Step |
Hyp |
Ref |
Expression |
1 |
|
simplr |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> 2 || N ) |
2 |
|
2prm |
|- 2 e. Prime |
3 |
|
simpll |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> N e. NN ) |
4 |
|
pcelnn |
|- ( ( 2 e. Prime /\ N e. NN ) -> ( ( 2 pCnt N ) e. NN <-> 2 || N ) ) |
5 |
2 3 4
|
sylancr |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 pCnt N ) e. NN <-> 2 || N ) ) |
6 |
1 5
|
mpbird |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. NN ) |
7 |
6
|
nnzd |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. ZZ ) |
8 |
7
|
peano2zd |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 pCnt N ) + 1 ) e. ZZ ) |
9 |
|
pcdvds |
|- ( ( 2 e. Prime /\ N e. NN ) -> ( 2 ^ ( 2 pCnt N ) ) || N ) |
10 |
2 3 9
|
sylancr |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) || N ) |
11 |
|
2nn |
|- 2 e. NN |
12 |
6
|
nnnn0d |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. NN0 ) |
13 |
|
nnexpcl |
|- ( ( 2 e. NN /\ ( 2 pCnt N ) e. NN0 ) -> ( 2 ^ ( 2 pCnt N ) ) e. NN ) |
14 |
11 12 13
|
sylancr |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) e. NN ) |
15 |
|
nndivdvds |
|- ( ( N e. NN /\ ( 2 ^ ( 2 pCnt N ) ) e. NN ) -> ( ( 2 ^ ( 2 pCnt N ) ) || N <-> ( N / ( 2 ^ ( 2 pCnt N ) ) ) e. NN ) ) |
16 |
3 14 15
|
syl2anc |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 ^ ( 2 pCnt N ) ) || N <-> ( N / ( 2 ^ ( 2 pCnt N ) ) ) e. NN ) ) |
17 |
10 16
|
mpbid |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( N / ( 2 ^ ( 2 pCnt N ) ) ) e. NN ) |
18 |
|
pcndvds2 |
|- ( ( 2 e. Prime /\ N e. NN ) -> -. 2 || ( N / ( 2 ^ ( 2 pCnt N ) ) ) ) |
19 |
2 3 18
|
sylancr |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> -. 2 || ( N / ( 2 ^ ( 2 pCnt N ) ) ) ) |
20 |
|
simpr |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 1 sigma N ) = ( 2 x. N ) ) |
21 |
|
nncn |
|- ( N e. NN -> N e. CC ) |
22 |
21
|
ad2antrr |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> N e. CC ) |
23 |
14
|
nncnd |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) e. CC ) |
24 |
14
|
nnne0d |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) =/= 0 ) |
25 |
22 23 24
|
divcan2d |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 ^ ( 2 pCnt N ) ) x. ( N / ( 2 ^ ( 2 pCnt N ) ) ) ) = N ) |
26 |
25
|
oveq2d |
|- ( ( ( 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 ) ) |
27 |
25
|
oveq2d |
|- ( ( ( 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 ) ) |
28 |
20 26 27
|
3eqtr4d |
|- ( ( ( 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 ) ) ) ) ) ) |
29 |
6 17 19 28
|
perfectlem2 |
|- ( ( ( 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 ) ) ) |
30 |
29
|
simprd |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( N / ( 2 ^ ( 2 pCnt N ) ) ) = ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) ) |
31 |
29
|
simpld |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( N / ( 2 ^ ( 2 pCnt N ) ) ) e. Prime ) |
32 |
30 31
|
eqeltrrd |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime ) |
33 |
6
|
nncnd |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) e. CC ) |
34 |
|
ax-1cn |
|- 1 e. CC |
35 |
|
pncan |
|- ( ( ( 2 pCnt N ) e. CC /\ 1 e. CC ) -> ( ( ( 2 pCnt N ) + 1 ) - 1 ) = ( 2 pCnt N ) ) |
36 |
33 34 35
|
sylancl |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( ( ( 2 pCnt N ) + 1 ) - 1 ) = ( 2 pCnt N ) ) |
37 |
36
|
eqcomd |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 pCnt N ) = ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) |
38 |
37
|
oveq2d |
|- ( ( ( N e. NN /\ 2 || N ) /\ ( 1 sigma N ) = ( 2 x. N ) ) -> ( 2 ^ ( 2 pCnt N ) ) = ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) ) |
39 |
38 30
|
oveq12d |
|- ( ( ( 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 ) ) ) |
40 |
25 39
|
eqtr3d |
|- ( ( ( 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 |
|- ( p = ( ( 2 pCnt N ) + 1 ) -> ( 2 ^ p ) = ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) ) |
