Step |
Hyp |
Ref |
Expression |
1 |
|
sge0split.a |
|
2 |
|
sge0split.b |
|
3 |
|
sge0split.u |
|
4 |
|
sge0split.in0 |
|
5 |
|
sge0split.f |
|
6 |
1
|
adantr |
|
7 |
2
|
adantr |
|
8 |
4
|
adantr |
|
9 |
5
|
adantr |
|
10 |
|
simpr |
|
11 |
6 7 3 8 9 10
|
sge0resplit |
|
12 |
|
unexg |
|
13 |
1 2 12
|
syl2anc |
|
14 |
3 13
|
eqeltrid |
|
15 |
14
|
adantr |
|
16 |
15 9 10
|
sge0ssre |
|
17 |
15 9 10
|
sge0ssre |
|
18 |
|
rexadd |
|
19 |
16 17 18
|
syl2anc |
|
20 |
19
|
eqcomd |
|
21 |
11 20
|
eqtrd |
|
22 |
|
simpl |
|
23 |
|
simpr |
|
24 |
14 5
|
sge0repnf |
|
25 |
24
|
adantr |
|
26 |
23 25
|
mtbid |
|
27 |
26
|
notnotrd |
|
28 |
14 5
|
sge0xrcl |
|
29 |
28
|
adantr |
|
30 |
|
ssun1 |
|
31 |
30 3
|
sseqtrri |
|
32 |
31
|
a1i |
|
33 |
5 32
|
fssresd |
|
34 |
1 33
|
sge0xrcl |
|
35 |
|
iccssxr |
|
36 |
|
ssun2 |
|
37 |
36 3
|
sseqtrri |
|
38 |
37
|
a1i |
|
39 |
5 38
|
fssresd |
|
40 |
2 39
|
sge0cl |
|
41 |
35 40
|
sseldi |
|
42 |
34 41
|
xaddcld |
|
43 |
42
|
adantr |
|
44 |
|
pnfxr |
|
45 |
|
eqid |
|
46 |
|
xreqle |
|
47 |
44 45 46
|
mp2an |
|
48 |
47
|
a1i |
|
49 |
14
|
adantr |
|
50 |
5
|
adantr |
|
51 |
|
rnresss |
|
52 |
51
|
sseli |
|
53 |
52
|
adantl |
|
54 |
49 50 53
|
sge0pnfval |
|
55 |
|
xrge0neqmnf |
|
56 |
40 55
|
syl |
|
57 |
|
xaddpnf2 |
|
58 |
41 56 57
|
syl2anc |
|
59 |
58
|
eqcomd |
|
60 |
59
|
adantr |
|
61 |
1
|
adantr |
|
62 |
33
|
adantr |
|
63 |
|
simpr |
|
64 |
61 62 63
|
sge0pnfval |
|
65 |
64
|
oveq1d |
|
66 |
60 54 65
|
3eqtr4d |
|
67 |
66 54
|
eqtr3d |
|
68 |
54 67
|
breq12d |
|
69 |
48 68
|
mpbird |
|
70 |
47
|
a1i |
|
71 |
14
|
adantr |
|
72 |
5
|
adantr |
|
73 |
|
rnresss |
|
74 |
73
|
sseli |
|
75 |
74
|
adantl |
|
76 |
71 72 75
|
sge0pnfval |
|
77 |
2
|
adantr |
|
78 |
39
|
adantr |
|
79 |
|
simpr |
|
80 |
77 78 79
|
sge0pnfval |
|
81 |
80
|
oveq2d |
|
82 |
1 33
|
sge0cl |
|
83 |
|
xrge0neqmnf |
|
84 |
82 83
|
syl |
|
85 |
|
xaddpnf1 |
|
86 |
34 84 85
|
syl2anc |
|
87 |
86
|
adantr |
|
88 |
81 87
|
eqtrd |
|
89 |
76 88
|
breq12d |
|
90 |
70 89
|
mpbird |
|
91 |
90
|
adantlr |
|
92 |
|
simpr |
|
93 |
|
vex |
|
94 |
|
eqid |
|
95 |
94
|
elrnmpt |
|
96 |
93 95
|
ax-mp |
|
97 |
92 96
|
sylib |
|
98 |
|
simp3 |
|
99 |
|
inss1 |
|
100 |
|
inss2 |
|
101 |
99 100
|
sstri |
|
102 |
|
inss2 |
|
103 |
|
inss2 |
|
104 |
102 103
|
sstri |
|
105 |
101 104
|
ssini |
|
106 |
105
|
a1i |
|
107 |
106 4
|
sseqtrd |
|
108 |
|
ss0 |
|
109 |
107 108
|
syl |
|
110 |
109
|
ad3antrrr |
|
111 |
|
indi |
|
112 |
111
|
eqcomi |
|
113 |
112
|
a1i |
|
114 |
3
|
eqcomi |
|
115 |
114
|
ineq2i |
|
116 |
115
|
a1i |
|
117 |
|
elinel1 |
|
118 |
|
elpwi |
|
119 |
117 118
|
syl |
|
120 |
|
df-ss |
|
121 |
120
|
biimpi |
|
122 |
119 121
|
syl |
|
123 |
113 116 122
|
3eqtrrd |
|
124 |
123
|
adantl |
|
125 |
|
elinel2 |
|
126 |
125
|
adantl |
|
127 |
|
rge0ssre |
|
128 |
5
|
ad2antrr |
|
129 |
|
pm4.56 |
|
130 |
129
|
biimpi |
|
131 |
|
elun |
|
132 |
130 131
|
sylnibr |
|
133 |
132
|
adantll |
|
134 |
|
rnresun |
|
135 |
134
|
eqcomi |
|
136 |
135
|
a1i |
|
137 |
114
|
reseq2i |
|
138 |
137
|
rneqi |
|
139 |
138
|
a1i |
|
140 |
|
ffn |
|
141 |
|
fnresdm |
|
142 |
5 140 141
|
3syl |
|
143 |
142
