Metamath Proof Explorer


Theorem pntlemo

Description: Lemma for pnt . Combine all the estimates to establish a smaller eventual bound on R ( Z ) / Z . (Contributed by Mario Carneiro, 14-Apr-2016)

Ref Expression
Hypotheses pntlem1.r R = a + ψ a a
pntlem1.a φ A +
pntlem1.b φ B +
pntlem1.l φ L 0 1
pntlem1.d D = A + 1
pntlem1.f F = 1 1 D L 32 B D 2
pntlem1.u φ U +
pntlem1.u2 φ U A
pntlem1.e E = U D
pntlem1.k K = e B E
pntlem1.y φ Y + 1 Y
pntlem1.x φ X + Y < X
pntlem1.c φ C +
pntlem1.w W = Y + 4 L E 2 + X K 2 4 + e 32 B U E L E 2 U 3 + C
pntlem1.z φ Z W +∞
pntlem1.m M = log X log K + 1
pntlem1.n N = log Z log K 2
pntlem1.U φ z Y +∞ R z z U
pntlem1.K φ y X +∞ z + y < z 1 + L E z < K y u z 1 + L E z R u u E
pntlem1.C φ z 1 +∞ R z log z 2 log z i = 1 z Y R z i log i z C
Assertion pntlemo φ R Z Z U F U 3

Proof

Step Hyp Ref Expression
1 pntlem1.r R = a + ψ a a
2 pntlem1.a φ A +
3 pntlem1.b φ B +
4 pntlem1.l φ L 0 1
5 pntlem1.d D = A + 1
6 pntlem1.f F = 1 1 D L 32 B D 2
7 pntlem1.u φ U +
8 pntlem1.u2 φ U A
9 pntlem1.e E = U D
10 pntlem1.k K = e B E
11 pntlem1.y φ Y + 1 Y
12 pntlem1.x φ X + Y < X
13 pntlem1.c φ C +
14 pntlem1.w W = Y + 4 L E 2 + X K 2 4 + e 32 B U E L E 2 U 3 + C
15 pntlem1.z φ Z W +∞
16 pntlem1.m M = log X log K + 1
17 pntlem1.n N = log Z log K 2
18 pntlem1.U φ z Y +∞ R z z U
19 pntlem1.K φ y X +∞ z + y < z 1 + L E z < K y u z 1 + L E z R u u E
20 pntlem1.C φ z 1 +∞ R z log z 2 log z i = 1 z Y R z i log i z C
21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 pntlemb φ Z + 1 < Z e Z Z Z Y 4 L E Z log X log K + 2 log Z log K 4 U 3 + C U E L E 2 32 B log Z
22 21 simp1d φ Z +
23 1 pntrf R : +
24 23 ffvelrni Z + R Z
25 22 24 syl φ R Z
26 25 22 rerpdivcld φ R Z Z
27 26 recnd φ R Z Z
28 27 abscld φ R Z Z
29 22 relogcld φ log Z
30 28 29 remulcld φ R Z Z log Z
31 7 rpred φ U
32 3re 3
33 32 a1i φ 3
34 29 33 readdcld φ log Z + 3
35 31 34 remulcld φ U log Z + 3
36 2re 2
37 36 a1i φ 2
38 1 2 3 4 5 6 7 8 9 10 pntlemc φ E + K + E 0 1 1 < K U E +
39 38 simp3d φ E 0 1 1 < K U E +
40 39 simp3d φ U E +
41 40 rpred φ U E
42 1 2 3 4 5 6 pntlemd φ L + D + F +
43 42 simp1d φ L +
44 38 simp1d φ E +
45 2z 2
46 rpexpcl E + 2 E 2 +
47 44 45 46 sylancl φ E 2 +
48 43 47 rpmulcld φ L E 2 +
49 3nn0 3 0
50 2nn 2
51 49 50 decnncl 32
52 nnrp 32 32 +
53 51 52 ax-mp 32 +
54 rpmulcl 32 + B + 32 B +
