Step |
Hyp |
Ref |
Expression |
1 |
|
rpvmasum.z |
|- Z = ( Z/nZ ` N ) |
2 |
|
rpvmasum.l |
|- L = ( ZRHom ` Z ) |
3 |
|
rpvmasum.a |
|- ( ph -> N e. NN ) |
4 |
|
rpvmasum2.g |
|- G = ( DChr ` N ) |
5 |
|
rpvmasum2.d |
|- D = ( Base ` G ) |
6 |
|
rpvmasum2.1 |
|- .1. = ( 0g ` G ) |
7 |
|
dchrisum0f.f |
|- F = ( b e. NN |-> sum_ v e. { q e. NN | q || b } ( X ` ( L ` v ) ) ) |
8 |
|
dchrisum0f.x |
|- ( ph -> X e. D ) |
9 |
|
dchrisum0flb.r |
|- ( ph -> X : ( Base ` Z ) --> RR ) |
10 |
|
dchrisum0flb.1 |
|- ( ph -> A e. ( ZZ>= ` 2 ) ) |
11 |
|
dchrisum0flb.2 |
|- ( ph -> P e. Prime ) |
12 |
|
dchrisum0flb.3 |
|- ( ph -> P || A ) |
13 |
|
dchrisum0flb.4 |
|- ( ph -> A. y e. ( 1 ..^ A ) if ( ( sqrt ` y ) e. NN , 1 , 0 ) <_ ( F ` y ) ) |
14 |
|
breq1 |
|- ( 1 = if ( ( sqrt ` A ) e. NN , 1 , 0 ) -> ( 1 <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) <-> if ( ( sqrt ` A ) e. NN , 1 , 0 ) <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) ) |
15 |
|
breq1 |
|- ( 0 = if ( ( sqrt ` A ) e. NN , 1 , 0 ) -> ( 0 <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) <-> if ( ( sqrt ` A ) e. NN , 1 , 0 ) <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) ) |
16 |
|
1t1e1 |
|- ( 1 x. 1 ) = 1 |
17 |
11
|
adantr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> P e. Prime ) |
18 |
|
nnq |
|- ( ( sqrt ` A ) e. NN -> ( sqrt ` A ) e. QQ ) |
19 |
18
|
adantl |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` A ) e. QQ ) |
20 |
|
nnne0 |
|- ( ( sqrt ` A ) e. NN -> ( sqrt ` A ) =/= 0 ) |
21 |
20
|
adantl |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` A ) =/= 0 ) |
22 |
|
2z |
|- 2 e. ZZ |
23 |
22
|
a1i |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> 2 e. ZZ ) |
24 |
|
pcexp |
|- ( ( P e. Prime /\ ( ( sqrt ` A ) e. QQ /\ ( sqrt ` A ) =/= 0 ) /\ 2 e. ZZ ) -> ( P pCnt ( ( sqrt ` A ) ^ 2 ) ) = ( 2 x. ( P pCnt ( sqrt ` A ) ) ) ) |
25 |
17 19 21 23 24
|
syl121anc |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P pCnt ( ( sqrt ` A ) ^ 2 ) ) = ( 2 x. ( P pCnt ( sqrt ` A ) ) ) ) |
26 |
|
eluz2nn |
|- ( A e. ( ZZ>= ` 2 ) -> A e. NN ) |
27 |
10 26
|
syl |
|- ( ph -> A e. NN ) |
28 |
27
|
nncnd |
|- ( ph -> A e. CC ) |
29 |
28
|
adantr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> A e. CC ) |
30 |
29
|
sqsqrtd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( ( sqrt ` A ) ^ 2 ) = A ) |
31 |
30
|
oveq2d |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P pCnt ( ( sqrt ` A ) ^ 2 ) ) = ( P pCnt A ) ) |
32 |
|
2cnd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> 2 e. CC ) |
33 |
|
simpr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` A ) e. NN ) |
34 |
17 33
|
pccld |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P pCnt ( sqrt ` A ) ) e. NN0 ) |
35 |
34
|
nn0cnd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P pCnt ( sqrt ` A ) ) e. CC ) |
36 |
