Step |
Hyp |
Ref |
Expression |
1 |
|
evennn2n |
|- ( N e. NN -> ( 2 || N <-> E. k e. NN ( 2 x. k ) = N ) ) |
2 |
1
|
3ad2ant3 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( 2 || N <-> E. k e. NN ( 2 x. k ) = N ) ) |
3 |
|
oveq2 |
|- ( N = ( 2 x. k ) -> ( 2 ^ N ) = ( 2 ^ ( 2 x. k ) ) ) |
4 |
3
|
eqcoms |
|- ( ( 2 x. k ) = N -> ( 2 ^ N ) = ( 2 ^ ( 2 x. k ) ) ) |
5 |
|
2cnd |
|- ( k e. NN -> 2 e. CC ) |
6 |
|
nncn |
|- ( k e. NN -> k e. CC ) |
7 |
5 6
|
mulcomd |
|- ( k e. NN -> ( 2 x. k ) = ( k x. 2 ) ) |
8 |
7
|
oveq2d |
|- ( k e. NN -> ( 2 ^ ( 2 x. k ) ) = ( 2 ^ ( k x. 2 ) ) ) |
9 |
|
2nn0 |
|- 2 e. NN0 |
10 |
9
|
a1i |
|- ( k e. NN -> 2 e. NN0 ) |
11 |
|
nnnn0 |
|- ( k e. NN -> k e. NN0 ) |
12 |
5 10 11
|
expmuld |
|- ( k e. NN -> ( 2 ^ ( k x. 2 ) ) = ( ( 2 ^ k ) ^ 2 ) ) |
13 |
8 12
|
eqtrd |
|- ( k e. NN -> ( 2 ^ ( 2 x. k ) ) = ( ( 2 ^ k ) ^ 2 ) ) |
14 |
13
|
adantl |
|- ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ k e. NN ) -> ( 2 ^ ( 2 x. k ) ) = ( ( 2 ^ k ) ^ 2 ) ) |
15 |
4 14
|
sylan9eqr |
|- ( ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ k e. NN ) /\ ( 2 x. k ) = N ) -> ( 2 ^ N ) = ( ( 2 ^ k ) ^ 2 ) ) |
16 |
15
|
oveq1d |
|- ( ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ k e. NN ) /\ ( 2 x. k ) = N ) -> ( ( 2 ^ N ) - 1 ) = ( ( ( 2 ^ k ) ^ 2 ) - 1 ) ) |
17 |
16
|
eqeq1d |
|- ( ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ k e. NN ) /\ ( 2 x. k ) = N ) -> ( ( ( 2 ^ N ) - 1 ) = ( P ^ M ) <-> ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) ) ) |
18 |
|
elnn1uz2 |
|- ( k e. NN <-> ( k = 1 \/ k e. ( ZZ>= ` 2 ) ) ) |
19 |
|
oveq2 |
|- ( k = 1 -> ( 2 ^ k ) = ( 2 ^ 1 ) ) |
20 |
|
2cn |
|- 2 e. CC |
21 |
|
exp1 |
|- ( 2 e. CC -> ( 2 ^ 1 ) = 2 ) |
22 |
20 21
|
ax-mp |
|- ( 2 ^ 1 ) = 2 |
23 |
19 22
|
eqtrdi |
|- ( k = 1 -> ( 2 ^ k ) = 2 ) |
24 |
23
|
oveq1d |
|- ( k = 1 -> ( ( 2 ^ k ) ^ 2 ) = ( 2 ^ 2 ) ) |
25 |
24
|
oveq1d |
|- ( k = 1 -> ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( ( 2 ^ 2 ) - 1 ) ) |
26 |
|
sq2 |
|- ( 2 ^ 2 ) = 4 |
27 |
26
|
oveq1i |
|- ( ( 2 ^ 2 ) - 1 ) = ( 4 - 1 ) |
28 |
|
4m1e3 |
|- ( 4 - 1 ) = 3 |
29 |
27 28
|
eqtri |
|- ( ( 2 ^ 2 ) - 1 ) = 3 |
30 |
25 29
|
eqtrdi |
|- ( k = 1 -> ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = 3 ) |
31 |
30
|
eqeq1d |
|- ( k = 1 -> ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) <-> 3 = ( P ^ M ) ) ) |
32 |
31
|
adantr |
|- ( ( k = 1 /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) <-> 3 = ( P ^ M ) ) ) |
33 |
|
eqcom |
|- ( 3 = ( P ^ M ) <-> ( P ^ M ) = 3 ) |
34 |
|
eldifi |
|- ( P e. ( Prime \ { 2 } ) -> P e. Prime ) |
35 |
|
prmnn |
|- ( P e. Prime -> P e. NN ) |