55 53 3 54 sylancr φ 32 B +
56 48 55 rpdivcld φ L E 2 32 B +
57 56 rpred φ L E 2 32 B
58 41 57 remulcld φ U E L E 2 32 B
59 58 29 remulcld φ U E L E 2 32 B log Z
60 37 59 remulcld φ 2 U E L E 2 32 B log Z
61 35 60 resubcld φ U log Z + 3 2 U E L E 2 32 B log Z
62 13 rpred φ C
63 61 62 readdcld φ U log Z + 3 - 2 U E L E 2 32 B log Z + C
64 7 rpcnd φ U
65 58 recnd φ U E L E 2 32 B
66 29 recnd φ log Z
67 64 65 66 subdird φ U U E L E 2 32 B log Z = U log Z U E L E 2 32 B log Z
68 43 rpcnd φ L
69 47 rpcnd φ E 2
70 55 rpcnne0d φ 32 B 32 B 0
71 div23 L E 2 32 B 32 B 0 L E 2 32 B = L 32 B E 2
72 68 69 70 71 syl3anc φ L E 2 32 B = L 32 B E 2
73 9 oveq1i E 2 = U D 2
74 42 simp2d φ D +
75 74 rpcnd φ D
76 74 rpne0d φ D 0
77 64 75 76 sqdivd φ U D 2 = U 2 D 2
78 73 77 syl5eq φ E 2 = U 2 D 2
79 78 oveq2d φ L 32 B E 2 = L 32 B U 2 D 2
80 43 55 rpdivcld φ L 32 B +
81 80 rpcnd φ L 32 B
82 64 sqcld φ U 2
83 rpexpcl D + 2 D 2 +
84 74 45 83 sylancl φ D 2 +
85 84 rpcnne0d φ D 2 D 2 0
86 divass L 32 B U 2 D 2 D 2 0 L 32 B U 2 D 2 = L 32 B U 2 D 2
87 div23 L 32 B U 2 D 2 D 2 0 L 32 B U 2 D 2 = L 32 B D 2 U 2
88 86 87 eqtr3d L 32 B U 2 D 2 D 2 0 L 32 B U 2 D 2 = L 32 B D 2 U 2
89 81 82 85 88 syl3anc φ L 32 B U 2 D 2 = L 32 B D 2 U 2
90 72 79 89 3eqtrd φ L E 2 32 B = L 32 B D 2 U 2
91 90 oveq2d φ U E L E 2 32 B = U E L 32 B D 2 U 2
92 df-3 3 = 2 + 1
93 92 oveq2i U 3 = U 2 + 1
94 2nn0 2 0
95 expp1 U 2 0 U 2 + 1 = U 2 U
96 64 94 95 sylancl φ U 2 + 1 = U 2 U
97 93 96 syl5eq φ U 3 = U 2 U
98 82 64 mulcomd φ U 2 U = U U 2
99 97 98 eqtrd φ U 3 = U U 2
100 99 oveq2d φ F U 3 = F U U 2
101 42 simp3d φ F +
102 101 rpcnd φ F
103 102 64 82 mulassd φ F U U 2 = F U U 2
104 1cnd φ 1
105 74 rpreccld φ 1 D +
106 105 rpcnd φ 1 D
107 104 106 64 subdird φ 1 1 D U = 1 U 1 D U
108 64 mulid2d φ 1 U = U
109 64 75 76 divrec2d φ U D = 1 D U
110 9 109 eqtr2id φ 1 D U = E
111 108 110 oveq12d φ 1 U 1 D U = U E
112 107 111 eqtr2d φ U E = 1 1 D U
113 112 oveq1d φ U E L 32 B D 2 = 1 1 D U L 32 B D 2
114 6 oveq1i F U = 1 1 D L 32 B D 2 U
115 104 106 subcld φ 1 1 D
116 80 84 rpdivcld φ L 32 B D 2 +
117 116 rpcnd φ L 32 B D 2
118 115 117 64 mul32d φ 1 1 D L 32 B D 2 U = 1 1 D U L 32 B D 2
119 114 118 syl5eq φ F U = 1 1 D U L 32 B D 2
120 113 119 eqtr4d φ U E L 32 B D 2 = F U
121 120 oveq1d φ U E L 32 B D 2 U 2 = F U U 2
122 40 rpcnd φ U E
