Step |
Hyp |
Ref |
Expression |
1 |
|
dirkertrigeq.d |
|
2 |
|
dirkertrigeq.n |
|
3 |
|
dirkertrigeq.f |
|
4 |
|
dirkertrigeq.h |
|
5 |
3
|
a1i |
|
6 |
1
|
dirkerval |
|
7 |
2 6
|
syl |
|
8 |
|
2cnd |
|
9 |
2
|
nncnd |
|
10 |
8 9
|
mulcld |
|
11 |
|
peano2cn |
|
12 |
10 11
|
syl |
|
13 |
|
picn |
|
14 |
13
|
a1i |
|
15 |
|
2ne0 |
|
16 |
15
|
a1i |
|
17 |
|
pire |
|
18 |
|
pipos |
|
19 |
17 18
|
gt0ne0ii |
|
20 |
19
|
a1i |
|
21 |
12 8 14 16 20
|
divdiv1d |
|
22 |
21
|
eqcomd |
|
23 |
22
|
ad2antrr |
|
24 |
|
iftrue |
|
25 |
24
|
adantl |
|
26 |
|
elfzelz |
|
27 |
26
|
zcnd |
|
28 |
27
|
adantl |
|
29 |
|
recn |
|
30 |
29
|
ad2antrr |
|
31 |
|
2cn |
|
32 |
31 13
|
mulcli |
|
33 |
32
|
a1i |
|
34 |
31 13 15 19
|
mulne0i |
|
35 |
34
|
a1i |
|
36 |
28 30 33 35
|
divassd |
|
37 |
26
|
adantl |
|
38 |
|
simpr |
|
39 |
|
simpl |
|
40 |
|
2rp |
|
41 |
|
pirp |
|
42 |
|
rpmulcl |
|
43 |
40 41 42
|
mp2an |
|
44 |
|
mod0 |
|
45 |
39 43 44
|
sylancl |
|
46 |
38 45
|
mpbid |
|
47 |
46
|
adantr |
|
48 |
37 47
|
zmulcld |
|
49 |
36 48
|
eqeltrd |
|
50 |
27
|
adantl |
|
51 |
29
|
adantr |
|
52 |
50 51
|
mulcld |
|
53 |
|
coseq1 |
|
54 |
52 53
|
syl |
|
55 |
54
|
adantlr |
|
56 |
49 55
|
mpbird |
|
57 |
56
|
ralrimiva |
|
58 |
57
|
adantll |
|
59 |
58
|
sumeq2d |
|
60 |
|
fzfid |
|
61 |
|
1cnd |
|
62 |
|
fsumconst |
|
63 |
60 61 62
|
syl2anc |
|
64 |
2
|
nnnn0d |
|
65 |
|
hashfz1 |
|
66 |
64 65
|
syl |
|
67 |
66
|
oveq1d |
|
68 |
9
|
mulid1d |
|
69 |
67 68
|
eqtrd |
|
70 |
69
|
ad2antrr |
|
71 |
59 63 70
|
3eqtrd |
|
72 |
71
|
oveq2d |
|
73 |
9
|
div1d |
|
74 |
73
|
eqcomd |
|
75 |
74
|
oveq2d |
|
76 |
|
1cnd |
|
77 |
|
ax-1ne0 |
|
78 |
77
|
a1i |
|
79 |
76 8 9 76 16 78
|
divadddivd |
|
80 |
76 76
|
mulcld |
|
81 |
9 8
|
mulcld |
|
82 |
80 81
|
addcomd |
|
83 |
9 8
|
mulcomd |
|
84 |
76
|
mulid1d |
|
85 |
83 84
|
oveq12d |
|
86 |
82 85
|
eqtrd |
|
87 |
8
|
mulid1d |
|
88 |
86 87
|
oveq12d |
|
89 |
75 79 88
|
3eqtrd |
|
90 |
89
|
ad2antrr |
|
91 |
72 90
|
eqtrd |
|
92 |
91
|
oveq1d |
|
93 |
23 25 92
|
3eqtr4rd |
|
94 |
|
iffalse |
|
95 |
94
|
adantl |
|
96 |
13
|
a1i |
|
97 |
19
|
a1i |
|
98 |
29 96 97
|
divcan1d |
|
99 |
98
|
eqcomd |
|
100 |
99
|
ad2antrr |
|
101 |
|
simpr |
|
102 |
|
simpl |
|
103 |
|
mod0 |
|
104 |