36 |
|
nnre |
|- ( P e. NN -> P e. RR ) |
37 |
34 35 36
|
3syl |
|- ( P e. ( Prime \ { 2 } ) -> P e. RR ) |
38 |
37
|
3ad2ant1 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> P e. RR ) |
39 |
|
nnnn0 |
|- ( M e. NN -> M e. NN0 ) |
40 |
39
|
3ad2ant2 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> M e. NN0 ) |
41 |
38 40
|
reexpcld |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( P ^ M ) e. RR ) |
42 |
41
|
adantr |
|- ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ ( P ^ M ) = 3 ) -> ( P ^ M ) e. RR ) |
43 |
|
simpr |
|- ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ ( P ^ M ) = 3 ) -> ( P ^ M ) = 3 ) |
44 |
42 43
|
eqled |
|- ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ ( P ^ M ) = 3 ) -> ( P ^ M ) <_ 3 ) |
45 |
44
|
ex |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( P ^ M ) = 3 -> ( P ^ M ) <_ 3 ) ) |
46 |
33 45
|
syl5bi |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( 3 = ( P ^ M ) -> ( P ^ M ) <_ 3 ) ) |
47 |
35
|
nnred |
|- ( P e. Prime -> P e. RR ) |
48 |
|
prmgt1 |
|- ( P e. Prime -> 1 < P ) |
49 |
47 48
|
jca |
|- ( P e. Prime -> ( P e. RR /\ 1 < P ) ) |
50 |
34 49
|
syl |
|- ( P e. ( Prime \ { 2 } ) -> ( P e. RR /\ 1 < P ) ) |
51 |
50
|
3ad2ant1 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( P e. RR /\ 1 < P ) ) |
52 |
|
nnz |
|- ( M e. NN -> M e. ZZ ) |
53 |
52
|
3ad2ant2 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> M e. ZZ ) |
54 |
|
3rp |
|- 3 e. RR+ |
55 |
54
|
a1i |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> 3 e. RR+ ) |
56 |
|
efexple |
|- ( ( ( P e. RR /\ 1 < P ) /\ M e. ZZ /\ 3 e. RR+ ) -> ( ( P ^ M ) <_ 3 <-> M <_ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) ) ) |
57 |
51 53 55 56
|
syl3anc |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( P ^ M ) <_ 3 <-> M <_ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) ) ) |
58 |
|
oddprmge3 |
|- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 3 ) ) |
59 |
|
eluzle |
|- ( P e. ( ZZ>= ` 3 ) -> 3 <_ P ) |
60 |
58 59
|
syl |
|- ( P e. ( Prime \ { 2 } ) -> 3 <_ P ) |
61 |
54
|
a1i |
|- ( P e. ( Prime \ { 2 } ) -> 3 e. RR+ ) |
62 |
|
nnrp |
|- ( P e. NN -> P e. RR+ ) |
63 |
34 35 62
|
3syl |
|- ( P e. ( Prime \ { 2 } ) -> P e. RR+ ) |
64 |
61 63
|
logled |
|- ( P e. ( Prime \ { 2 } ) -> ( 3 <_ P <-> ( log ` 3 ) <_ ( log ` P ) ) ) |
65 |
60 64
|
mpbid |
|- ( P e. ( Prime \ { 2 } ) -> ( log ` 3 ) <_ ( log ` P ) ) |
66 |
65
|
3ad2ant1 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( log ` 3 ) <_ ( log ` P ) ) |
67 |
|
relogcl |
|- ( 3 e. RR+ -> ( log ` 3 ) e. RR ) |
68 |
54 67
|
ax-mp |
|- ( log ` 3 ) e. RR |
69 |
|
rplogcl |