123 122 117 82 mulassd φ U E L 32 B D 2 U 2 = U E L 32 B D 2 U 2
124 121 123 eqtr3d φ F U U 2 = U E L 32 B D 2 U 2
125 100 103 124 3eqtr2d φ F U 3 = U E L 32 B D 2 U 2
126 91 125 eqtr4d φ U E L E 2 32 B = F U 3
127 126 oveq2d φ U U E L E 2 32 B = U F U 3
128 127 oveq1d φ U U E L E 2 32 B log Z = U F U 3 log Z
129 67 128 eqtr3d φ U log Z U E L E 2 32 B log Z = U F U 3 log Z
130 31 29 remulcld φ U log Z
131 130 59 resubcld φ U log Z U E L E 2 32 B log Z
132 129 131 eqeltrrd φ U F U 3 log Z
133 22 rpred φ Z
134 21 simp2d φ 1 < Z e Z Z Z Y
135 134 simp1d φ 1 < Z
136 133 135 rplogcld φ log Z +
137 37 136 rerpdivcld φ 2 log Z
138 fzfid φ 1 Z Y Fin
139 22 adantr φ n 1 Z Y Z +
140 elfznn n 1 Z Y n
141 140 adantl φ n 1 Z Y n
142 141 nnrpd φ n 1 Z Y n +
143 139 142 rpdivcld φ n 1 Z Y Z n +
144 23 ffvelrni Z n + R Z n
145 143 144 syl φ n 1 Z Y R Z n
146 145 139 rerpdivcld φ n 1 Z Y R Z n Z
147 146 recnd φ n 1 Z Y R Z n Z
148 147 abscld φ n 1 Z Y R Z n Z
149 142 relogcld φ n 1 Z Y log n
150 148 149 remulcld φ n 1 Z Y R Z n Z log n
151 138 150 fsumrecl φ n = 1 Z Y R Z n Z log n
152 137 151 remulcld φ 2 log Z n = 1 Z Y R Z n Z log n
153 152 62 readdcld φ 2 log Z n = 1 Z Y R Z n Z log n + C
154 25 recnd φ R Z
155 154 abscld φ R Z
156 155 recnd φ R Z
157 156 66 mulcld φ R Z log Z
158 137 recnd φ 2 log Z
159 145 recnd φ n 1 Z Y R Z n
160 159 abscld φ n 1 Z Y R Z n
161 160 149 remulcld φ n 1 Z Y R Z n log n
162 138 161 fsumrecl φ n = 1 Z Y R Z n log n
163 162 recnd φ n = 1 Z Y R Z n log n
164 158 163 mulcld φ 2 log Z n = 1 Z Y R Z n log n
165 22 rpcnd φ Z
166 22 rpne0d φ Z 0
167 157 164 165 166 divsubdird φ R Z log Z 2 log Z n = 1 Z Y R Z n log n Z = R Z log Z Z 2 log Z n = 1 Z Y R Z n log n Z
168 156 66 165 166 div23d φ R Z log Z Z = R Z Z log Z
169 154 165 166 absdivd φ R Z Z = R Z Z
170 22 rprege0d φ Z 0 Z
171 absid Z 0 Z Z = Z
172 170 171 syl φ Z = Z
173 172 oveq2d φ R Z Z = R Z Z
174 169 173 eqtrd φ R Z Z = R Z Z
175 174 oveq1d φ R Z Z log Z = R Z Z log Z
176 168 175 eqtr4d φ R Z log Z Z = R Z Z log Z
177 158 163 165 166 divassd φ 2 log Z n = 1 Z Y R Z n log n Z = 2 log Z n = 1 Z Y R Z n log n Z
178 165 adantr φ n 1 Z Y Z
179 166 adantr φ n 1 Z Y Z 0
180 159 178 179 absdivd φ n 1 Z Y R Z n Z = R Z n Z
181 172 adantr φ n 1 Z Y Z = Z
182 181 oveq2d φ n 1 Z Y R Z n Z = R Z n Z
183 180 182 eqtrd φ n 1 Z Y R Z n Z = R Z n Z
