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Theorem perfectALTV 38990
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.) (Revised by AV, 1-Jul-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
perfectALTV  |-  ( ( N  e.  NN  /\  N  e. Even  )  ->  ( ( 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 perfectALTV
StepHypRef Expression
1 2dvdseven 38928 . . . . . . . 8  |-  ( N  e. Even  ->  2  ||  N
)
21ad2antlr 741 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
2  ||  N )
3 2prm 14719 . . . . . . . 8  |-  2  e.  Prime
4 simpll 768 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  NN )
5 pcelnn 14898 . . . . . . . 8  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  (
( 2  pCnt  N
)  e.  NN  <->  2  ||  N ) )
63, 4, 5sylancr 676 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( ( 2  pCnt 
N )  e.  NN  <->  2 
||  N ) )
72, 6mpbird 240 . . . . . 6  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2  pCnt  N
)  e.  NN )
87nnzd 11062 . . . . 5  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2  pCnt  N
)  e.  ZZ )
98peano2zd 11066 . . . 4  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( ( 2  pCnt 
N )  +  1 )  e.  ZZ )
10 pcdvds 14892 . . . . . . . . 9  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  (
2 ^ ( 2 
pCnt  N ) )  ||  N )
113, 4, 10sylancr 676 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2 ^ (
2  pCnt  N )
)  ||  N )
12 2nn 10790 . . . . . . . . . 10  |-  2  e.  NN
137nnnn0d 10949 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2  pCnt  N
)  e.  NN0 )
14 nnexpcl 12323 . . . . . . . . . 10  |-  ( ( 2  e.  NN  /\  ( 2  pCnt  N
)  e.  NN0 )  ->  ( 2 ^ (
2  pCnt  N )
)  e.  NN )
1512, 13, 14sylancr 676 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2 ^ (
2  pCnt  N )
)  e.  NN )
16 nndivdvds 14388 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  ( 2 ^ (
2  pCnt  N )
)  e.  NN )  ->  ( ( 2 ^ ( 2  pCnt 
N ) )  ||  N 
<->  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) )  e.  NN ) )
174, 15, 16syl2anc 673 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( ( 2 ^ ( 2  pCnt  N
) )  ||  N  <->  ( N  /  ( 2 ^ ( 2  pCnt 
N ) ) )  e.  NN ) )
1811, 17mpbid 215 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( N  /  (
2 ^ ( 2 
pCnt  N ) ) )  e.  NN )
1918nnzd 11062 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( N  /  (
2 ^ ( 2 
pCnt  N ) ) )  e.  ZZ )
20 pcndvds2 14896 . . . . . . . . 9  |-  ( ( 2  e.  Prime  /\  N  e.  NN )  ->  -.  2  ||  ( N  / 
( 2 ^ (
2  pCnt  N )
) ) )
213, 4, 20sylancr 676 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  -.  2  ||  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) ) )
22 isodd3 38927 . . . . . . . 8  |-  ( ( N  /  ( 2 ^ ( 2  pCnt 
N ) ) )  e. Odd 
<->  ( ( N  / 
( 2 ^ (
2  pCnt  N )
) )  e.  ZZ  /\ 
-.  2  ||  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) ) ) )
2319, 21, 22sylanbrc 677 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( N  /  (
2 ^ ( 2 
pCnt  N ) ) )  e. Odd  )
24 simpr 468 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 1  sigma  N )  =  ( 2  x.  N ) )
25 nncn 10639 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  CC )
2625ad2antrr 740 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  e.  CC )
2715nncnd 10647 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2 ^ (
2  pCnt  N )
)  e.  CC )
2815nnne0d 10676 . . . . . . . . . 10  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2 ^ (
2  pCnt  N )
)  =/=  0 )
2926, 27, 28divcan2d 10407 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( ( 2 ^ ( 2  pCnt  N
) )  x.  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) ) )  =  N )