|- ( ( P e. RR /\ 1 < P ) -> ( log ` P ) e. RR+ ) |
70 |
34 49 69
|
3syl |
|- ( P e. ( Prime \ { 2 } ) -> ( log ` P ) e. RR+ ) |
71 |
70
|
3ad2ant1 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( log ` P ) e. RR+ ) |
72 |
|
divle1le |
|- ( ( ( log ` 3 ) e. RR /\ ( log ` P ) e. RR+ ) -> ( ( ( log ` 3 ) / ( log ` P ) ) <_ 1 <-> ( log ` 3 ) <_ ( log ` P ) ) ) |
73 |
68 71 72
|
sylancr |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( ( log ` 3 ) / ( log ` P ) ) <_ 1 <-> ( log ` 3 ) <_ ( log ` P ) ) ) |
74 |
66 73
|
mpbird |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( log ` 3 ) / ( log ` P ) ) <_ 1 ) |
75 |
|
fldivle |
|- ( ( ( log ` 3 ) e. RR /\ ( log ` P ) e. RR+ ) -> ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) <_ ( ( log ` 3 ) / ( log ` P ) ) ) |
76 |
68 71 75
|
sylancr |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) <_ ( ( log ` 3 ) / ( log ` P ) ) ) |
77 |
|
nnre |
|- ( M e. NN -> M e. RR ) |
78 |
77
|
3ad2ant2 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> M e. RR ) |
79 |
68
|
a1i |
|- ( P e. ( Prime \ { 2 } ) -> ( log ` 3 ) e. RR ) |
80 |
62
|
relogcld |
|- ( P e. NN -> ( log ` P ) e. RR ) |
81 |
34 35 80
|
3syl |
|- ( P e. ( Prime \ { 2 } ) -> ( log ` P ) e. RR ) |
82 |
35
|
nnrpd |
|- ( P e. Prime -> P e. RR+ ) |
83 |
|
1red |
|- ( P e. Prime -> 1 e. RR ) |
84 |
83 48
|
gtned |
|- ( P e. Prime -> P =/= 1 ) |
85 |
82 84
|
jca |
|- ( P e. Prime -> ( P e. RR+ /\ P =/= 1 ) ) |
86 |
|
logne0 |
|- ( ( P e. RR+ /\ P =/= 1 ) -> ( log ` P ) =/= 0 ) |
87 |
34 85 86
|
3syl |
|- ( P e. ( Prime \ { 2 } ) -> ( log ` P ) =/= 0 ) |
88 |
79 81 87
|
redivcld |
|- ( P e. ( Prime \ { 2 } ) -> ( ( log ` 3 ) / ( log ` P ) ) e. RR ) |
89 |
88
|
flcld |
|- ( P e. ( Prime \ { 2 } ) -> ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) e. ZZ ) |
90 |
89
|
zred |
|- ( P e. ( Prime \ { 2 } ) -> ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) e. RR ) |
91 |
90
|
3ad2ant1 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) e. RR ) |
92 |
88
|
3ad2ant1 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( log ` 3 ) / ( log ` P ) ) e. RR ) |
93 |
|
letr |
|- ( ( M e. RR /\ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) e. RR /\ ( ( log ` 3 ) / ( log ` P ) ) e. RR ) -> ( ( M <_ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) /\ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) <_ ( ( log ` 3 ) / ( log ` P ) ) ) -> M <_ ( ( log ` 3 ) / ( log ` P ) ) ) ) |
94 |
78 91 92 93
|
syl3anc |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( M <_ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) /\ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) <_ ( ( log ` 3 ) / ( log ` P ) ) ) -> M <_ ( ( log ` 3 ) / ( log ` P ) ) ) ) |
