Step |
Hyp |
Ref |
Expression |
1 |
|
cosne0 |
|
2 |
|
atandmtan |
|
3 |
1 2
|
syldan |
|
4 |
|
atanval |
|
5 |
3 4
|
syl |
|
6 |
|
ax-1cn |
|
7 |
|
ax-icn |
|
8 |
|
tancl |
|
9 |
1 8
|
syldan |
|
10 |
|
mulcl |
|
11 |
7 9 10
|
sylancr |
|
12 |
|
addcl |
|
13 |
6 11 12
|
sylancr |
|
14 |
|
atandm2 |
|
15 |
3 14
|
sylib |
|
16 |
15
|
simp3d |
|
17 |
13 16
|
logcld |
|
18 |
|
subcl |
|
19 |
6 11 18
|
sylancr |
|
20 |
15
|
simp2d |
|
21 |
19 20
|
logcld |
|
22 |
17 21
|
negsubdi2d |
|
23 |
|
efsub |
|
24 |
17 21 23
|
syl2anc |
|
25 |
|
coscl |
|
26 |
25
|
adantr |
|
27 |
|
sincl |
|
28 |
27
|
adantr |
|
29 |
|
mulcl |
|
30 |
7 28 29
|
sylancr |
|
31 |
26 30 26 1
|
divdird |
|
32 |
26 1
|
dividd |
|
33 |
7
|
a1i |
|
34 |
33 28 26 1
|
divassd |
|
35 |
|
tanval |
|
36 |
1 35
|
syldan |
|
37 |
36
|
oveq2d |
|
38 |
34 37
|
eqtr4d |
|
39 |
32 38
|
oveq12d |
|
40 |
31 39
|
eqtrd |
|
41 |
|
efival |
|
42 |
41
|
adantr |
|
43 |
42
|
oveq1d |
|
44 |
|
eflog |
|
45 |
13 16 44
|
syl2anc |
|
46 |
40 43 45
|
3eqtr4d |
|
47 |
26 30 26 1
|
divsubdird |
|
48 |
32 38
|
oveq12d |
|
49 |
47 48
|
eqtrd |
|
50 |
|
negcl |
|
51 |
50
|
adantr |
|
52 |
|
efival |
|
53 |
51 52
|
syl |
|
54 |
|
cosneg |
|
55 |
54
|
adantr |
|
56 |
|
sinneg |
|
57 |
56
|
adantr |
|
58 |
57
|
oveq2d |
|
59 |
|
mulneg2 |
|
60 |
7 28 59
|
sylancr |
|
61 |
58 60
|
eqtrd |
|
62 |
55 61
|
oveq12d |
|
63 |
53 62
|
eqtrd |
|
64 |
|
simpl |
|
65 |
|
mulneg2 |
|
66 |
7 64 65
|
sylancr |
|
67 |
66
|
fveq2d |
|
68 |
26 30
|
negsubd |
|
69 |
63 67 68
|
3eqtr3d |
|
70 |
69
|
oveq1d |
|
71 |
|
eflog |
|
72 |
19 20 71
|
syl2anc |
|
73 |
49 70 72
|
3eqtr4d |
|
74 |
46 73
|
oveq12d |
|
75 |
|
mulcl |
|
76 |
7 64 75
|
sylancr |
|
77 |
|
efcl |
|
78 |
76 77
|
syl |
|
79 |
76
|
negcld |
|
80 |
|
efcl |
|
81 |
79 80
|
syl |
|
82 |
|
efne0 |
|
83 |
79 82
|
syl |
|
84 |
78 81 26 83 1
|
divcan7d |
|
85 |
|
efsub |
|
86 |
76 79 85
|
syl2anc |
|
87 |
76 76
|
subnegd |
|
88 |
76
|
2timesd |
|
89 |
87 88
|
eqtr4d |
|
90 |
89
|
fveq2d |
|
91 |
84 86 90
|
3eqtr2d |
|
92 |
24 74 91
|
3eqtr2d |
|
93 |
92
|
fveq2d |
|
94 |
64
|
adantr |
|
95 |
94
|
renegd |
|
96 |
94
|
recld |
|
97 |
96
|
renegcld |
|
98 |
|
simpr |
|
99 |
96
|
lt0neg1d |
|
100 |
98 99
|
mpbid |
|
101 |
|
eliooord |
|
102 |
101
|
adantl |
|
103 |
102
|
simpld |
|
104 |
103
|
adantr |
|
105 |
|
halfpire |
|
106 |
|
ltnegcon1 |
|
107 |
105 96 106
|
sylancr |
|
108 |
104 107
|
mpbid |
|
109 |
|
0xr |
|
110 |
105
|
rexri |
|
111 |
|
elioo2 |
|
112 |
109 110 111
|
mp2an |
|
113 |
97 100 108 112
|
syl3anbrc |
|
114 |
95 113
|
eqeltrd |
|
115 |
|
tanregt0 |
|
116 |
51 114 115
|
syl2an2r |
|
117 |
|
tanneg |
|
118 |
1 117
|
syldan |
|
119 |
118
|
adantr |
|
120 |
119
|
fveq2d |
|
121 |
9
|
adantr |
|
122 |
121
|
renegd |
|
123 |
120 122
|
eqtrd |
|
124 |
116 123
|
breqtrd |
|
125 |
9
|
recld |
|
126 |
125
|
adantr |
|
127 |
126
|
lt0neg1d |
|
128 |
124 127
|
mpbird |
|
129 |
128
|
lt0ne0d |
|
130 |
|
atanlogsub |
|
131 |
3 129 130
|
syl2an2r |
|
132 |
|
1re |
|
133 |
|
ioossre |
|
134 |
7
|
a1i |
|
135 |
11
|
adantr |
|
136 |
|
ine0 |
|
137 |
136
|
a1i |
|
138 |
|
ixi |
|
139 |
138
|
oveq1i |
|
140 |
9
|