32 35
|
mulcomd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( 2 x. ( P pCnt ( sqrt ` A ) ) ) = ( ( P pCnt ( sqrt ` A ) ) x. 2 ) ) |
37 |
25 31 36
|
3eqtr3rd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( ( P pCnt ( sqrt ` A ) ) x. 2 ) = ( P pCnt A ) ) |
38 |
37
|
oveq2d |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P ^ ( ( P pCnt ( sqrt ` A ) ) x. 2 ) ) = ( P ^ ( P pCnt A ) ) ) |
39 |
|
prmnn |
|- ( P e. Prime -> P e. NN ) |
40 |
17 39
|
syl |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> P e. NN ) |
41 |
40
|
nncnd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> P e. CC ) |
42 |
|
2nn0 |
|- 2 e. NN0 |
43 |
42
|
a1i |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> 2 e. NN0 ) |
44 |
41 43 34
|
expmuld |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P ^ ( ( P pCnt ( sqrt ` A ) ) x. 2 ) ) = ( ( P ^ ( P pCnt ( sqrt ` A ) ) ) ^ 2 ) ) |
45 |
38 44
|
eqtr3d |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P ^ ( P pCnt A ) ) = ( ( P ^ ( P pCnt ( sqrt ` A ) ) ) ^ 2 ) ) |
46 |
45
|
fveq2d |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` ( P ^ ( P pCnt A ) ) ) = ( sqrt ` ( ( P ^ ( P pCnt ( sqrt ` A ) ) ) ^ 2 ) ) ) |
47 |
40 34
|
nnexpcld |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P ^ ( P pCnt ( sqrt ` A ) ) ) e. NN ) |
48 |
47
|
nnrpd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P ^ ( P pCnt ( sqrt ` A ) ) ) e. RR+ ) |
49 |
48
|
rprege0d |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( ( P ^ ( P pCnt ( sqrt ` A ) ) ) e. RR /\ 0 <_ ( P ^ ( P pCnt ( sqrt ` A ) ) ) ) ) |
50 |
|
sqrtsq |
|- ( ( ( P ^ ( P pCnt ( sqrt ` A ) ) ) e. RR /\ 0 <_ ( P ^ ( P pCnt ( sqrt ` A ) ) ) ) -> ( sqrt ` ( ( P ^ ( P pCnt ( sqrt ` A ) ) ) ^ 2 ) ) = ( P ^ ( P pCnt ( sqrt ` A ) ) ) ) |
51 |
49 50
|
syl |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` ( ( P ^ ( P pCnt ( sqrt ` A ) ) ) ^ 2 ) ) = ( P ^ ( P pCnt ( sqrt ` A ) ) ) ) |
52 |
46 51
|
eqtrd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` ( P ^ ( P pCnt A ) ) ) = ( P ^ ( P pCnt ( sqrt ` A ) ) ) ) |
53 |
52 47
|
eqeltrd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN ) |
54 |
53
|
iftrued |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) = 1 ) |
55 |
11 27
|
pccld |
|- ( ph -> ( P pCnt A ) e. NN0 ) |
56 |
1 2 3 4 5 6 7 8 9 11 55
|
dchrisum0flblem1 |
|- ( ph -> if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) <_ ( F ` ( P ^ ( P pCnt A ) ) ) ) |
57 |
56
|
adantr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) <_ ( F ` ( P ^ ( P pCnt A ) ) ) ) |
58 |
54 57
|
eqbrtrrd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> 1 <_ ( F ` ( P ^ ( P pCnt A ) ) ) ) |
59 |
|
pcdvds |
|- ( ( P e. Prime /\ A e. NN ) -> ( P ^ ( P pCnt A ) ) || A ) |
60 |
11 27 59
|
syl2anc |
|- ( ph -> ( P ^ ( P pCnt A ) ) || A ) |
61 |
11 39
|
syl |
|- ( ph -> P e. NN ) |
62 |
61 55
|