95 |
|
1red |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> 1 e. RR ) |
96 |
|
letr |
|- ( ( M e. RR /\ ( ( log ` 3 ) / ( log ` P ) ) e. RR /\ 1 e. RR ) -> ( ( M <_ ( ( log ` 3 ) / ( log ` P ) ) /\ ( ( log ` 3 ) / ( log ` P ) ) <_ 1 ) -> M <_ 1 ) ) |
97 |
78 92 95 96
|
syl3anc |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( M <_ ( ( log ` 3 ) / ( log ` P ) ) /\ ( ( log ` 3 ) / ( log ` P ) ) <_ 1 ) -> M <_ 1 ) ) |
98 |
|
nnge1 |
|- ( M e. NN -> 1 <_ M ) |
99 |
|
eqcom |
|- ( M = 1 <-> 1 = M ) |
100 |
|
1red |
|- ( M e. NN -> 1 e. RR ) |
101 |
100 77
|
letri3d |
|- ( M e. NN -> ( 1 = M <-> ( 1 <_ M /\ M <_ 1 ) ) ) |
102 |
99 101
|
bitr2id |
|- ( M e. NN -> ( ( 1 <_ M /\ M <_ 1 ) <-> M = 1 ) ) |
103 |
102
|
biimpd |
|- ( M e. NN -> ( ( 1 <_ M /\ M <_ 1 ) -> M = 1 ) ) |
104 |
98 103
|
mpand |
|- ( M e. NN -> ( M <_ 1 -> M = 1 ) ) |
105 |
104
|
3ad2ant2 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( M <_ 1 -> M = 1 ) ) |
106 |
97 105
|
syld |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( M <_ ( ( log ` 3 ) / ( log ` P ) ) /\ ( ( log ` 3 ) / ( log ` P ) ) <_ 1 ) -> M = 1 ) ) |
107 |
106
|
expd |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( M <_ ( ( log ` 3 ) / ( log ` P ) ) -> ( ( ( log ` 3 ) / ( log ` P ) ) <_ 1 -> M = 1 ) ) ) |
108 |
94 107
|
syld |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( M <_ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) /\ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) <_ ( ( log ` 3 ) / ( log ` P ) ) ) -> ( ( ( log ` 3 ) / ( log ` P ) ) <_ 1 -> M = 1 ) ) ) |
109 |
76 108
|
mpan2d |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( M <_ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) -> ( ( ( log ` 3 ) / ( log ` P ) ) <_ 1 -> M = 1 ) ) ) |
110 |
74 109
|
mpid |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( M <_ ( |_ ` ( ( log ` 3 ) / ( log ` P ) ) ) -> M = 1 ) ) |
111 |
57 110
|
sylbid |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( P ^ M ) <_ 3 -> M = 1 ) ) |
112 |
46 111
|
syld |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( 3 = ( P ^ M ) -> M = 1 ) ) |
113 |
112
|
adantl |
|- ( ( k = 1 /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( 3 = ( P ^ M ) -> M = 1 ) ) |
114 |
32 113
|
sylbid |
|- ( ( k = 1 /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) -> M = 1 ) ) |
115 |
114
|
ex |
|- ( k = 1 -> ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) -> M = 1 ) ) ) |
116 |
|
sq1 |
|- ( 1 ^ 2 ) = 1 |
117 |
116
|
eqcomi |
|- 1 = ( 1 ^ 2 ) |
118 |
117
|
oveq2i |
|- ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( ( ( 2 ^ k ) ^ 2 ) - ( 1 ^ 2 ) ) |
119 |
118
|
eqeq1i |
|- ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) <-> ( ( ( 2 ^ k ) ^ 2 ) - ( 1 ^ 2 ) ) = ( P ^ M ) ) |
120 |
|
eqcom |
|- ( ( ( ( 2 ^ k ) ^ 2 ) - ( 1 ^ 2 ) ) = ( P ^ M ) <-> ( P ^ M ) = ( ( ( 2 ^ k ) ^ 2 ) - ( 1 ^ 2 ) ) ) |
121 |
9
|
a1i |
|- ( k e. ( ZZ>= ` 2 ) -> 2 e. NN0 ) |
122 |
|
eluzge2nn0 |
|- ( k e. ( ZZ>= ` 2 ) -> k e. NN0 ) |
123 |
121 122
|
nn0expcld |
|- ( k e. ( ZZ>= ` 2 ) -> ( 2 ^ k ) e. NN0 ) |
124 |
123
|
adantr |
|- ( ( k e. ( ZZ>= ` 2 ) /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( 2 ^ k ) e. NN0 ) |
125 |
|
1nn0 |
|- 1 e. NN0 |
126 |
125
|
a1i |
|- ( ( k e. ( ZZ>= ` 2 ) /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> 1 e. NN0 ) |
127 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
128 |
22
|
eqcomi |
|- 2 = ( 2 ^ 1 ) |
129 |
127 128
|
eqtri |
|- ( 1 + 1 ) = ( 2 ^ 1 ) |
130 |
|
eluz2gt1 |
|- ( k e. ( ZZ>= ` 2 ) -> 1 < k ) |
131 |
|
2re |
|- 2 e. RR |
132 |
131
|
a1i |
|- ( k e. ( ZZ>= ` 2 ) -> 2 e. RR ) |
133 |
|
1zzd |
|- ( k e. ( ZZ>= ` 2 ) -> 1 e. ZZ ) |
134 |
|
eluzelz |
|- ( k e. ( ZZ>= ` 2 ) -> k e. ZZ ) |
135 |
|
1lt2 |
|- 1 < 2 |
136 |
135
|
a1i |
|- ( k e. ( ZZ>= ` 2 ) -> 1 < 2 ) |
137 |
132 133 134 136
|
ltexp2d |
|- ( k e. ( ZZ>= ` 2 ) -> ( 1 < k <-> ( 2 ^ 1 ) < ( 2 ^ k ) ) ) |
138 |
130 137
|
mpbid |
|- ( k e. ( ZZ>= ` 2 ) -> ( 2 ^ 1 ) < ( 2 ^ k ) ) |
139 |
129 138
|
eqbrtrid |
|- ( k e. ( ZZ>= ` 2 ) -> ( 1 + 1 ) < ( 2 ^ k ) ) |
140 |
139
|
adantr |
|- ( ( k e. ( ZZ>= ` 2 ) /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( 1 + 1 ) < ( 2 ^ k ) ) |
141 |
34 39
|
anim12i |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN ) -> ( P e. Prime /\ M e. NN0 ) ) |
142 |
141
|
3adant3 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( P e. Prime /\ M e. NN0 ) ) |
143 |
142
|
adantl |
|- ( ( k e. ( ZZ>= ` 2 ) /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( P e. Prime /\ M e. NN0 ) ) |
144 |
|
difsqpwdvds |
|- ( ( ( ( 2 ^ k ) e. NN0 /\ 1 e. NN0 /\ ( 1 + 1 ) < ( 2 ^ k ) ) /\ ( P e. Prime /\ M e. NN0 ) ) -> ( ( P ^ M ) = ( ( ( 2 ^ k ) ^ 2 ) - ( 1 ^ 2 ) ) -> P || ( 2 x. 1 ) ) ) |
145 |
124 126 140 143 144
|
syl31anc |
|- ( ( k e. ( ZZ>= ` 2 ) /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( ( P ^ M ) = ( ( ( 2 ^ k ) ^ 2 ) - ( 1 ^ 2 ) ) -> P || ( 2 x. 1 ) ) ) |
146 |
|
2t1e2 |
|- ( 2 x. 1 ) = 2 |
147 |
146
|
breq2i |
|- ( P || ( 2 x. 1 ) <-> P || 2 ) |
148 |
|
prmuz2 |
|- ( P e. Prime -> P e. ( ZZ>= ` 2 ) ) |
149 |
34 148
|
syl |
|- ( P e. ( Prime \ { 2 } ) -> P e. ( ZZ>= ` 2 ) ) |
150 |
|
2prm |
|- 2 e. Prime |
151 |
|
dvdsprm |