42 |
41
|
oveq1d |
|- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( 2 ^ p ) - 1 ) = ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) ) |
43 |
42
|
eleq1d |
|- ( p = ( ( 2 pCnt N ) + 1 ) -> ( ( ( 2 ^ p ) - 1 ) e. Prime <-> ( ( 2 ^ ( ( 2 pCnt N ) + 1 ) ) - 1 ) e. Prime ) ) |
44 |
|
oveq1 |
|- ( p = ( ( 2 pCnt N ) + 1 ) -> ( p - 1 ) = ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) |
45 |
44
|
oveq2d |
|- ( p = ( ( 2 pCnt N ) + 1 ) -> ( 2 ^ ( p - 1 ) ) = ( 2 ^ ( ( ( 2 pCnt N ) + 1 ) - 1 ) ) ) |
46 |
45 42
|
oveq12d |
|- ( 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 ) ) ) |
47 |
46
|
eqeq2d |
|- ( 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 ) ) ) ) |
48 |
43 47
|
anbi12d |
|- ( 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 ) ) ) ) ) |
49 |
48
|
rspcev |
|- ( ( ( ( 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 ) ) ) ) |
50 |
8 32 40 49
|
syl12anc |
|- ( ( ( 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 ) ) ) ) |
51 |
50
|
ex |
|- ( ( 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 |
|- ( ( 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 |
|- 2 e. CC |
54 |
|
mersenne |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> p e. Prime ) |
55 |
|
prmnn |
|- ( p e. Prime -> p e. NN ) |
56 |
54 55
|
syl |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> p e. NN ) |
57 |
|
expm1t |
|- ( ( 2 e. CC /\ p e. NN ) -> ( 2 ^ p ) = ( ( 2 ^ ( p - 1 ) ) x. 2 ) ) |
58 |
53 56 57
|
sylancr |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ p ) = ( ( 2 ^ ( p - 1 ) ) x. 2 ) ) |
59 |
|
nnm1nn0 |
|- ( p e. NN -> ( p - 1 ) e. NN0 ) |
60 |
56 59
|
syl |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( p - 1 ) e. NN0 ) |
61 |
|
expcl |
|- ( ( 2 e. CC /\ ( p - 1 ) e. NN0 ) -> ( 2 ^ ( p - 1 ) ) e. CC ) |
62 |
53 60 61
|
sylancr |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ ( p - 1 ) ) e. CC ) |
63 |
|
mulcom |
|- ( ( ( 2 ^ ( p - 1 ) ) e. CC /\ 2 e. CC ) -> ( ( 2 ^ ( p - 1 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) ) |
64 |
62 53 63
|
sylancl |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ ( p - 1 ) ) x. 2 ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) ) |
65 |
58 64
|
eqtrd |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( 2 ^ p ) = ( 2 x. ( 2 ^ ( p - 1 ) ) ) ) |
66 |
65
|
oveq1d |
|- ( ( 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 |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> 2 e. CC ) |
68 |
|
prmnn |
|- ( ( ( 2 ^ p ) - 1 ) e. Prime -> ( ( 2 ^ p ) - 1 ) e. NN ) |
69 |
68
|
adantl |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ p ) - 1 ) e. NN ) |
70 |
69
|
nncnd |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( ( 2 ^ p ) - 1 ) e. CC ) |
71 |
67 62 70
|
mulassd |
|- ( ( 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 ) ) ) ) |
72 |
52 66 71
|
3eqtrd |
|- ( ( 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 |
|- ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) -> ( 1 sigma N ) = ( 1 sigma ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) ) |
74 |
|
oveq2 |
|- ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) -> ( 2 x. N ) = ( 2 x. ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) ) |
75 |
73 74
|
eqeq12d |
|- ( 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 ) ) ) ) ) |
76 |
72 75
|
syl5ibrcom |
|- ( ( p e. ZZ /\ ( ( 2 ^ p ) - 1 ) e. Prime ) -> ( N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) -> ( 1 sigma N ) = ( 2 x. N ) ) ) |
77 |
76
|
impr |
|- ( ( p e. ZZ /\ ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) ) -> ( 1 sigma N ) = ( 2 x. N ) ) |
78 |
77
|
rexlimiva |
|- ( E. p e. ZZ ( ( ( 2 ^ p ) - 1 ) e. Prime /\ N = ( ( 2 ^ ( p - 1 ) ) x. ( ( 2 ^ p ) - 1 ) ) ) -> ( 1 sigma N ) = ( 2 x. N ) ) |
79 |
51 78
|
impbid1 |
|- ( ( 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 ) ) ) ) ) |