Step |
Hyp |
Ref |
Expression |
1 |
|
sge0tsms.g |
|
2 |
|
sge0tsms.x |
|
3 |
|
sge0tsms.f |
|
4 |
|
eqid |
|
5 |
4
|
a1i |
|
6 |
|
xrltso |
|
7 |
6
|
supex |
|
8 |
7
|
a1i |
|
9 |
|
elsng |
|
10 |
8 9
|
syl |
|
11 |
5 10
|
mpbird |
|
12 |
2
|
adantr |
|
13 |
3
|
adantr |
|
14 |
|
simpr |
|
15 |
12 13 14
|
sge0pnfval |
|
16 |
3
|
ffnd |
|
17 |
16
|
adantr |
|
18 |
|
fvelrnb |
|
19 |
17 18
|
syl |
|
20 |
14 19
|
mpbid |
|
21 |
|
iccssxr |
|
22 |
|
simpr |
|
23 |
3
|
adantr |
|
24 |
|
elinel1 |
|
25 |
|
elpwi |
|
26 |
24 25
|
syl |
|
27 |
26
|
adantl |
|
28 |
|
fssres |
|
29 |
23 27 28
|
syl2anc |
|
30 |
|
elinel2 |
|
31 |
30
|
adantl |
|
32 |
|
0red |
|
33 |
29 31 32
|
fdmfifsupp |
|
34 |
1 22 29 33
|
gsumge0cl |
|
35 |
21 34
|
sseldi |
|
36 |
35
|
ralrimiva |
|
37 |
36
|
3ad2ant1 |
|
38 |
|
eqid |
|
39 |
38
|
rnmptss |
|
40 |
37 39
|
syl |
|
41 |
|
snelpwi |
|
42 |
|
snfi |
|
43 |
42
|
a1i |
|
44 |
41 43
|
elind |
|
45 |
44
|
3ad2ant2 |
|
46 |
3
|
adantr |
|
47 |
|
snssi |
|
48 |
47
|
adantl |
|
49 |
46 48
|
fssresd |
|
50 |
49
|
feqmptd |
|
51 |
|
fvres |
|
52 |
51
|
mpteq2ia |
|
53 |
52
|
a1i |
|
54 |
50 53
|
eqtrd |
|
55 |
54
|
oveq2d |
|
56 |
55
|
3adant3 |
|
57 |
|
xrge0cmn |
|
58 |
1 57
|
eqeltri |
|
59 |
|
cmnmnd |
|
60 |
58 59
|
ax-mp |
|
61 |
60
|
a1i |
|
62 |
|
simp2 |
|
63 |
3
|
ffvelrnda |
|
64 |
63
|
3adant3 |
|
65 |
|
df-ss |
|
66 |
21 65
|
mpbi |
|
67 |
66
|
eqcomi |
|
68 |
|
ovex |
|
69 |
|
xrsbas |
|
70 |
1 69
|
ressbas |
|
71 |
68 70
|
ax-mp |
|
72 |
67 71
|
eqtri |
|
73 |
|
fveq2 |
|
74 |
72 73
|
gsumsn |
|
75 |
61 62 64 74
|
syl3anc |
|
76 |
|
simp3 |
|
77 |
56 75 76
|
3eqtrrd |
|
78 |
|
reseq2 |
|
79 |
78
|
oveq2d |
|
80 |
79
|
rspceeqv |
|
81 |
45 77 80
|
syl2anc |
|
82 |
|
pnfxr |
|
83 |
82
|
a1i |
|
84 |
38
|
elrnmpt |
|
85 |
83 84
|
syl |
|
86 |
81 85
|
mpbird |
|
87 |
|
supxrpnf |
|
88 |
40 86 87
|
syl2anc |
|
89 |
88
|
3exp |
|
90 |
89
|
adantr |
|
91 |
90
|
rexlimdv |
|
92 |
20 91
|
mpd |
|
93 |
15 92
|
eqtr4d |
|
94 |
2
|
adantr |
|
95 |
3
|
adantr |
|
96 |
|
simpr |
|
97 |
95 96
|
fge0iccico |
|
98 |
94 97
|
sge0reval |
|
99 |
23 27
|
feqresmpt |
|
100 |
99
|
adantlr |
|
101 |
100
|
oveq2d |
|
102 |
1
|
fveq2i |
|
103 |
|
eqid |
|
104 |
|
xrsadd |
|
105 |
103 104
|
ressplusg |
|
106 |
68 105
|
ax-mp |
|
107 |
106
|
eqcomi |
|
108 |
102 107
|
eqtr2i |
|
109 |
1
|
oveq1i |
|
110 |
|
icossicc |
|
111 |
68 110
|
pm3.2i |
|
112 |
|
ressabs |
|
113 |
111 112
|
ax-mp |
|
114 |
109 113
|
eqtr2i |
|
115 |
58
|
elexi |
|
116 |
115
|
a1i |
|
117 |
|
simpr |
|
118 |
110
|
a1i |
|
119 |
|
0xr |
|
120 |
119
|
a1i |
|
121 |
82
|
a1i |
|
122 |
95
|
ad2antrr |
|
123 |
26
|
sselda |
|
124 |
123
|
adantll |
|
125 |
122 124
|
ffvelrnd |
|
126 |
21 125
|
sseldi |
|
127 |
|
iccgelb |
|
128 |
120 121 125 127
|
syl3anc |
