Step |
Hyp |
Ref |
Expression |
1 |
|
mertens.1 |
|
2 |
|
mertens.2 |
|
3 |
|
mertens.3 |
|
4 |
|
mertens.4 |
|
5 |
|
mertens.5 |
|
6 |
|
mertens.6 |
|
7 |
|
mertens.7 |
|
8 |
|
mertens.8 |
|
9 |
|
nn0uz |
|
10 |
|
0zd |
|
11 |
|
seqex |
|
12 |
11
|
a1i |
|
13 |
|
fzfid |
|
14 |
|
simpl |
|
15 |
|
elfznn0 |
|
16 |
14 15 3
|
syl2an |
|
17 |
|
fveq2 |
|
18 |
17
|
eleq1d |
|
19 |
4 5
|
eqeltrd |
|
20 |
19
|
ralrimiva |
|
21 |
|
fveq2 |
|
22 |
21
|
eleq1d |
|
23 |
22
|
cbvralvw |
|
24 |
20 23
|
sylib |
|
25 |
24
|
ad2antrr |
|
26 |
|
fznn0sub |
|
27 |
26
|
adantl |
|
28 |
18 25 27
|
rspcdva |
|
29 |
16 28
|
mulcld |
|
30 |
13 29
|
fsumcl |
|
31 |
6 30
|
eqeltrd |
|
32 |
9 10 31
|
serf |
|
33 |
32
|
ffvelrnda |
|
34 |
1
|
adantlr |
|
35 |
2
|
adantlr |
|
36 |
3
|
adantlr |
|
37 |
4
|
adantlr |
|
38 |
5
|
adantlr |
|
39 |
6
|
adantlr |
|
40 |
7
|
adantr |
|
41 |
8
|
adantr |
|
42 |
|
simpr |
|
43 |
|
fveq2 |
|
44 |
43
|
cbvsumv |
|
45 |
|
fvoveq1 |
|
46 |
45
|
sumeq1d |
|
47 |
44 46
|
syl5eq |
|
48 |
47
|
fveq2d |
|
49 |
48
|
eqeq2d |
|
50 |
49
|
cbvrexvw |
|
51 |
|
eqeq1 |
|
52 |
51
|
rexbidv |
|
53 |
50 52
|
syl5bb |
|
54 |
53
|
cbvabv |
|
55 |
|
fveq2 |
|
56 |
55
|
cbvsumv |
|
57 |
56
|
oveq1i |
|
58 |
57
|
oveq2i |
|
59 |
58
|
breq2i |
|
60 |
|
fveq2 |
|
61 |
60
|
cbvsumv |
|
62 |
|
fvoveq1 |
|
63 |
62
|
sumeq1d |
|
64 |
61 63
|
syl5eq |
|
65 |
64
|
fveq2d |
|
66 |
65
|
breq1d |
|
67 |
59 66
|
syl5bb |
|
68 |
67
|
cbvralvw |
|
69 |
68
|
anbi2i |
|
70 |
34 35 36 37 38 39 40 41 42 54 69
|
mertenslem2 |
|
71 |
|
eluznn0 |
|
72 |
|
fzfid |
|
73 |
|
simpll |
|
74 |
|
elfznn0 |
|
75 |
74
|
adantl |
|
76 |
9 10 4 5 8
|
isumcl |
|
77 |
76
|
adantr |
|
78 |
1 3
|
eqeltrd |
|
79 |
77 78
|
mulcld |
|
80 |
73 75 79
|
syl2anc |
|
81 |
|
fzfid |
|
82 |
|
simplll |
|
83 |
74
|
ad2antlr |
|
84 |
82 83 3
|
syl2anc |
|
85 |
|
elfznn0 |
|
86 |
85
|
adantl |
|
87 |
82 86 19
|
syl2anc |
|
88 |
84 87
|
mulcld |
|
89 |
81 88
|
fsumcl |
|
90 |
72 80 89
|
fsumsub |
|
91 |
73 75 3
|
syl2anc |
|
92 |
76
|
ad2antrr |
|
93 |
81 87
|
fsumcl |
|
94 |
91 92 93
|
subdid |
|
95 |
|
eqid |
|
96 |
|
fznn0sub |
|
97 |
96
|
adantl |
|
98 |
|
peano2nn0 |
|
99 |
97 98
|
syl |
|
100 |
99
|
nn0zd |
|
101 |
|