184 183 oveq1d φ n 1 Z Y R Z n Z log n = R Z n Z log n
185 160 recnd φ n 1 Z Y R Z n
186 149 recnd φ n 1 Z Y log n
187 22 rpcnne0d φ Z Z 0
188 187 adantr φ n 1 Z Y Z Z 0
189 div23 R Z n log n Z Z 0 R Z n log n Z = R Z n Z log n
190 185 186 188 189 syl3anc φ n 1 Z Y R Z n log n Z = R Z n Z log n
191 184 190 eqtr4d φ n 1 Z Y R Z n Z log n = R Z n log n Z
192 191 sumeq2dv φ n = 1 Z Y R Z n Z log n = n = 1 Z Y R Z n log n Z
193 161 recnd φ n 1 Z Y R Z n log n
194 138 165 193 166 fsumdivc φ n = 1 Z Y R Z n log n Z = n = 1 Z Y R Z n log n Z
195 192 194 eqtr4d φ n = 1 Z Y R Z n Z log n = n = 1 Z Y R Z n log n Z
196 195 oveq2d φ 2 log Z n = 1 Z Y R Z n Z log n = 2 log Z n = 1 Z Y R Z n log n Z
197 177 196 eqtr4d φ 2 log Z n = 1 Z Y R Z n log n Z = 2 log Z n = 1 Z Y R Z n Z log n
198 176 197 oveq12d φ R Z log Z Z 2 log Z n = 1 Z Y R Z n log n Z = R Z Z log Z 2 log Z n = 1 Z Y R Z n Z log n
199 167 198 eqtrd φ R Z log Z 2 log Z n = 1 Z Y R Z n log n Z = R Z Z log Z 2 log Z n = 1 Z Y R Z n Z log n
200 2fveq3 z = Z R z = R Z
201 fveq2 z = Z log z = log Z
202 200 201 oveq12d z = Z R z log z = R Z log Z
203 201 oveq2d z = Z 2 log z = 2 log Z
204 oveq2 i = n z i = z n
205 204 fveq2d i = n R z i = R z n
206 205 fveq2d i = n R z i = R z n
207 fveq2 i = n log i = log n
208 206 207 oveq12d i = n R z i log i = R z n log n
209 208 cbvsumv i = 1 z Y R z i log i = n = 1 z Y R z n log n
210 fvoveq1 z = Z z Y = Z Y
211 210 oveq2d z = Z 1 z Y = 1 Z Y
212 simpl z = Z n 1 Z Y z = Z
213 212 fvoveq1d z = Z n 1 Z Y R z n = R Z n
214 213 fveq2d z = Z n 1 Z Y R z n = R Z n
215 214 oveq1d z = Z n 1 Z Y R z n log n = R Z n log n
216 211 215 sumeq12rdv z = Z n = 1 z Y R z n log n = n = 1 Z Y R Z n log n
217 209 216 syl5eq z = Z i = 1 z Y R z i log i = n = 1 Z Y R Z n log n
218 203 217 oveq12d z = Z 2 log z i = 1 z Y R z i log i = 2 log Z n = 1 Z Y R Z n log n
219 202 218 oveq12d z = Z R z log z 2 log z i = 1 z Y R z i log i = R Z log Z 2 log Z n = 1 Z Y R Z n log n
220 id z = Z z = Z
221 219 220 oveq12d z = Z R z log z 2 log z i = 1 z Y R z i log i z = R Z log Z 2 log Z n = 1 Z Y R Z n log n Z
222 221 breq1d z = Z R z log z 2 log z i = 1 z Y R z i log i z C R Z log Z 2 log Z n = 1 Z Y R Z n log n Z C
223 1re 1
224 rexr 1 1 *
225 elioopnf 1 * Z 1 +∞ Z 1 < Z
226 223 224 225 mp2b Z 1 +∞ Z 1 < Z