nnexpcld |
|- ( ph -> ( P ^ ( P pCnt A ) ) e. NN ) |
63 |
|
nndivdvds |
|- ( ( A e. NN /\ ( P ^ ( P pCnt A ) ) e. NN ) -> ( ( P ^ ( P pCnt A ) ) || A <-> ( A / ( P ^ ( P pCnt A ) ) ) e. NN ) ) |
64 |
27 62 63
|
syl2anc |
|- ( ph -> ( ( P ^ ( P pCnt A ) ) || A <-> ( A / ( P ^ ( P pCnt A ) ) ) e. NN ) ) |
65 |
60 64
|
mpbid |
|- ( ph -> ( A / ( P ^ ( P pCnt A ) ) ) e. NN ) |
66 |
65
|
nnzd |
|- ( ph -> ( A / ( P ^ ( P pCnt A ) ) ) e. ZZ ) |
67 |
66
|
adantr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( A / ( P ^ ( P pCnt A ) ) ) e. ZZ ) |
68 |
27
|
adantr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> A e. NN ) |
69 |
68
|
nnrpd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> A e. RR+ ) |
70 |
69
|
rprege0d |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( A e. RR /\ 0 <_ A ) ) |
71 |
62
|
adantr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P ^ ( P pCnt A ) ) e. NN ) |
72 |
71
|
nnrpd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( P ^ ( P pCnt A ) ) e. RR+ ) |
73 |
|
sqrtdiv |
|- ( ( ( A e. RR /\ 0 <_ A ) /\ ( P ^ ( P pCnt A ) ) e. RR+ ) -> ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) = ( ( sqrt ` A ) / ( sqrt ` ( P ^ ( P pCnt A ) ) ) ) ) |
74 |
70 72 73
|
syl2anc |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) = ( ( sqrt ` A ) / ( sqrt ` ( P ^ ( P pCnt A ) ) ) ) ) |
75 |
|
nnz |
|- ( ( sqrt ` A ) e. NN -> ( sqrt ` A ) e. ZZ ) |
76 |
|
znq |
|- ( ( ( sqrt ` A ) e. ZZ /\ ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN ) -> ( ( sqrt ` A ) / ( sqrt ` ( P ^ ( P pCnt A ) ) ) ) e. QQ ) |
77 |
75 53 76
|
syl2an2 |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( ( sqrt ` A ) / ( sqrt ` ( P ^ ( P pCnt A ) ) ) ) e. QQ ) |
78 |
74 77
|
eqeltrd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. QQ ) |
79 |
|
zsqrtelqelz |
|- ( ( ( A / ( P ^ ( P pCnt A ) ) ) e. ZZ /\ ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. QQ ) -> ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. ZZ ) |
80 |
67 78 79
|
syl2anc |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. ZZ ) |
81 |
65
|
adantr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( A / ( P ^ ( P pCnt A ) ) ) e. NN ) |
82 |
81
|
nnrpd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( A / ( P ^ ( P pCnt A ) ) ) e. RR+ ) |
83 |
82
|
sqrtgt0d |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> 0 < ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) |
84 |
|
elnnz |
|- ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN <-> ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. ZZ /\ 0 < ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) |
85 |
80 83 84
|
sylanbrc |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN ) |
86 |
85
|
iftrued |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) = 1 ) |
87 |
|
fveq2 |