adantr |
|
141 |
140
|
mulm1d |
|
142 |
118
|
adantr |
|
143 |
141 142
|
eqtr4d |
|
144 |
139 143
|
syl5eq |
|
145 |
134 134 140
|
mulassd |
|
146 |
138
|
oveq1i |
|
147 |
64
|
adantr |
|
148 |
147
|
mulm1d |
|
149 |
146 148
|
syl5eq |
|
150 |
134 134 147
|
mulassd |
|
151 |
149 150
|
eqtr3d |
|
152 |
151
|
fveq2d |
|
153 |
144 145 152
|
3eqtr3d |
|
154 |
134 135 137 153
|
mvllmuld |
|
155 |
76
|
adantr |
|
156 |
|
reim |
|
157 |
156
|
adantr |
|
158 |
157
|
eqeq1d |
|
159 |
158
|
biimpa |
|
160 |
155 159
|
reim0bd |
|
161 |
|
tanhbnd |
|
162 |
160 161
|
syl |
|
163 |
154 162
|
eqeltrd |
|
164 |
133 163
|
sselid |
|
165 |
|
readdcl |
|
166 |
132 164 165
|
sylancr |
|
167 |
|
df-neg |
|
168 |
|
eliooord |
|
169 |
163 168
|
syl |
|
170 |
169
|
simpld |
|
171 |
167 170
|
eqbrtrrid |
|
172 |
|
0red |
|
173 |
132
|
a1i |
|
174 |
172 173 164
|
ltsubadd2d |
|
175 |
171 174
|
mpbid |
|
176 |
166 175
|
elrpd |
|
177 |
176
|
relogcld |
|
178 |
169
|
simprd |
|
179 |
|
difrp |
|
180 |
164 132 179
|
sylancl |
|
181 |
178 180
|
mpbid |
|
182 |
181
|
relogcld |
|
183 |
177 182
|
resubcld |
|
184 |
|
relogrn |
|
185 |
183 184
|
syl |
|
186 |
64
|
adantr |
|
187 |
186
|
recld |
|
188 |
|
simpr |
|
189 |
102
|
simprd |
|
190 |
189
|
adantr |
|
191 |
|
elioo2 |
|
192 |
109 110 191
|
mp2an |
|
193 |
187 188 190 192
|
syl3anbrc |
|
194 |
|
tanregt0 |
|
195 |
64 193 194
|
syl2an2r |
|
196 |
195
|
gt0ne0d |
|
197 |
3 196 130
|
syl2an2r |
|
198 |
|
recl |
|
199 |
198
|
adantr |
|
200 |
|
0re |
|
201 |
|
lttri4 |
|
202 |
199 200 201
|
sylancl |
|
203 |
131 185 197 202
|
mpjao3dan |
|
204 |
|
logef |
|
205 |
203 204
|
syl |
|
206 |
|
2cn |
|
207 |
|
mulcl |
|
208 |
206 76 207
|
sylancr |
|
209 |
|
picn |
|
210 |
|
2ne0 |
|
211 |
|
divneg |
|
212 |
209 206 210 211
|
mp3an |
|
213 |
212 103
|
eqbrtrrid |
|
214 |
|
pire |
|
215 |
214
|
renegcli |
|
216 |
215
|
a1i |
|
217 |
|
2re |
|
218 |
217
|
a1i |
|
219 |
|
2pos |
|
220 |
219
|
a1i |
|
221 |
|
ltdivmul |
|
222 |
216 199 218 220 221
|
syl112anc |
|
223 |
213 222
|
mpbid |
|
224 |
|
immul2 |
|
225 |
217 76 224
|
sylancr |
|
226 |
157
|
oveq2d |
|
227 |
225 226
|
eqtr4d |
|
228 |
223 227
|
breqtrrd |
|
229 |
|
remulcl |
|
230 |
217 199 229
|
sylancr |
|
231 |
214
|
a1i |
|
232 |
|
ltmuldiv2 |
|
233 |
199 231 218 220 232
|
syl112anc |
|
234 |
189 233
|
mpbird |
|
235 |
230 231 234
|
ltled |
|
236 |
227 235
|
eqbrtrd |
|
237 |
|
ellogrn |
|
238 |
208 228 236 237
|
syl3anbrc |
|
239 |
|
logef |
|
240 |
238 239
|
syl |
|
241 |
93 205 240
|
3eqtr3d |
|
242 |
241
|
negeqd |
|
243 |
22 242
|
eqtr3d |
|
244 |
243
|
oveq2d |
|
245 |
|
halfcl |
|
246 |
7 245
|
mp1i |
|
247 |
206
|
a1i |
|
248 |
246 247 79
|
mulassd |
|
249 |
7 206 210
|
divcan1i |
|
250 |
249
|
oveq1i |
|
251 |
33 33 51
|
mulassd |
|
252 |
138
|
oveq1i |
|
253 |
|
mul2neg |
|
254 |
6 64 253
|
sylancr |
|
255 |
|
mulid2 |
|
256 |
255
|
adantr |
|
257 |
254 256
|
eqtrd |
|
258 |
252 257
|
syl5eq |
|
259 |
66
|
oveq2d |
|
260 |
251 258 259
|
3eqtr3rd |
|
261 |
250 260
|
syl5eq |
|
262 |
|
mulneg2 |
|
263 |
206 76 262
|
sylancr |
|
264 |
263
|
oveq2d |
|
265 |
248 261 264
|
3eqtr3rd |
|
266 |
5 244 265
|
3eqtrd |
|