227 133 135 226 sylanbrc φ Z 1 +∞
228 222 20 227 rspcdva φ R Z log Z 2 log Z n = 1 Z Y R Z n log n Z C
229 199 228 eqbrtrrd φ R Z Z log Z 2 log Z n = 1 Z Y R Z n Z log n C
230 30 152 62 lesubadd2d φ R Z Z log Z 2 log Z n = 1 Z Y R Z n Z log n C R Z Z log Z 2 log Z n = 1 Z Y R Z n Z log n + C
231 229 230 mpbid φ R Z Z log Z 2 log Z n = 1 Z Y R Z n Z log n + C
232 2cnd φ 2
233 148 recnd φ n 1 Z Y R Z n Z
234 233 186 mulcld φ n 1 Z Y R Z n Z log n
235 138 234 fsumcl φ n = 1 Z Y R Z n Z log n
236 136 rpne0d φ log Z 0
237 232 235 66 236 div23d φ 2 n = 1 Z Y R Z n Z log n log Z = 2 log Z n = 1 Z Y R Z n Z log n
238 29 resqcld φ log Z 2
239 57 238 remulcld φ L E 2 32 B log Z 2
240 41 239 remulcld φ U E L E 2 32 B log Z 2
241 remulcl 2 U E L E 2 32 B log Z 2 2 U E L E 2 32 B log Z 2
242 36 240 241 sylancr φ 2 U E L E 2 32 B log Z 2
243 35 29 remulcld φ U log Z + 3 log Z
244 remulcl 2 n = 1 Z Y R Z n Z log n 2 n = 1 Z Y R Z n Z log n
245 36 151 244 sylancr φ 2 n = 1 Z Y R Z n Z log n
246 31 adantr φ n 1 Z Y U
247 246 141 nndivred φ n 1 Z Y U n
248 247 148 resubcld φ n 1 Z Y U n R Z n Z
249 248 149 remulcld φ n 1 Z Y U n R Z n Z log n
250 138 249 fsumrecl φ n = 1 Z Y U n R Z n Z log n
251 37 250 remulcld φ 2 n = 1 Z Y U n R Z n Z log n
252 243 245 resubcld φ U log Z + 3 log Z 2 n = 1 Z Y R Z n Z log n
253 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 pntlemf φ U E L E 2 32 B log Z 2 n = 1 Z Y U n R Z n Z log n
254 2pos 0 < 2
255 254 a1i φ 0 < 2
256 lemul2 U E L E 2 32 B log Z 2 n = 1 Z Y U n R Z n Z log n 2 0 < 2 U E L E 2 32 B log Z 2 n = 1 Z Y U n R Z n Z log n 2 U E L E 2 32 B log Z 2 2 n = 1 Z Y U n R Z n Z log n
257 240 250 37 255 256 syl112anc φ U E L E 2 32 B log Z 2 n = 1 Z Y U n R Z n Z log n 2 U E L E 2 32 B log Z 2 2 n = 1 Z Y U n R Z n Z log n
258 253 257 mpbid φ 2 U E L E 2 32 B log Z 2 2 n = 1 Z Y U n R Z n Z log n
259 247 recnd φ n 1 Z Y U n
260 259 233 186 subdird φ n 1 Z Y U n R Z n Z log n = U n log n R Z n Z log n
261 260 sumeq2dv φ n = 1 Z Y U n R Z n Z log n = n = 1 Z Y U n log n R Z n Z log n
262 247 149 remulcld φ n 1 Z Y U n log n
263 262 recnd φ n 1 Z Y U n log n
264 138 263 234 fsumsub φ n = 1 Z Y U n log n R Z n Z log n = n = 1 Z Y U n log n n = 1 Z Y R Z n Z log n
265 261 264 eqtrd φ n = 1 Z Y U n R Z n Z log n = n = 1 Z Y U n log n n = 1 Z Y R Z n Z log n