|
rneqd |
|
144 |
136 139 143
|
3eqtrd |
|
145 |
144
|
ad2antrr |
|
146 |
133 145
|
neleqtrd |
|
147 |
128 146
|
fge0iccico |
|
148 |
147
|
ad2antrr |
|
149 |
119
|
adantr |
|
150 |
|
simpr |
|
151 |
149 150
|
sseldd |
|
152 |
151
|
adantll |
|
153 |
148 152
|
ffvelrnd |
|
154 |
127 153
|
sseldi |
|
155 |
154
|
recnd |
|
156 |
110 124 126 155
|
fsumsplit |
|
157 |
|
infi |
|
158 |
125 157
|
syl |
|
159 |
158
|
adantl |
|
160 |
|
simpl |
|
161 |
|
elinel1 |
|
162 |
161
|
adantl |
|
163 |
160 162 154
|
syl2anc |
|
164 |
159 163
|
fsumrecl |
|
165 |
|
infi |
|
166 |
125 165
|
syl |
|
167 |
166
|
adantl |
|
168 |
|
simpl |
|
169 |
|
elinel1 |
|
170 |
169
|
adantl |
|
171 |
168 170 154
|
syl2anc |
|
172 |
167 171
|
fsumrecl |
|
173 |
|
rexadd |
|
174 |
164 172 173
|
syl2anc |
|
175 |
174
|
eqcomd |
|
176 |
156 175
|
eqtrd |
|
177 |
|
ressxr |
|
178 |
177 164
|
sseldi |
|
179 |
177 172
|
sseldi |
|
180 |
1
|
adantr |
|
181 |
33
|
adantr |
|
182 |
|
simpr |
|
183 |
181 182
|
fge0iccico |
|
184 |
180 183
|
sge0reval |
|
185 |
184
|
eqcomd |
|
186 |
34
|
adantr |
|
187 |
185 186
|
eqeltrd |
|
188 |
187
|
adantr |
|
189 |
2
|
adantr |
|
190 |
39
|
adantr |
|
191 |
|
simpr |
|
192 |
190 191
|
fge0iccico |
|
193 |
189 192
|
sge0reval |
|
194 |
193
|
eqcomd |
|
195 |
41
|
adantr |
|
196 |
194 195
|
eqeltrd |
|
197 |
196
|
adantlr |
|
198 |
188 197
|
jca |
|
199 |
198
|
adantr |
|
200 |
178 179 199
|
jca31 |
|
201 |
180
|
adantr |
|
202 |
181
|
adantr |
|
203 |
182
|
adantr |
|
204 |
202 203
|
fge0iccico |
|
205 |
100
|
a1i |
|
206 |
158
|
adantl |
|
207 |
201 204 205 206
|
fsumlesge0 |
|
208 |
100
|
sseli |
|
209 |
|
fvres |
|
210 |
208 209
|
syl |
|
211 |
210
|
adantl |
|
212 |
211
|
sumeq2dv |
|
213 |
184
|
adantr |
|
214 |
212 213
|
breq12d |
|
215 |
207 214
|
mpbid |
|
216 |
215
|
adantlr |
|
217 |
189
|
adantr |
|
218 |
190
|
adantr |
|
219 |
191
|
adantr |
|
220 |
218 219
|
fge0iccico |
|
221 |
103
|
a1i |
|
222 |
166
|
adantl |
|
223 |
217 220 221 222
|
fsumlesge0 |
|
224 |
103
|
sseli |
|
225 |
|
fvres |
|
226 |
224 225
|
syl |
|
227 |
226
|
adantl |
|
228 |
227
|
sumeq2dv |
|
229 |
193
|
adantr |
|
230 |
228 229
|
breq12d |
|
231 |
223 230
|
mpbid |
|
232 |
231
|
adantllr |
|
233 |
216 232
|
jca |
|
234 |
|
xle2add |
|
235 |
200 233 234
|
sylc |
|
236 |
176 235
|
eqbrtrd |
|
237 |
236
|
3adant3 |
|
238 |
98 237
|
eqbrtrd |
|
239 |
238
|
3exp |
|
240 |
239
|
rexlimdv |
|
241 |
240
|
adantr |
|
242 |
97 241
|
mpd |
|
243 |
242
|
ralrimiva |
|
244 |
147
|
sge0rnre |
|
245 |
177
|
a1i |
|
246 |
244 245
|
sstrd |
|
247 |
188 197
|
xaddcld |
|
248 |
|
supxrleub |
|
249 |
246 247 248
|
syl2anc |
|
250 |
243 249
|
mpbird |
|
251 |
14
|
ad2antrr |
|
252 |
251 147
|
sge0reval |
|
253 |
184
|
adantr |
|
254 |
193
|
adantlr |
|
255 |
253 254
|
oveq12d |
|
256 |
250 252 255
|
3brtr4d |
|
257 |
91 256
|
pm2.61dan |
|
258 |
69 257
|
pm2.61dan |
|
259 |
258
|
adantr |
|
260 |
|
pnfge |
|
261 |
42 260
|
syl |
|
262 |
261
|
adantr |
|
263 |
|
id |
|
264 |
263
|
eqcomd |
|
265 |
264
|
adantl |
|
266 |
262 265
|
breqtrd |
|
267 |
29 43 259 266
|
xrletrid |
|
268 |
22 27 267
|
syl2anc |
|
269 |
21 268
|
pm2.61dan |
|