|
129 |
|
id |
|
130 |
129
|
eqcomd |
|
131 |
130
|
adantl |
|
132 |
3
|
ffund |
|
133 |
132
|
ad2antrr |
|
134 |
22 123
|
sylan |
|
135 |
3
|
fdmd |
|
136 |
135
|
eqcomd |
|
137 |
136
|
ad2antrr |
|
138 |
134 137
|
eleqtrd |
|
139 |
|
fvelrn |
|
140 |
133 138 139
|
syl2anc |
|
141 |
140
|
adantr |
|
142 |
131 141
|
eqeltrd |
|
143 |
142
|
adantl3r |
|
144 |
96
|
ad3antrrr |
|
145 |
143 144
|
pm2.65da |
|
146 |
145
|
neqned |
|
147 |
|
ge0xrre |
|
148 |
125 146 147
|
syl2anc |
|
149 |
148
|
ltpnfd |
|
150 |
120 121 126 128 149
|
elicod |
|
151 |
|
eqid |
|
152 |
150 151
|
fmptd |
|
153 |
|
0e0icopnf |
|
154 |
153
|
a1i |
|
155 |
|
eliccxr |
|
156 |
|
xaddid2 |
|
157 |
|
xaddid1 |
|
158 |
156 157
|
jca |
|
159 |
155 158
|
syl |
|
160 |
159
|
adantl |
|
161 |
72 108 114 116 117 118 152 154 160
|
gsumress |
|
162 |
|
rege0subm |
|
163 |
162
|
a1i |
|
164 |
|
eqid |
|
165 |
117 163 152 164
|
gsumsubm |
|
166 |
|
eqidd |
|
167 |
|
vex |
|
168 |
167
|
mptex |
|
169 |
168
|
a1i |
|
170 |
|
ovexd |
|
171 |
|
ovexd |
|
172 |
|
rge0ssre |
|
173 |
|
ax-resscn |
|
174 |
172 173
|
sstri |
|
175 |
|
cnfldbas |
|
176 |
164 175
|
ressbas2 |
|
177 |
174 176
|
ax-mp |
|
178 |
177
|
eqcomi |
|
179 |
110 21
|
sstri |
|
180 |
|
eqid |
|
181 |
180 69
|
ressbas2 |
|
182 |
179 181
|
ax-mp |
|
183 |
178 182
|
eqtri |
|
184 |
183
|
a1i |
|
185 |
|
rge0srg |
|
186 |
185
|
a1i |
|
187 |
|
simpl |
|
188 |
|
simpr |
|
189 |
|
eqid |
|
190 |
|
eqid |
|
191 |
189 190
|
srgacl |
|
192 |
186 187 188 191
|
syl3anc |
|
193 |
192
|
adantl |
|
194 |
172
|
a1i |
|
195 |
|
id |
|
196 |
195 178
|
eleqtrdi |
|
197 |
194 196
|
sseldd |
|
198 |
197
|
adantr |
|
199 |
172
|
a1i |
|
200 |
|
id |
|
201 |
200 178
|
eleqtrdi |
|
202 |
199 201
|
sseldd |
|
203 |
202
|
adantl |
|
204 |
|
rexadd |
|
205 |
204
|
eqcomd |
|
206 |
162
|
elexi |
|
207 |
|
cnfldadd |
|
208 |
164 207
|
ressplusg |
|
209 |
206 208
|
ax-mp |
|
210 |
209 207
|
eqtr3i |
|
211 |
210 207
|
eqtr4i |
|
212 |
211
|
oveqi |
|
213 |
212
|
a1i |
|
214 |
180 104
|
ressplusg |
|
215 |
206 214
|
ax-mp |
|
216 |
215
|
eqcomi |
|
217 |
216
|
oveqi |
|
218 |
217
|
a1i |
|
219 |
205 213 218
|
3eqtr4d |
|
220 |
198 203 219
|
syl2anc |
|
221 |
220
|
adantl |
|
222 |
|
funmpt |
|
223 |
222
|
a1i |
|
224 |
150 177
|
eleqtrdi |
|
225 |
224
|
ralrimiva |
|
226 |
151
|
rnmptss |
|
227 |
225 226
|
syl |
|
228 |
169 170 171 184 193 221 223 227
|
gsumpropd2 |
|
229 |
165 166 228
|
3eqtrd |
|
230 |
30
|
adantl |
|
231 |
148
|
recnd |
|
232 |
230 231
|
gsumfsum |
|
233 |
229 232
|
eqtr3d |
|
234 |
101 161 233
|
3eqtrrd |
|
235 |
234
|
mpteq2dva |
|
236 |
235
|
rneqd |
|
237 |
236
|
supeq1d |
|
238 |
98 237
|
eqtrd |
|
239 |
93 238
|
pm2.61dan |
|
240 |
1 2 3 4
|
xrge0tsms |
|
241 |
239 240
|
eleq12d |
|
242 |
11 241
|
mpbird |
|