3029oveq2d 6324 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 1  sigma  ( ( 2 ^ ( 2 
pCnt  N ) )  x.  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) ) ) )  =  ( 1  sigma  N )
)
3129oveq2d 6324 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2  x.  (
( 2 ^ (
2  pCnt  N )
)  x.  ( N  /  ( 2 ^ ( 2  pCnt  N
) ) ) ) )  =  ( 2  x.  N ) )
3224, 30, 313eqtr4d 2515 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 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 ) ) ) ) ) )
337, 18, 23, 32perfectALTVlem2 38989 . . . . . 6  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( ( N  / 
( 2 ^ (
2  pCnt  N )
) )  e.  Prime  /\  ( N  /  (
2 ^ ( 2 
pCnt  N ) ) )  =  ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 ) ) )
3433simprd 470 . . . . 5  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( N  /  (
2 ^ ( 2 
pCnt  N ) ) )  =  ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 ) )
3533simpld 466 . . . . 5  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( N  /  (
2 ^ ( 2 
pCnt  N ) ) )  e.  Prime )
3634, 35eqeltrrd 2550 . . . 4  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 )  e.  Prime )
377nncnd 10647 . . . . . . . . 9  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2  pCnt  N
)  e.  CC )
38 ax-1cn 9615 . . . . . . . . 9  |-  1  e.  CC
39 pncan 9901 . . . . . . . . 9  |-  ( ( ( 2  pCnt  N
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( 2 
pCnt  N )  +  1 )  -  1 )  =  ( 2  pCnt 
N ) )
4037, 38, 39sylancl 675 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( ( ( 2 
pCnt  N )  +  1 )  -  1 )  =  ( 2  pCnt 
N ) )
4140eqcomd 2477 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2  pCnt  N
)  =  ( ( ( 2  pCnt  N
)  +  1 )  -  1 ) )
4241oveq2d 6324 . . . . . 6  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  -> 
( 2 ^ (
2  pCnt  N )
)  =  ( 2 ^ ( ( ( 2  pCnt  N )  +  1 )  - 
1 ) ) )
4342, 34oveq12d 6326 . . . . 5  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 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 ) ) )
4429, 43eqtr3d 2507 . . . 4  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  N  =  ( (
2 ^ ( ( ( 2  pCnt  N
)  +  1 )  -  1 ) )  x.  ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 ) ) )
45 oveq2 6316 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ p )  =  ( 2 ^ (
( 2  pCnt  N
)  +  1 ) ) )
4645oveq1d 6323 . . . . . . 7  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
2 ^ p )  -  1 )  =  ( ( 2 ^ ( ( 2  pCnt 
N )  +  1 ) )  -  1 ) )
4746eleq1d 2533 . . . . . 6  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( (
( 2 ^ p
)  -  1 )  e.  Prime  <->  ( ( 2 ^ ( ( 2 
pCnt  N )  +  1 ) )  -  1 )  e.  Prime )
)
48 oveq1 6315 . . . . . . . . 9  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( p  -  1 )  =  ( ( ( 2 
pCnt  N )  +  1 )  -  1 ) )
4948oveq2d 6324 . . . . . . . 8  |-  ( p  =  ( ( 2 
pCnt  N )  +  1 )  ->  ( 2 ^ ( p  - 
1 ) )  =  ( 2 ^ (
( ( 2  pCnt 
N )  +  1 )  -  1 ) ) )
5049, 46oveq12d 6326 . . . . . . 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 ) ) )
5150eqeq2d 2481 . . . . . 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 ) ) ) )
5247, 51anbi12d 725 . . . . 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 ) ) ) ) )
5352rspcev 3136 . . . 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 ) ) ) )
549, 36, 44, 53syl12anc 1290 . . 3  |-  ( ( ( N  e.  NN  /\  N  e. Even  )  /\  ( 1  sigma  N )  =  ( 2  x.  N ) )  ->  E. p  e.  ZZ  ( ( ( 2 ^ p )  - 
1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )
5554ex 441 . 2  |-  ( ( N  e.  NN  /\  N  e. Even  )  ->  ( ( 1  sigma  N )  =  ( 2  x.  N )  ->  E. p  e.  ZZ  ( ( ( 2 ^ p )  -  1 )  e. 