266 265 oveq2d φ 2 n = 1 Z Y U n R Z n Z log n = 2 n = 1 Z Y U n log n n = 1 Z Y R Z n Z log n
267 138 262 fsumrecl φ n = 1 Z Y U n log n
268 267 recnd φ n = 1 Z Y U n log n
269 232 268 235 subdid φ 2 n = 1 Z Y U n log n n = 1 Z Y R Z n Z log n = 2 n = 1 Z Y U n log n 2 n = 1 Z Y R Z n Z log n
270 266 269 eqtrd φ 2 n = 1 Z Y U n R Z n Z log n = 2 n = 1 Z Y U n log n 2 n = 1 Z Y R Z n Z log n
271 remulcl 2 n = 1 Z Y U n log n 2 n = 1 Z Y U n log n
272 36 267 271 sylancr φ 2 n = 1 Z Y U n log n
273 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 pntlemk φ 2 n = 1 Z Y U n log n U log Z + 3 log Z
274 272 243 245 273 lesub1dd φ 2 n = 1 Z Y U n log n 2 n = 1 Z Y R Z n Z log n U log Z + 3 log Z 2 n = 1 Z Y R Z n Z log n
275 270 274 eqbrtrd φ 2 n = 1 Z Y U n R Z n Z log n U log Z + 3 log Z 2 n = 1 Z Y R Z n Z log n
276 242 251 252 258 275 letrd φ 2 U E L E 2 32 B log Z 2 U log Z + 3 log Z 2 n = 1 Z Y R Z n Z log n
277 242 243 245 276 lesubd φ 2 n = 1 Z Y R Z n Z log n U log Z + 3 log Z 2 U E L E 2 32 B log Z 2
278 35 recnd φ U log Z + 3
279 60 recnd φ 2 U E L E 2 32 B log Z
280 278 279 66 subdird φ U log Z + 3 2 U E L E 2 32 B log Z log Z = U log Z + 3 log Z 2 U E L E 2 32 B log Z log Z
281 59 recnd φ U E L E 2 32 B log Z
282 232 281 66 mulassd φ 2 U E L E 2 32 B log Z log Z = 2 U E L E 2 32 B log Z log Z
283 65 66 66 mulassd φ U E L E 2 32 B log Z log Z = U E L E 2 32 B log Z log Z
284 66 sqvald φ log Z 2 = log Z log Z
285 284 oveq2d φ U E L E 2 32 B log Z 2 = U E L E 2 32 B log Z log Z
286 56 rpcnd φ L E 2 32 B
287 238 recnd φ log Z 2
288 122 286 287 mulassd φ U E L E 2 32 B log Z 2 = U E L E 2 32 B log Z 2
289 283 285 288 3eqtr2d φ U E L E 2 32 B log Z log Z = U E L E 2 32 B log Z 2
290 289 oveq2d φ 2 U E L E 2 32 B log Z log Z = 2 U E L E 2 32 B log Z 2
291 282 290 eqtrd φ 2 U E L E 2 32 B log Z log Z = 2 U E L E 2 32 B log Z 2
292 291 oveq2d φ U log Z + 3 log Z 2 U E L E 2 32 B log Z log Z = U log Z + 3 log Z 2 U E L E 2 32 B log Z 2
293 280 292 eqtrd φ U log Z + 3 2 U E L E 2 32 B log Z log Z = U log Z + 3 log Z 2 U E L E 2 32 B log Z 2
294 277 293 breqtrrd φ 2 n = 1 Z Y R Z n Z log n U log Z + 3 2 U E L E 2 32 B log Z log Z
295 245 61 136 ledivmul2d φ 2 n = 1 Z Y R Z n Z log n log Z U log Z + 3 2 U E L E 2 32 B log Z 2 n = 1 Z Y R Z n Z log n U log Z + 3 2 U E L E 2 32 B log Z log Z
