Step |
Hyp |
Ref |
Expression |
1 |
|
vieta1.1 |
|
2 |
|
vieta1.2 |
|
3 |
|
vieta1.3 |
|
4 |
|
vieta1.4 |
|
5 |
|
vieta1.5 |
|
6 |
|
vieta1.6 |
|
7 |
|
fveq2 |
|
8 |
7
|
eqeq2d |
|
9 |
|
cnveq |
|
10 |
9
|
imaeq1d |
|
11 |
10 3
|
eqtr4di |
|
12 |
11
|
fveq2d |
|
13 |
7 2
|
eqtr4di |
|
14 |
12 13
|
eqeq12d |
|
15 |
8 14
|
anbi12d |
|
16 |
2
|
biantrur |
|
17 |
15 16
|
bitr4di |
|
18 |
11
|
sumeq1d |
|
19 |
|
fveq2 |
|
20 |
19 1
|
eqtr4di |
|
21 |
13
|
oveq1d |
|
22 |
20 21
|
fveq12d |
|
23 |
20 13
|
fveq12d |
|
24 |
22 23
|
oveq12d |
|
25 |
24
|
negeqd |
|
26 |
18 25
|
eqeq12d |
|
27 |
17 26
|
imbi12d |
|
28 |
|
eqeq1 |
|
29 |
28
|
anbi1d |
|
30 |
29
|
imbi1d |
|
31 |
30
|
ralbidv |
|
32 |
|
eqeq1 |
|
33 |
32
|
anbi1d |
|
34 |
33
|
imbi1d |
|
35 |
34
|
ralbidv |
|
36 |
|
eqeq1 |
|
37 |
36
|
anbi1d |
|
38 |
37
|
imbi1d |
|
39 |
38
|
ralbidv |
|
40 |
|
eqeq1 |
|
41 |
40
|
anbi1d |
|
42 |
41
|
imbi1d |
|
43 |
42
|
ralbidv |
|
44 |
|
eqid |
|
45 |
44
|
coef3 |
|
46 |
45
|
adantr |
|
47 |
|
0nn0 |
|
48 |
|
ffvelrn |
|
49 |
46 47 48
|
sylancl |
|
50 |
|
1nn0 |
|
51 |
|
ffvelrn |
|
52 |
46 50 51
|
sylancl |
|
53 |
|
simpr |
|
54 |
53
|
fveq2d |
|
55 |
|
ax-1ne0 |
|
56 |
55
|
a1i |
|
57 |
53 56
|
eqnetrrd |
|
58 |
|
fveq2 |
|
59 |
|
dgr0 |
|
60 |
58 59
|
eqtrdi |
|
61 |
60
|
necon3i |
|
62 |
57 61
|
syl |
|
63 |
|
eqid |
|
64 |
63 44
|
dgreq0 |
|
65 |
64
|
necon3bid |
|
66 |
65
|
adantr |
|
67 |
62 66
|
mpbid |
|
68 |
54 67
|
eqnetrd |
|
69 |
49 52 68
|
divcld |
|
70 |
69
|
negcld |
|
71 |
|
id |
|
72 |
71
|
sumsn |
|
73 |
70 70 72
|
syl2anc |
|
74 |
73
|
adantrr |
|
75 |
|
eqid |
|
76 |
75
|
fta1 |
|
77 |
62 76
|
syldan |
|
78 |
77
|
simpld |
|
79 |
78
|
adantrr |
|
80 |
44 63
|
coeid2 |
|
81 |
70 80
|
syldan |
|
82 |
53
|
oveq2d |
|
83 |
82
|
sumeq1d |
|
84 |
|
nn0uz |
|
85 |
|
1e0p1 |
|
86 |
|
fveq2 |
|
87 |
|
oveq2 |
|
88 |
86 87
|
oveq12d |
|
89 |
46
|
ffvelrnda |
|
90 |
|
expcl |
|
91 |
70 90
|
sylan |
|
92 |
89 91
|
mulcld |
|
93 |
|
0z |
|
94 |
70
|
exp0d |
|
95 |
94
|
oveq2d |
|
96 |
49
|
mulid1d |
|
97 |
95 96
|
eqtrd |
|
98 |
97 49
|
eqeltrd |
|
99 |
|
fveq2 |
|
100 |
|
oveq2 |
|
101 |
99 100
|
oveq12d |
|
102 |
101
|
fsum1 |
|
103 |