|- ( y = ( A / ( P ^ ( P pCnt A ) ) ) -> ( sqrt ` y ) = ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) |
88 |
87
|
eleq1d |
|- ( y = ( A / ( P ^ ( P pCnt A ) ) ) -> ( ( sqrt ` y ) e. NN <-> ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN ) ) |
89 |
88
|
ifbid |
|- ( y = ( A / ( P ^ ( P pCnt A ) ) ) -> if ( ( sqrt ` y ) e. NN , 1 , 0 ) = if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) ) |
90 |
|
fveq2 |
|- ( y = ( A / ( P ^ ( P pCnt A ) ) ) -> ( F ` y ) = ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) |
91 |
89 90
|
breq12d |
|- ( y = ( A / ( P ^ ( P pCnt A ) ) ) -> ( if ( ( sqrt ` y ) e. NN , 1 , 0 ) <_ ( F ` y ) <-> if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) <_ ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) |
92 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
93 |
65 92
|
eleqtrdi |
|- ( ph -> ( A / ( P ^ ( P pCnt A ) ) ) e. ( ZZ>= ` 1 ) ) |
94 |
27
|
nnzd |
|- ( ph -> A e. ZZ ) |
95 |
61
|
nnred |
|- ( ph -> P e. RR ) |
96 |
|
pcelnn |
|- ( ( P e. Prime /\ A e. NN ) -> ( ( P pCnt A ) e. NN <-> P || A ) ) |
97 |
11 27 96
|
syl2anc |
|- ( ph -> ( ( P pCnt A ) e. NN <-> P || A ) ) |
98 |
12 97
|
mpbird |
|- ( ph -> ( P pCnt A ) e. NN ) |
99 |
|
prmuz2 |
|- ( P e. Prime -> P e. ( ZZ>= ` 2 ) ) |
100 |
|
eluz2gt1 |
|- ( P e. ( ZZ>= ` 2 ) -> 1 < P ) |
101 |
11 99 100
|
3syl |
|- ( ph -> 1 < P ) |
102 |
|
expgt1 |
|- ( ( P e. RR /\ ( P pCnt A ) e. NN /\ 1 < P ) -> 1 < ( P ^ ( P pCnt A ) ) ) |
103 |
95 98 101 102
|
syl3anc |
|- ( ph -> 1 < ( P ^ ( P pCnt A ) ) ) |
104 |
|
1red |
|- ( ph -> 1 e. RR ) |
105 |
|
0lt1 |
|- 0 < 1 |
106 |
105
|
a1i |
|- ( ph -> 0 < 1 ) |
107 |
62
|
nnred |
|- ( ph -> ( P ^ ( P pCnt A ) ) e. RR ) |
108 |
62
|
nngt0d |
|- ( ph -> 0 < ( P ^ ( P pCnt A ) ) ) |
109 |
27
|
nnred |
|- ( ph -> A e. RR ) |
110 |
27
|
nngt0d |
|- ( ph -> 0 < A ) |
111 |
|
ltdiv2 |
|- ( ( ( 1 e. RR /\ 0 < 1 ) /\ ( ( P ^ ( P pCnt A ) ) e. RR /\ 0 < ( P ^ ( P pCnt A ) ) ) /\ ( A e. RR /\ 0 < A ) ) -> ( 1 < ( P ^ ( P pCnt A ) ) <-> ( A / ( P ^ ( P pCnt A ) ) ) < ( A / 1 ) ) ) |
112 |
104 106 107 108 109 110 111
|
syl222anc |
|- ( ph -> ( 1 < ( P ^ ( P pCnt A ) ) <-> ( A / ( P ^ ( P pCnt A ) ) ) < ( A / 1 ) ) ) |
113 |
103 112
|
mpbid |
|- ( ph -> ( A / ( P ^ ( P pCnt A ) ) ) < ( A / 1 ) ) |
114 |
28
|
div1d |
|- ( ph -> ( A / 1 ) = A ) |
115 |
113 114
|
breqtrd |
|- ( ph -> ( A / ( P ^ ( P pCnt A ) ) ) < A ) |
116 |
|
elfzo2 |
|- ( ( A / ( P ^ ( P pCnt A ) ) ) e. ( 1 ..^ A ) <-> ( ( A / ( P ^ ( P pCnt A ) ) ) e. ( ZZ>= ` 1 ) /\ A e. ZZ /\ ( A / ( P ^ ( P pCnt A ) ) ) < A ) ) |
117 |
93 94 115 116
|
syl3anbrc |
|- ( ph -> ( A / ( P ^ ( P pCnt A ) ) ) e. ( 1 ..^ A ) ) |
118 |
91 13 117
|
rspcdva |
|- ( ph -> if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) <_ ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) |
119 |
118
|