|- ( ( P e. ( ZZ>= ` 2 ) /\ 2 e. Prime ) -> ( P || 2 <-> P = 2 ) ) |
152 |
149 150 151
|
sylancl |
|- ( P e. ( Prime \ { 2 } ) -> ( P || 2 <-> P = 2 ) ) |
153 |
147 152
|
syl5bb |
|- ( P e. ( Prime \ { 2 } ) -> ( P || ( 2 x. 1 ) <-> P = 2 ) ) |
154 |
|
eldifsn |
|- ( P e. ( Prime \ { 2 } ) <-> ( P e. Prime /\ P =/= 2 ) ) |
155 |
|
eqneqall |
|- ( P = 2 -> ( P =/= 2 -> M = 1 ) ) |
156 |
155
|
com12 |
|- ( P =/= 2 -> ( P = 2 -> M = 1 ) ) |
157 |
154 156
|
simplbiim |
|- ( P e. ( Prime \ { 2 } ) -> ( P = 2 -> M = 1 ) ) |
158 |
153 157
|
sylbid |
|- ( P e. ( Prime \ { 2 } ) -> ( P || ( 2 x. 1 ) -> M = 1 ) ) |
159 |
158
|
3ad2ant1 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( P || ( 2 x. 1 ) -> M = 1 ) ) |
160 |
159
|
adantl |
|- ( ( k e. ( ZZ>= ` 2 ) /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( P || ( 2 x. 1 ) -> M = 1 ) ) |
161 |
145 160
|
syld |
|- ( ( k e. ( ZZ>= ` 2 ) /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( ( P ^ M ) = ( ( ( 2 ^ k ) ^ 2 ) - ( 1 ^ 2 ) ) -> M = 1 ) ) |
162 |
120 161
|
syl5bi |
|- ( ( k e. ( ZZ>= ` 2 ) /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( ( ( ( 2 ^ k ) ^ 2 ) - ( 1 ^ 2 ) ) = ( P ^ M ) -> M = 1 ) ) |
163 |
119 162
|
syl5bi |
|- ( ( k e. ( ZZ>= ` 2 ) /\ ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) ) -> ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) -> M = 1 ) ) |
164 |
163
|
ex |
|- ( k e. ( ZZ>= ` 2 ) -> ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) -> M = 1 ) ) ) |
165 |
115 164
|
jaoi |
|- ( ( k = 1 \/ k e. ( ZZ>= ` 2 ) ) -> ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) -> M = 1 ) ) ) |
166 |
18 165
|
sylbi |
|- ( k e. NN -> ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) -> M = 1 ) ) ) |
167 |
166
|
impcom |
|- ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ k e. NN ) -> ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) -> M = 1 ) ) |
168 |
167
|
adantr |
|- ( ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ k e. NN ) /\ ( 2 x. k ) = N ) -> ( ( ( ( 2 ^ k ) ^ 2 ) - 1 ) = ( P ^ M ) -> M = 1 ) ) |
169 |
17 168
|
sylbid |
|- ( ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ k e. NN ) /\ ( 2 x. k ) = N ) -> ( ( ( 2 ^ N ) - 1 ) = ( P ^ M ) -> M = 1 ) ) |
170 |
169
|
rexlimdva2 |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( E. k e. NN ( 2 x. k ) = N -> ( ( ( 2 ^ N ) - 1 ) = ( P ^ M ) -> M = 1 ) ) ) |
171 |
2 170
|
sylbid |
|- ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) -> ( 2 || N -> ( ( ( 2 ^ N ) - 1 ) = ( P ^ M ) -> M = 1 ) ) ) |
172 |
171
|
3imp |
|- ( ( ( P e. ( Prime \ { 2 } ) /\ M e. NN /\ N e. NN ) /\ 2 || N /\ ( ( 2 ^ N ) - 1 ) = ( P ^ M ) ) -> M = 1 ) |