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Theorem perfect 23371
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.) This is Metamath 100 proof #70. (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 754 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  2  ||  N )
2 2prm 14105 . . . . . . . 8  |-  2  e.  Prime
3 simpll 753 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  NN )
4 pcelnn 14265 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  (
( 2  pCnt  N
)  e.  NN  <->  2  ||  N ) )
52, 3, 4sylancr 663 . . . . . . 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 10968 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  ZZ )
87peano2zd 10972 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2  pCnt  N )  +  1 )  e.  ZZ )
9 pcdvds 14259 . . . . . . . . 9  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  (
2 ^ ( 2 
pCnt  N ) )  ||  N )
102, 3, 9sylancr 663 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  ||  N )
11 2nn 10694 . . . . . . . . . 10  |-  2  e.  NN
126nnnn0d 10853 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  NN0 )
13 nnexpcl 12153 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  ( 2  pCnt  N
)  e.  NN0 )  ->  ( 2 ^ (
2  pCnt  N )
)  e.  NN )
1411, 12, 13sylancr 663 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  e.  NN )
15 nndivdvds 13864 . . . . . . . . 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 661 . . . . . . . 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 14263 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  -.  2  ||  ( N  / 
( 2 ^ (
2  pCnt  N )
) ) )
192, 3, 18sylancr 663 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  -.  2  ||  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) )
20 simpr 461 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 1 
sigma  N )  =  ( 2  x.  N ) )
21 nncn 10545 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
2221ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  CC )
2314nncnd 10553 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  e.  CC )
2414nnne0d 10581 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  =/=  0 )
2522, 23, 24divcan2d 10323 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( 2 
pCnt  N ) )  x.  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) )  =  N )
2625oveq2d 6293 . . . . . . . 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 6293 . . . . . . . 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 2492 . . . . . . 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 23370 . . . . . 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 463 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  =  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) )
3129simpld 459 . . . . 5  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) )  e. 
Prime )
3230, 31eqeltrrd 2530 . . . 4  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
2 ^ ( ( 2  pCnt  N )  +  1 ) )  -  1 )  e. 
Prime )
336nncnd 10553 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  e.  CC )
34 ax-1cn 9548 . . . . . . . . 9  |-  1  e.  CC
35 pncan 9826 . . . . . . . . 9  |-  ( ( ( 2  pCnt  N
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2 
pCnt  N )  +  1 )  -  1 )  =  ( 2  pCnt 
N ) )
3633, 34, 35sylancl 662 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( (
( 2  pCnt  N
)  +  1 )  -  1 )  =  ( 2  pCnt  N
) )
3736eqcomd 2449 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 
pCnt  N )  =  ( ( ( 2  pCnt 
N )  +  1 )  -  1 ) )
3837oveq2d 6293 . . . . . 6  |-  ( ( ( N  e.  NN  /\  2  ||  N )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  ( 2 ^ ( 2  pCnt 
N ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
3938, 30oveq12d 6295 . . . . 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 2484 . . . 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 6285 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ p )  =  ( 2 ^ (
( 2  pCnt  N
)  +  1 ) ) )
4241oveq1d 6292 . . . . . . 7  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
2 ^ p )  -  1 )  =  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) )
4342eleq1d 2510 . . . . . 6  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
( 2 ^ p
)  -  1 )  e.  Prime  <->  ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 )  e.  Prime )
)
44 oveq1 6284 . . . . . . . . 9  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( p  -  1 )  =  ( ( ( 2 
pCnt  N )  +  1 )  -  1 ) )
4544oveq2d 6293 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ ( p  - 
1 ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
4645, 42oveq12d 6295 . . . . . . 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 2455 . . . . . 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 710 . . . . 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 3194 . . . 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 1225 . . 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 23368 . . . . . 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 10607 . . . . . . . . 9  |-  2  e.  CC
