 4280  (class class class)co 6080   CCcc 9268   1c1 9271    + caddc 9273    x. cmul 9275    - cmin 9583    / cdiv 9981   NNcn 10310   2c2 10359   NN0cn0 10567   ZZcz 10634   ^cexp 11849    || cdivides 13518   Primecprime 13746    pCnt cpc 13886    sigma csgm 22318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ioc 11293  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-fl 11626  df-mod 11693  df-seq 11791  df-exp 11850  df-fac 12036  df-bc 12063  df-hash 12088  df-shft 12540  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-limsup 12933  df-clim 12950  df-rlim 12951  df-sum 13148  df-ef 13336  df-sin 13338  df-cos 13339  df-pi 13341  df-dvds 13519  df-gcd 13674  df-prm 13747  df-pc 13887  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184  df-log 21893  df-cxp 21894  df-sgm 22324
This theorem is referenced by: (None)
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