Step |
Hyp |
Ref |
Expression |
1 |
|
perfectlem.1 |
|- ( ph -> A e. NN ) |
2 |
|
perfectlem.2 |
|- ( ph -> B e. NN ) |
3 |
|
perfectlem.3 |
|- ( ph -> -. 2 || B ) |
4 |
|
perfectlem.4 |
|- ( ph -> ( 1 sigma ( ( 2 ^ A ) x. B ) ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) ) |
5 |
|
2nn |
|- 2 e. NN |
6 |
1
|
nnnn0d |
|- ( ph -> A e. NN0 ) |
7 |
|
peano2nn0 |
|- ( A e. NN0 -> ( A + 1 ) e. NN0 ) |
8 |
6 7
|
syl |
|- ( ph -> ( A + 1 ) e. NN0 ) |
9 |
|
nnexpcl |
|- ( ( 2 e. NN /\ ( A + 1 ) e. NN0 ) -> ( 2 ^ ( A + 1 ) ) e. NN ) |
10 |
5 8 9
|
sylancr |
|- ( ph -> ( 2 ^ ( A + 1 ) ) e. NN ) |
11 |
|
2re |
|- 2 e. RR |
12 |
1
|
peano2nnd |
|- ( ph -> ( A + 1 ) e. NN ) |
13 |
|
1lt2 |
|- 1 < 2 |
14 |
13
|
a1i |
|- ( ph -> 1 < 2 ) |
15 |
|
expgt1 |
|- ( ( 2 e. RR /\ ( A + 1 ) e. NN /\ 1 < 2 ) -> 1 < ( 2 ^ ( A + 1 ) ) ) |
16 |
11 12 14 15
|
mp3an2i |
|- ( ph -> 1 < ( 2 ^ ( A + 1 ) ) ) |
17 |
|
1nn |
|- 1 e. NN |
18 |
|
nnsub |
|- ( ( 1 e. NN /\ ( 2 ^ ( A + 1 ) ) e. NN ) -> ( 1 < ( 2 ^ ( A + 1 ) ) <-> ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN ) ) |
19 |
17 10 18
|
sylancr |
|- ( ph -> ( 1 < ( 2 ^ ( A + 1 ) ) <-> ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN ) ) |
20 |
16 19
|
mpbid |
|- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN ) |
21 |
10
|
nnzd |
|- ( ph -> ( 2 ^ ( A + 1 ) ) e. ZZ ) |
22 |
|
peano2zm |
|- ( ( 2 ^ ( A + 1 ) ) e. ZZ -> ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ ) |
23 |
21 22
|
syl |
|- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ ) |
24 |
|
1nn0 |
|- 1 e. NN0 |
25 |
|
sgmnncl |
|- ( ( 1 e. NN0 /\ B e. NN ) -> ( 1 sigma B ) e. NN ) |
26 |
24 2 25
|
sylancr |
|- ( ph -> ( 1 sigma B ) e. NN ) |
27 |
26
|
nnzd |
|- ( ph -> ( 1 sigma B ) e. ZZ ) |
28 |
|
dvdsmul1 |
|- ( ( ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ /\ ( 1 sigma B ) e. ZZ ) -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) ) |
29 |
23 27 28
|
syl2anc |
|- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) ) |
30 |
|
2cn |
|- 2 e. CC |
31 |
|
expp1 |
|- ( ( 2 e. CC /\ A e. NN0 ) -> ( 2 ^ ( A + 1 ) ) = ( ( 2 ^ A ) x. 2 ) ) |
32 |
30 6 31
|
sylancr |
|- ( ph -> ( 2 ^ ( A + 1 ) ) = ( ( 2 ^ A ) x. 2 ) ) |
33 |
|
nnexpcl |
|- ( ( 2 e. NN /\ A e. NN0 ) -> ( 2 ^ A ) e. NN ) |
34 |
5 6 33
|
sylancr |
|- ( ph -> ( 2 ^ A ) e. NN ) |
35 |
34
|
nncnd |
|- ( ph -> ( 2 ^ A ) e. CC ) |
36 |
|
mulcom |
|- ( ( ( 2 ^ A ) e. CC /\ 2 e. CC ) -> ( ( 2 ^ A ) x. 2 ) = ( 2 x. ( 2 ^ A ) ) ) |
37 |
35 30 36
|
sylancl |
|- ( ph -> ( ( 2 ^ A ) x. 2 ) = ( 2 x. ( 2 ^ A ) ) ) |
38 |
32 37
|
eqtrd |
|- ( ph -> ( 2 ^ ( A + 1 ) ) = ( 2 x. ( 2 ^ A ) ) ) |
39 |
38
|
oveq1d |
|- ( ph -> ( ( 2 ^ ( A + 1 ) ) x. B ) = ( ( 2 x. ( 2 ^ A ) ) x. B ) ) |
40 |
30
|
a1i |
|- ( ph -> 2 e. CC ) |