adantr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) <_ ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) |
120 |
86 119
|
eqbrtrrd |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> 1 <_ ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) |
121 |
|
1re |
|- 1 e. RR |
122 |
|
0le1 |
|- 0 <_ 1 |
123 |
121 122
|
pm3.2i |
|- ( 1 e. RR /\ 0 <_ 1 ) |
124 |
123
|
a1i |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( 1 e. RR /\ 0 <_ 1 ) ) |
125 |
1 2 3 4 5 6 7 8 9
|
dchrisum0ff |
|- ( ph -> F : NN --> RR ) |
126 |
125 62
|
ffvelrnd |
|- ( ph -> ( F ` ( P ^ ( P pCnt A ) ) ) e. RR ) |
127 |
126
|
adantr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( F ` ( P ^ ( P pCnt A ) ) ) e. RR ) |
128 |
125 65
|
ffvelrnd |
|- ( ph -> ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. RR ) |
129 |
128
|
adantr |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. RR ) |
130 |
|
lemul12a |
|- ( ( ( ( 1 e. RR /\ 0 <_ 1 ) /\ ( F ` ( P ^ ( P pCnt A ) ) ) e. RR ) /\ ( ( 1 e. RR /\ 0 <_ 1 ) /\ ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. RR ) ) -> ( ( 1 <_ ( F ` ( P ^ ( P pCnt A ) ) ) /\ 1 <_ ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) -> ( 1 x. 1 ) <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) ) |
131 |
124 127 124 129 130
|
syl22anc |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( ( 1 <_ ( F ` ( P ^ ( P pCnt A ) ) ) /\ 1 <_ ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) -> ( 1 x. 1 ) <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) ) |
132 |
58 120 131
|
mp2and |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> ( 1 x. 1 ) <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) |
133 |
16 132
|
eqbrtrrid |
|- ( ( ph /\ ( sqrt ` A ) e. NN ) -> 1 <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) |
134 |
|
0red |
|- ( ph -> 0 e. RR ) |
135 |
|
0re |
|- 0 e. RR |
136 |
121 135
|
ifcli |
|- if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) e. RR |
137 |
136
|
a1i |
|- ( ph -> if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) e. RR ) |
138 |
|
breq2 |
|- ( 1 = if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) -> ( 0 <_ 1 <-> 0 <_ if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) ) ) |
139 |
|
breq2 |
|- ( 0 = if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) -> ( 0 <_ 0 <-> 0 <_ if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) ) ) |
140 |
|
0le0 |
|- 0 <_ 0 |
141 |
138 139 122 140
|
keephyp |
|- 0 <_ if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) |
142 |
141
|
a1i |
|- ( ph -> 0 <_ if ( ( sqrt ` ( P ^ ( P pCnt A ) ) ) e. NN , 1 , 0 ) ) |
143 |
134 137 126 142 56
|
letrd |
|- ( ph -> 0 <_ ( F ` ( P ^ ( P pCnt A ) ) ) ) |
144 |
121 135
|
ifcli |
|- if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) e. RR |
145 |
144
|
a1i |
|- ( ph -> if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) e. RR ) |
146 |
|
breq2 |