Prime  /\  N  =  ( ( 2 ^ (
p  -  1 ) )  x.  ( ( 2 ^ p )  -  1 ) ) ) ) )
56 perfect1 24235 . . . . . 6  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 1  sigma  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) )  =  ( ( 2 ^ p )  x.  ( ( 2 ^ p )  -  1 ) ) )
57 2cn 10702 . . . . . . . . 9  |-  2  e.  CC
58 mersenne 24234 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  Prime )
59 prmnn 14704 . . . . . . . . . 10  |-  ( p  e.  Prime  ->  p  e.  NN )
6058, 59syl 17 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  p  e.  NN )
61 expm1t 12338 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  p  e.  NN )  ->  ( 2 ^ p
)  =  ( ( 2 ^ ( p  -  1 ) )  x.  2 ) )
6257, 60, 61sylancr 676 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( ( 2 ^ ( p  -  1 ) )  x.  2 ) )
63 nnm1nn0 10935 . . . . . . . . . . 11  |-  ( p  e.  NN  ->  (
p  -  1 )  e.  NN0 )
6460, 63syl 17 . . . . . . . . . 10  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( p  -  1 )  e.  NN0 )
65 expcl 12328 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  ( p  -  1
)  e.  NN0 )  ->  ( 2 ^ (
p  -  1 ) )  e.  CC )
6657, 64, 65sylancr 676 . . . . . . . . 9  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ (
p  -  1 ) )  e.  CC )
67 mulcom 9643 . . . . . . . . 9  |-  ( ( ( 2 ^ (
p  -  1 ) )  e.  CC  /\  2  e.  CC )  ->  ( ( 2 ^ ( p  -  1 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6866, 57, 67sylancl 675 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ ( p  -  1 ) )  x.  2 )  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
6962, 68eqtrd 2505 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( 2 ^ p
)  =  ( 2  x.  ( 2 ^ ( p  -  1 ) ) ) )
7069oveq1d 6323 . . . . . 6  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  x.  (
( 2 ^ p
)  -  1 ) )  =  ( ( 2  x.  ( 2 ^ ( p  - 
1 ) ) )  x.  ( ( 2 ^ p )  - 
1 ) ) )
71 2cnd 10704 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  2  e.  CC )
72 prmnn 14704 . . . . . . . . 9  |-  ( ( ( 2 ^ p
)  -  1 )  e.  Prime  ->  ( ( 2 ^ p )  -  1 )  e.  NN )
7372adantl 473 . . . . . . . 8  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  NN )
7473nncnd 10647 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( ( 2 ^ p )  -  1 )  e.  CC )
7571, 66, 74mulassd 9684 . . . . . 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 ) ) ) )
7656, 70, 753eqtrd 2509 . . . . 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 ) ) ) )
77 oveq2 6316 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
1  sigma  N )  =  ( 1  sigma  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )
78 oveq2 6316 . . . . . 6  |-  ( N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  (
2  x.  N )  =  ( 2  x.  ( ( 2 ^ ( p  -  1 ) )  x.  (
( 2 ^ p
)  -  1 ) ) ) )
7977, 78eqeq12d 2486 . . . . 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 ) ) ) ) )
8076, 79syl5ibrcom 230 . . . 4  |-  ( ( p  e.  ZZ  /\  ( ( 2 ^ p )  -  1 )  e.  Prime )  ->  ( N  =  ( ( 2 ^ (
p  -  1 ) )  x.  ( ( 2 ^ p )  -  1 ) )  ->  ( 1  sigma  N )  =  ( 2  x.  N ) ) )
8180impr 631 . . 3  |-  ( ( p  e.  ZZ  /\  ( ( ( 2 ^ p )  - 
1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  -  1 ) )  x.  ( ( 2 ^ p )  - 
1 ) ) ) )  ->  ( 1 
sigma  N )  =  ( 2  x.  N ) )
8281rexlimiva 2868 . 2  |-  ( E. p  e.  ZZ  (
( ( 2 ^ p )  -  1 )  e.  Prime  /\  N  =  ( ( 2 ^ ( p  - 
1 ) )  x.  ( ( 2 ^ p )  -  1 ) ) )  -> 
( 1  sigma  N )  =  ( 2  x.  N ) )
8355, 82impbid1 208 1  |-  ( ( N  e.  NN  /\  N  e. Even  )  ->  ( ( 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 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757   class class class wbr 4395  (class class class)co 6308   CCcc 9555   1c1 9558    + caddc 9560    x. cmul 9562    - cmin 9880    / cdiv 10291   NNcn 10631   2c2 10681   NN0cn0 10893   ZZcz 10961   ^cexp 12310    || cdvds 14382   Primecprime 14701    pCnt cpc 14865    sigma csgm 24101   Even ceven 38898   Odd codd 38899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ioc 11665  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-pi 14203  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901  df-log 23585  df-cxp 23586  df-sgm 24107  df-even 38900  df-odd 38901
This theorem is referenced by: (None)
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