54 mersenne 23367 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  Prime )
55 prmnn 14092 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
5654, 55syl 16 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  NN )
57 expm1t 12168 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  p  e.  NN )  ->  ( 2 ^ p
)  =  ( ( 2 ^ ( p  -  1 ) )  x.  2 ) )
5853, 56, 57sylancr 663 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( ( 2 ^ ( p  -  1 ) )  x.  2 ) )
59 nnm1nn0 10838 . . . . . . . . . . 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 12158 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  ( p  -  1
)  e.  NN0 )  ->  ( 2 ^ (
p  -  1 ) )  e.  CC )
6253, 60, 61sylancr 663 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ (
p  -  1 ) )  e.  CC )
63 mulcom 9576 . . . . . . . . 9  |-  ( ( ( 2 ^ (
p  -  1 ) )  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^ ( p  -  1 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6462, 53, 63sylancl 662 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( p  -  1 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6558, 64eqtrd 2482 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6665oveq1d 6292 . . . . . 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 10609 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  2  e.  CC )
68 prmnn 14092 . . . . . . . . 9  |-  ( ( ( 2 ^ p
)  -  1 )  e.  Prime  ->  ( ( 2 ^ p )  -  1 )  e.  NN )
6968adantl 466 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  NN )
7069nncnd 10553 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  CC )
7167, 62, 70mulassd 9617 . . . . . 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 2486 . . . . 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 6285 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
1  sigma  N )  =  ( 1  sigma  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )
74 oveq2 6285 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
2  x.  N )  =  ( 2  x.  ( ( 2 ^ ( p  -  1 ) )  x.  (
( 2 ^ p
)  -  1 ) ) ) )
7573, 74eqeq12d 2463 . . . . 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 619 . . 3  |-  ( ( p  e.  ZZ  /\  ( ( ( 2 ^ p )  - 
1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )  ->  ( 1 
sigma  N )  =  ( 2  x.  N ) )
7877rexlimiva 2929 . 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 1381    e. wcel 1802   E.wrex 2792   class class class wbr 4433  (class class class)co 6277   CCcc 9488   1c1 9491    + caddc 9493    x. cmul 9495    - cmin 9805    / cdiv 10207   NNcn 10537   2c2 10586   NN0cn0 10796   ZZcz 10865   ^cexp 12140    || cdvds 13858   Primecprime 14089    pCnt cpc 14232    sigma csgm 23234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569  ax-mulf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-iin 4314  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-supp 6900  df-recs 7040  df-rdg 7074  df-1o 7128  df-2o 7129  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-ixp 7468  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-fsupp 7828  df-fi 7869  df-sup 7899  df-oi 7933  df-card 8318  df-cda 8546  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-ioo 11537  df-ioc 11538  df-ico 11539  df-icc 11540  df-fz 11677  df-fzo 11799  df-fl 11903  df-mod 11971  df-seq 12082  df-exp 12141  df-fac 12328  df-bc 12355  df-hash 12380  df-shft 12874  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-limsup 13268  df-clim 13285  df-rlim 13286  df-sum 13483  df-ef 13676  df-sin 13678  df-cos 13679  df-pi 13681  df-dvds 13859  df-gcd 14017  df-prm 14090  df-pc 14233  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-mulr 14583  df-starv 14584  df-sca 14585  df-vsca 14586  df-ip 14587  df-tset 14588  df-ple 14589  df-ds 14591  df-unif 14592  df-hom 14593  df-cco 14594  df-rest 14692  df-topn 14693  df-0g 14711  df-gsum 14712  df-topgen 14713  df-pt 14714  df-prds 14717  df-xrs 14771  df-qtop 14776  df-imas 14777  df-xps 14779  df-mre 14855  df-mrc 14856  df-acs 14858  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-submnd 15836  df-mulg 15929  df-cntz 16224  df-cmn 16669  df-psmet 18279  df-xmet 18280  df-met 18281  df-bl 18282  df-mopn 18283  df-fbas 18284  df-fg 18285  df-cnfld 18289  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-cld 19386  df-ntr 19387  df-cls 19388  df-nei 19465  df-lp 19503  df-perf 19504  df-cn 19594  df-cnp 19595  df-haus 19682  df-tx 19929  df-hmeo 20122  df-fil 20213  df-fm 20305  df-flim 20306  df-flf 20307  df-xms 20689  df-ms 20690  df-tms 20691  df-cncf 21248  df-limc 22136  df-dv 22137  df-log 22809  df-cxp 22810  df-sgm 23240
This theorem is referenced by: (None)
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