93 98 102
|
sylancr |
|
104 |
103 97
|
eqtrd |
|
105 |
104 47
|
jctil |
|
106 |
70
|
exp1d |
|
107 |
106
|
oveq2d |
|
108 |
52 69
|
mulneg2d |
|
109 |
49 52 68
|
divcan2d |
|
110 |
109
|
negeqd |
|
111 |
107 108 110
|
3eqtrd |
|
112 |
111
|
oveq2d |
|
113 |
49
|
negidd |
|
114 |
112 113
|
eqtrd |
|
115 |
84 85 88 92 105 114
|
fsump1i |
|
116 |
115
|
simprd |
|
117 |
81 83 116
|
3eqtr2d |
|
118 |
|
plyf |
|
119 |
118
|
ffnd |
|
120 |
119
|
adantr |
|
121 |
|
fniniseg |
|
122 |
120 121
|
syl |
|
123 |
70 117 122
|
mpbir2and |
|
124 |
123
|
snssd |
|
125 |
124
|
adantrr |
|
126 |
|
hashsng |
|
127 |
70 126
|
syl |
|
128 |
127 53
|
eqtrd |
|
129 |
128
|
adantrr |
|
130 |
|
simprr |
|
131 |
129 130
|
eqtr4d |
|
132 |
|
snfi |
|
133 |
|
hashen |
|
134 |
132 78 133
|
sylancr |
|
135 |
134
|
adantrr |
|
136 |
131 135
|
mpbid |
|
137 |
|
fisseneq |
|
138 |
79 125 136 137
|
syl3anc |
|
139 |
138
|
sumeq1d |
|
140 |
|
1m1e0 |
|
141 |
53
|
oveq1d |
|
142 |
140 141
|
syl5eqr |
|
143 |
142
|
fveq2d |
|
144 |
143 54
|
oveq12d |
|
145 |
144
|
negeqd |
|
146 |
145
|
adantrr |
|
147 |
74 139 146
|
3eqtr3d |
|
148 |
147
|
ex |
|
149 |
148
|
rgen |
|
150 |
|
id |
|
151 |
150
|
cbvsumv |
|
152 |
151
|
eqeq1i |
|
153 |
152
|
imbi2i |
|
154 |
153
|
ralbii |
|
155 |
|
eqid |
|
156 |
|
eqid |
|
157 |
|
eqid |
|
158 |
|
simp1r |
|
159 |
|
simp3r |
|
160 |
|
simp1l |
|
161 |
|
simp3l |
|
162 |
|
simp2 |
|
163 |
162 154
|
sylib |
|
164 |
|
eqid |
|
165 |
155 156 157 158 159 160 161 163 164
|
vieta1lem2 |
|
166 |
165
|
3exp |
|
167 |
154 166
|
syl5bir |
|
168 |
167
|
ralrimdva |
|
169 |
|
fveq2 |
|
170 |
169
|
eqeq2d |
|
171 |
|
cnveq |
|
172 |
171
|
imaeq1d |
|
173 |
172
|
fveq2d |
|
174 |
173 169
|
eqeq12d |
|
175 |
170 174
|
anbi12d |
|
176 |
172
|
sumeq1d |
|
177 |
|
fveq2 |
|
178 |
169
|
oveq1d |
|
179 |
177 178
|
fveq12d |
|
180 |
177 169
|
fveq12d |
|
181 |
179 180
|
oveq12d |
|
182 |
181
|
negeqd |
|
183 |
176 182
|
eqeq12d |
|
184 |
175 183
|
imbi12d |
|
185 |
184
|
cbvralvw |
|
186 |
168 185
|
syl6ib |
|
187 |
31 35 39 43 149 186
|
nnind |
|
188 |
6 187
|
syl |
|
189 |
|
plyssc |
|
190 |
189 4
|
sseldi |
|
191 |
27 188 190
|
rspcdva |
|
192 |
5 191
|
mpd |
|