simplll |
|
102 |
|
eluznn0 |
|
103 |
99 102
|
sylan |
|
104 |
101 103 4
|
syl2anc |
|
105 |
101 103 5
|
syl2anc |
|
106 |
8
|
ad2antrr |
|
107 |
73 4
|
sylan |
|
108 |
73 5
|
sylan |
|
109 |
107 108
|
eqeltrd |
|
110 |
9 99 109
|
iserex |
|
111 |
106 110
|
mpbid |
|
112 |
95 100 104 105 111
|
isumcl |
|
113 |
9 95 99 107 108 106
|
isumsplit |
|
114 |
97
|
nn0cnd |
|
115 |
|
ax-1cn |
|
116 |
|
pncan |
|
117 |
114 115 116
|
sylancl |
|
118 |
117
|
oveq2d |
|
119 |
118
|
sumeq1d |
|
120 |
82 86 4
|
syl2anc |
|
121 |
120
|
sumeq2dv |
|
122 |
119 121
|
eqtr4d |
|
123 |
122
|
oveq1d |
|
124 |
113 123
|
eqtrd |
|
125 |
93 112 124
|
mvrladdd |
|
126 |
125
|
oveq2d |
|
127 |
3 77
|
mulcomd |
|
128 |
1
|
oveq2d |
|
129 |
127 128
|
eqtr4d |
|
130 |
73 75 129
|
syl2anc |
|
131 |
81 91 87
|
fsummulc2 |
|
132 |
130 131
|
oveq12d |
|
133 |
94 126 132
|
3eqtr3rd |
|
134 |
133
|
sumeq2dv |
|
135 |
|
fveq2 |
|
136 |
135
|
oveq2d |
|
137 |
|
eqid |
|
138 |
|
ovex |
|
139 |
136 137 138
|
fvmpt |
|
140 |
75 139
|
syl |
|
141 |
|
simpr |
|
142 |
141 9
|
eleqtrdi |
|
143 |
140 142 80
|
fsumser |
|
144 |
|
fveq2 |
|
145 |
144
|
oveq2d |
|
146 |
|
fveq2 |
|
147 |
146
|
oveq2d |
|
148 |
88
|
anasss |
|
149 |
145 147 148
|
fsum0diag2 |
|
150 |
|
simpll |
|
151 |
|
elfznn0 |
|
152 |
151
|
adantl |
|
153 |
150 152 6
|
syl2anc |
|
154 |
150 152 30
|
syl2anc |
|
155 |
153 142 154
|
fsumser |
|
156 |
149 155
|
eqtrd |
|
157 |
143 156
|
oveq12d |
|
158 |
90 134 157
|
3eqtr3rd |
|
159 |
158
|
fveq2d |
|
160 |
159
|
breq1d |
|
161 |
71 160
|
sylan2 |
|
162 |
161
|
anassrs |
|
163 |
162
|
ralbidva |
|
164 |
163
|
rexbidva |
|
165 |
164
|
adantr |
|
166 |
70 165
|
mpbird |
|
167 |
166
|
ralrimiva |
|
168 |
1
|
fveq2d |
|
169 |
2 168
|
eqtr4d |
|
170 |
9 10 169 78 7
|
abscvgcvg |
|
171 |
9 10 1 3 170
|
isumclim2 |
|
172 |
78
|
ralrimiva |
|
173 |
|
fveq2 |
|
174 |
173
|
eleq1d |
|
175 |
174
|
rspccva |
|
176 |
172 175
|
sylan |
|
177 |
|
fveq2 |
|
178 |
177
|
oveq2d |
|
179 |
|
ovex |
|
180 |
178 137 179
|
fvmpt |
|
181 |
180
|
adantl |
|
182 |
9 10 76 171 176 181
|
isermulc2 |
|
183 |
9 10 1 3 170
|
isumcl |
|
184 |
76 183
|
mulcomd |
|
185 |
182 184
|
breqtrd |
|
186 |
9 10 12 33 167 185
|
2clim |
|