41 |
2
|
nncnd |
|- ( ph -> B e. CC ) |
42 |
40 35 41
|
mulassd |
|- ( ph -> ( ( 2 x. ( 2 ^ A ) ) x. B ) = ( 2 x. ( ( 2 ^ A ) x. B ) ) ) |
43 |
|
ax-1cn |
|- 1 e. CC |
44 |
43
|
a1i |
|- ( ph -> 1 e. CC ) |
45 |
|
2prm |
|- 2 e. Prime |
46 |
2
|
nnzd |
|- ( ph -> B e. ZZ ) |
47 |
|
coprm |
|- ( ( 2 e. Prime /\ B e. ZZ ) -> ( -. 2 || B <-> ( 2 gcd B ) = 1 ) ) |
48 |
45 46 47
|
sylancr |
|- ( ph -> ( -. 2 || B <-> ( 2 gcd B ) = 1 ) ) |
49 |
3 48
|
mpbid |
|- ( ph -> ( 2 gcd B ) = 1 ) |
50 |
|
2z |
|- 2 e. ZZ |
51 |
|
rpexp1i |
|- ( ( 2 e. ZZ /\ B e. ZZ /\ A e. NN0 ) -> ( ( 2 gcd B ) = 1 -> ( ( 2 ^ A ) gcd B ) = 1 ) ) |
52 |
50 46 6 51
|
mp3an2i |
|- ( ph -> ( ( 2 gcd B ) = 1 -> ( ( 2 ^ A ) gcd B ) = 1 ) ) |
53 |
49 52
|
mpd |
|- ( ph -> ( ( 2 ^ A ) gcd B ) = 1 ) |
54 |
|
sgmmul |
|- ( ( 1 e. CC /\ ( ( 2 ^ A ) e. NN /\ B e. NN /\ ( ( 2 ^ A ) gcd B ) = 1 ) ) -> ( 1 sigma ( ( 2 ^ A ) x. B ) ) = ( ( 1 sigma ( 2 ^ A ) ) x. ( 1 sigma B ) ) ) |
55 |
44 34 2 53 54
|
syl13anc |
|- ( ph -> ( 1 sigma ( ( 2 ^ A ) x. B ) ) = ( ( 1 sigma ( 2 ^ A ) ) x. ( 1 sigma B ) ) ) |
56 |
1
|
nncnd |
|- ( ph -> A e. CC ) |
57 |
|
pncan |
|- ( ( A e. CC /\ 1 e. CC ) -> ( ( A + 1 ) - 1 ) = A ) |
58 |
56 43 57
|
sylancl |
|- ( ph -> ( ( A + 1 ) - 1 ) = A ) |
59 |
58
|
oveq2d |
|- ( ph -> ( 2 ^ ( ( A + 1 ) - 1 ) ) = ( 2 ^ A ) ) |
60 |
59
|
oveq2d |
|- ( ph -> ( 1 sigma ( 2 ^ ( ( A + 1 ) - 1 ) ) ) = ( 1 sigma ( 2 ^ A ) ) ) |
61 |
|
1sgm2ppw |
|- ( ( A + 1 ) e. NN -> ( 1 sigma ( 2 ^ ( ( A + 1 ) - 1 ) ) ) = ( ( 2 ^ ( A + 1 ) ) - 1 ) ) |
62 |
12 61
|
syl |
|- ( ph -> ( 1 sigma ( 2 ^ ( ( A + 1 ) - 1 ) ) ) = ( ( 2 ^ ( A + 1 ) ) - 1 ) ) |
63 |
60 62
|
eqtr3d |
|- ( ph -> ( 1 sigma ( 2 ^ A ) ) = ( ( 2 ^ ( A + 1 ) ) - 1 ) ) |
64 |
63
|
oveq1d |
|- ( ph -> ( ( 1 sigma ( 2 ^ A ) ) x. ( 1 sigma B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) ) |
65 |
55 4 64
|
3eqtr3d |
|- ( ph -> ( 2 x. ( ( 2 ^ A ) x. B ) ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) ) |
66 |
39 42 65
|
3eqtrd |
|- ( ph -> ( ( 2 ^ ( A + 1 ) ) x. B ) = ( ( ( 2 ^ ( A + 1 ) ) - 1 ) x. ( 1 sigma B ) ) ) |
67 |
29 66
|
breqtrrd |
|- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( 2 ^ ( A + 1 ) ) x. B ) ) |
68 |
23 21
|
gcdcomd |
|- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) ) |
69 |
|
iddvdsexp |
|- ( ( 2 e. ZZ /\ ( A + 1 ) e. NN ) -> 2 || ( 2 ^ ( A + 1 ) ) ) |
70 |
50 12 69
|
sylancr |
|- ( ph -> 2 || ( 2 ^ ( A + 1 ) ) ) |
71 |
|
n2dvds1 |
|- -. 2 || 1 |
72 |
50
|
a1i |
|- ( ph -> 2 e. ZZ ) |
73 |
|
1zzd |
|- ( ph -> 1 e. ZZ ) |
74 |
72 21 73
|
3jca |
|- ( ph -> ( 2 e. ZZ /\ ( 2 ^ ( A + 1 ) ) e. ZZ /\ 1 e. ZZ ) ) |
75 |
|
dvdssub2 |
|- ( ( ( 2 e. ZZ /\ ( 2 ^ ( A + 1 ) ) e. ZZ /\ 1 e. ZZ ) /\ 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) ) -> ( 2 || ( 2 ^ ( A + 1 ) ) <-> 2 || 1 ) ) |