102 41 103
|
sylancl |
|
105 |
101 104
|
mpbid |
|
106 |
105
|
adantlr |
|
107 |
|
rpreccl |
|
108 |
41 107
|
ax-mp |
|
109 |
|
moddi |
|
110 |
108 43 109
|
mp3an13 |
|
111 |
29 96 97
|
divrec2d |
|
112 |
111
|
eqcomd |
|
113 |
96 97
|
reccld |
|
114 |
32
|
a1i |
|
115 |
113 114
|
mulcomd |
|
116 |
|
2cnd |
|
117 |
116 96 113
|
mulassd |
|
118 |
13 19
|
recidi |
|
119 |
118
|
oveq2i |
|
120 |
116
|
mulid1d |
|
121 |
119 120
|
syl5eq |
|
122 |
115 117 121
|
3eqtrd |
|
123 |
112 122
|
oveq12d |
|
124 |
110 123
|
eqtr2d |
|
125 |
124
|
adantr |
|
126 |
113
|
adantr |
|
127 |
|
id |
|
128 |
43
|
a1i |
|
129 |
127 128
|
modcld |
|
130 |
129
|
recnd |
|
131 |
130
|
adantr |
|
132 |
|
ax-1cn |
|
133 |
132 13 77 19
|
divne0i |
|
134 |
133
|
a1i |
|
135 |
|
neqne |
|
136 |
135
|
adantl |
|
137 |
126 131 134 136
|
mulne0d |
|
138 |
125 137
|
eqnetrd |
|
139 |
138
|
adantr |
|
140 |
|
oddfl |
|
141 |
106 139 140
|
syl2anc |
|
142 |
141
|
oveq1d |
|
143 |
100 142
|
eqtrd |
|
144 |
143
|
oveq2d |
|
145 |
144
|
fveq2d |
|
146 |
145
|
sumeq2sdv |
|
147 |
146
|
oveq2d |
|
148 |
147
|
oveq1d |
|
149 |
148
|
adantlll |
|
150 |
2
|
ad2antrr |
|
151 |
17
|
a1i |
|
152 |
127 151 97
|
redivcld |
|
153 |
152
|
rehalfcld |
|
154 |
153
|
flcld |
|
155 |
154
|
ad2antlr |
|
156 |
|
eqid |
|
157 |
150 155 156
|
dirkertrigeqlem3 |
|
158 |
157
|
adantlr |
|
159 |
141
|
adantlll |
|
160 |
159
|
eqcomd |
|
161 |
160
|
oveq1d |
|
162 |
161
|
oveq2d |
|
163 |
162
|
fveq2d |
|
164 |
161
|
fvoveq1d |
|
165 |
164
|
oveq2d |
|
166 |
163 165
|
oveq12d |
|
167 |
98
|
oveq2d |
|
168 |
167
|
fveq2d |
|
169 |
98
|
fvoveq1d |
|
170 |
169
|
oveq2d |
|
171 |
168 170
|
oveq12d |
|
172 |
171
|
adantl |
|
173 |
172
|
ad2antrr |
|
174 |
166 173
|
eqtrd |
|
175 |
149 158 174
|
3eqtrrd |
|
176 |
|
simplr |
|
177 |
|
simpr |
|
178 |
176 41 103
|
sylancl |
|
179 |
177 178
|
mtbid |
|
180 |
176
|
recnd |
|
181 |
|
sineq0 |
|
182 |
180 181
|
syl |
|
183 |
179 182
|
mtbird |
|
184 |
183
|
neqned |
|
185 |
2
|
ad2antrr |
|
186 |
176 184 185
|
dirkertrigeqlem2 |
|
187 |
186
|
eqcomd |
|
188 |
187
|
adantlr |
|
189 |
175 188
|
pm2.61dan |
|
190 |
95 189
|
eqtr2d |
|
191 |
93 190
|
pm2.61dan |
|
192 |
191
|
mpteq2dva |
|
193 |
4 192
|
syl5req |
|
194 |
5 7 193
|
3eqtrd |
|