296 294 295 mpbird φ 2 n = 1 Z Y R Z n Z log n log Z U log Z + 3 2 U E L E 2 32 B log Z
297 237 296 eqbrtrrd φ 2 log Z n = 1 Z Y R Z n Z log n U log Z + 3 2 U E L E 2 32 B log Z
298 152 61 62 297 leadd1dd φ 2 log Z n = 1 Z Y R Z n Z log n + C U log Z + 3 - 2 U E L E 2 32 B log Z + C
299 30 153 63 231 298 letrd φ R Z Z log Z U log Z + 3 - 2 U E L E 2 32 B log Z + C
300 remulcl U 3 U 3
301 31 32 300 sylancl φ U 3
302 301 62 readdcld φ U 3 + C
303 21 simp3d φ 4 L E Z log X log K + 2 log Z log K 4 U 3 + C U E L E 2 32 B log Z
304 303 simp3d φ U 3 + C U E L E 2 32 B log Z
305 302 59 130 304 leadd2dd φ U log Z + U 3 + C U log Z + U E L E 2 32 B log Z
306 33 recnd φ 3
307 64 66 306 adddid φ U log Z + 3 = U log Z + U 3
308 307 oveq1d φ U log Z + 3 + C = U log Z + U 3 + C
309 130 recnd φ U log Z
310 64 306 mulcld φ U 3
311 13 rpcnd φ C
312 309 310 311 addassd φ U log Z + U 3 + C = U log Z + U 3 + C
313 308 312 eqtrd φ U log Z + 3 + C = U log Z + U 3 + C
314 281 2timesd φ 2 U E L E 2 32 B log Z = U E L E 2 32 B log Z + U E L E 2 32 B log Z
315 314 oveq2d φ U log Z - U E L E 2 32 B log Z + 2 U E L E 2 32 B log Z = U log Z U E L E 2 32 B log Z + U E L E 2 32 B log Z + U E L E 2 32 B log Z
316 309 281 281 nppcan3d φ U log Z U E L E 2 32 B log Z + U E L E 2 32 B log Z + U E L E 2 32 B log Z = U log Z + U E L E 2 32 B log Z
317 315 316 eqtrd φ U log Z - U E L E 2 32 B log Z + 2 U E L E 2 32 B log Z = U log Z + U E L E 2 32 B log Z
318 305 313 317 3brtr4d φ U log Z + 3 + C U log Z - U E L E 2 32 B log Z + 2 U E L E 2 32 B log Z
319 35 62 readdcld φ U log Z + 3 + C
320 319 60 131 lesubaddd φ U log Z + 3 + C - 2 U E L E 2 32 B log Z U log Z U E L E 2 32 B log Z U log Z + 3 + C U log Z - U E L E 2 32 B log Z + 2 U E L E 2 32 B log Z
321 318 320 mpbird φ U log Z + 3 + C - 2 U E L E 2 32 B log Z U log Z U E L E 2 32 B log Z
322 278 311 279 addsubd φ U log Z + 3 + C - 2 U E L E 2 32 B log Z = U log Z + 3 - 2 U E L E 2 32 B log Z + C
323 321 322 129 3brtr3d φ U log Z + 3 - 2 U E L E 2 32 B log Z + C U F U 3 log Z
324 30 63 132 299 323 letrd φ R Z Z log Z U F U 3 log Z
325 3z 3
326 rpexpcl U + 3 U 3 +
327 7 325 326 sylancl φ U 3 +
328 101 327 rpmulcld φ F U 3 +
329 328 rpred φ F U 3
330 31 329 resubcld φ U F U 3
331 28 330 136 lemul1d φ R Z Z U F U 3 R Z Z log Z U F U 3 log Z
332 324 331 mpbird φ R Z Z U F U 3