76 |
74 75
|
sylan |
|- ( ( ph /\ 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) ) -> ( 2 || ( 2 ^ ( A + 1 ) ) <-> 2 || 1 ) ) |
77 |
71 76
|
mtbiri |
|- ( ( ph /\ 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) ) -> -. 2 || ( 2 ^ ( A + 1 ) ) ) |
78 |
77
|
ex |
|- ( ph -> ( 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) -> -. 2 || ( 2 ^ ( A + 1 ) ) ) ) |
79 |
70 78
|
mt2d |
|- ( ph -> -. 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) ) |
80 |
|
coprm |
|- ( ( 2 e. Prime /\ ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ ) -> ( -. 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) <-> ( 2 gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 ) ) |
81 |
45 23 80
|
sylancr |
|- ( ph -> ( -. 2 || ( ( 2 ^ ( A + 1 ) ) - 1 ) <-> ( 2 gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 ) ) |
82 |
79 81
|
mpbid |
|- ( ph -> ( 2 gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 ) |
83 |
|
rpexp1i |
|- ( ( 2 e. ZZ /\ ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ /\ ( A + 1 ) e. NN0 ) -> ( ( 2 gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 -> ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 ) ) |
84 |
50 23 8 83
|
mp3an2i |
|- ( ph -> ( ( 2 gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 -> ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 ) ) |
85 |
82 84
|
mpd |
|- ( ph -> ( ( 2 ^ ( A + 1 ) ) gcd ( ( 2 ^ ( A + 1 ) ) - 1 ) ) = 1 ) |
86 |
68 85
|
eqtrd |
|- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = 1 ) |
87 |
|
coprmdvds |
|- ( ( ( ( 2 ^ ( A + 1 ) ) - 1 ) e. ZZ /\ ( 2 ^ ( A + 1 ) ) e. ZZ /\ B e. ZZ ) -> ( ( ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( 2 ^ ( A + 1 ) ) x. B ) /\ ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = 1 ) -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || B ) ) |
88 |
23 21 46 87
|
syl3anc |
|- ( ph -> ( ( ( ( 2 ^ ( A + 1 ) ) - 1 ) || ( ( 2 ^ ( A + 1 ) ) x. B ) /\ ( ( ( 2 ^ ( A + 1 ) ) - 1 ) gcd ( 2 ^ ( A + 1 ) ) ) = 1 ) -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || B ) ) |
89 |
67 86 88
|
mp2and |
|- ( ph -> ( ( 2 ^ ( A + 1 ) ) - 1 ) || B ) |
90 |
|
nndivdvds |
|- ( ( B e. NN /\ ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN ) -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) || B <-> ( B / ( ( 2 ^ ( A + 1 ) ) - 1 ) ) e. NN ) ) |
91 |
2 20 90
|
syl2anc |
|- ( ph -> ( ( ( 2 ^ ( A + 1 ) ) - 1 ) || B <-> ( B / ( ( 2 ^ ( A + 1 ) ) - 1 ) ) e. NN ) ) |
92 |
89 91
|
mpbid |
|- ( ph -> ( B / ( ( 2 ^ ( A + 1 ) ) - 1 ) ) e. NN ) |
93 |
10 20 92
|
3jca |
|- ( ph -> ( ( 2 ^ ( A + 1 ) ) e. NN /\ ( ( 2 ^ ( A + 1 ) ) - 1 ) e. NN /\ ( B / ( ( 2 ^ ( A + 1 ) ) - 1 ) ) e. NN ) ) |