|- ( 1 = if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) -> ( 0 <_ 1 <-> 0 <_ if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) ) ) |
147 |
|
breq2 |
|- ( 0 = if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) -> ( 0 <_ 0 <-> 0 <_ if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) ) ) |
148 |
146 147 122 140
|
keephyp |
|- 0 <_ if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) |
149 |
148
|
a1i |
|- ( ph -> 0 <_ if ( ( sqrt ` ( A / ( P ^ ( P pCnt A ) ) ) ) e. NN , 1 , 0 ) ) |
150 |
134 145 128 149 118
|
letrd |
|- ( ph -> 0 <_ ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) |
151 |
126 128 143 150
|
mulge0d |
|- ( ph -> 0 <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) |
152 |
151
|
adantr |
|- ( ( ph /\ -. ( sqrt ` A ) e. NN ) -> 0 <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) |
153 |
14 15 133 152
|
ifbothda |
|- ( ph -> if ( ( sqrt ` A ) e. NN , 1 , 0 ) <_ ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) |
154 |
62
|
nncnd |
|- ( ph -> ( P ^ ( P pCnt A ) ) e. CC ) |
155 |
62
|
nnne0d |
|- ( ph -> ( P ^ ( P pCnt A ) ) =/= 0 ) |
156 |
28 154 155
|
divcan2d |
|- ( ph -> ( ( P ^ ( P pCnt A ) ) x. ( A / ( P ^ ( P pCnt A ) ) ) ) = A ) |
157 |
156
|
fveq2d |
|- ( ph -> ( F ` ( ( P ^ ( P pCnt A ) ) x. ( A / ( P ^ ( P pCnt A ) ) ) ) ) = ( F ` A ) ) |
158 |
|
pcndvds2 |
|- ( ( P e. Prime /\ A e. NN ) -> -. P || ( A / ( P ^ ( P pCnt A ) ) ) ) |
159 |
11 27 158
|
syl2anc |
|- ( ph -> -. P || ( A / ( P ^ ( P pCnt A ) ) ) ) |
160 |
|
coprm |
|- ( ( P e. Prime /\ ( A / ( P ^ ( P pCnt A ) ) ) e. ZZ ) -> ( -. P || ( A / ( P ^ ( P pCnt A ) ) ) <-> ( P gcd ( A / ( P ^ ( P pCnt A ) ) ) ) = 1 ) ) |
161 |
11 66 160
|
syl2anc |
|- ( ph -> ( -. P || ( A / ( P ^ ( P pCnt A ) ) ) <-> ( P gcd ( A / ( P ^ ( P pCnt A ) ) ) ) = 1 ) ) |
162 |
159 161
|
mpbid |
|- ( ph -> ( P gcd ( A / ( P ^ ( P pCnt A ) ) ) ) = 1 ) |
163 |
|
prmz |
|- ( P e. Prime -> P e. ZZ ) |
164 |
11 163
|
syl |
|- ( ph -> P e. ZZ ) |
165 |
|
rpexp1i |
|- ( ( P e. ZZ /\ ( A / ( P ^ ( P pCnt A ) ) ) e. ZZ /\ ( P pCnt A ) e. NN0 ) -> ( ( P gcd ( A / ( P ^ ( P pCnt A ) ) ) ) = 1 -> ( ( P ^ ( P pCnt A ) ) gcd ( A / ( P ^ ( P pCnt A ) ) ) ) = 1 ) ) |
166 |
164 66 55 165
|
syl3anc |
|- ( ph -> ( ( P gcd ( A / ( P ^ ( P pCnt A ) ) ) ) = 1 -> ( ( P ^ ( P pCnt A ) ) gcd ( A / ( P ^ ( P pCnt A ) ) ) ) = 1 ) ) |
167 |
162 166
|
mpd |
|- ( ph -> ( ( P ^ ( P pCnt A ) ) gcd ( A / ( P ^ ( P pCnt A ) ) ) ) = 1 ) |
168 |
1 2 3 4 5 6 7 8 62 65 167
|
dchrisum0fmul |
|- ( ph -> ( F ` ( ( P ^ ( P pCnt A ) ) x. ( A / ( P ^ ( P pCnt A ) ) ) ) ) = ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) |
169 |
157 168
|
eqtr3d |
|- ( ph -> ( F ` A ) = ( ( F ` ( P ^ ( P pCnt A ) ) ) x. ( F ` ( A / ( P ^ ( P pCnt A ) ) ) ) ) ) |
170 |
153 169
|
breqtrrd |
|- ( ph -> if ( ( sqrt ` A ) e. NN , 1 , 0 ) <_ ( F ` A ) ) |