Step |
Hyp |
Ref |
Expression |
1 |
|
rlimcnp.a |
|
2 |
|
rlimcnp.0 |
|
3 |
|
rlimcnp.b |
|
4 |
|
rlimcnp.r |
|
5 |
|
rlimcnp.d |
|
6 |
|
rlimcnp.c |
|
7 |
|
rlimcnp.s |
|
8 |
|
rlimcnp.j |
|
9 |
|
rlimcnp.k |
|
10 |
|
rpreccl |
|
11 |
10
|
adantl |
|
12 |
|
rpreccl |
|
13 |
|
rpcnne0 |
|
14 |
13
|
adantl |
|
15 |
|
recrec |
|
16 |
14 15
|
syl |
|
17 |
16
|
eqcomd |
|
18 |
|
oveq2 |
|
19 |
18
|
rspceeqv |
|
20 |
12 17 19
|
syl2an2 |
|
21 |
|
simpr |
|
22 |
21
|
breq1d |
|
23 |
22
|
imbi1d |
|
24 |
23
|
ralbidv |
|
25 |
11 20 24
|
rexxfrd |
|
26 |
25
|
adantr |
|
27 |
|
simplr |
|
28 |
3
|
sselda |
|
29 |
28
|
adantlr |
|
30 |
|
elrp |
|
31 |
|
elrp |
|
32 |
|
ltrec1 |
|
33 |
30 31 32
|
syl2anb |
|
34 |
27 29 33
|
syl2anc |
|
35 |
34
|
imbi1d |
|
36 |
35
|
ralbidva |
|
37 |
36
|
adantlr |
|
38 |
|
rpcn |
|
39 |
|
rpne0 |
|
40 |
38 39
|
recrecd |
|
41 |
28 40
|
syl |
|
42 |
|
simpr |
|
43 |
41 42
|
eqeltrd |
|
44 |
|
eleq1 |
|
45 |
|
oveq2 |
|
46 |
45
|
eleq1d |
|
47 |
44 46
|
bibi12d |
|
48 |
5
|
ralrimiva |
|
49 |
48
|
adantr |
|
50 |
28
|
rpreccld |
|
51 |
47 49 50
|
rspcdva |
|
52 |
43 51
|
mpbird |
|
53 |
50
|
rpne0d |
|
54 |
|
eldifsn |
|
55 |
52 53 54
|
sylanbrc |
|
56 |
|
eldifi |
|
57 |
56
|
adantl |
|
58 |
|
rge0ssre |
|
59 |
1
|
ssdifssd |
|
60 |
59
|
sselda |
|
61 |
58 60
|
sseldi |
|
62 |
|
0re |
|
63 |
|
pnfxr |
|
64 |
|
elico2 |
|
65 |
62 63 64
|
mp2an |
|
66 |
65
|
simp2bi |
|
67 |
60 66
|
syl |
|
68 |
|
eldifsni |
|
69 |
68
|
adantl |
|
70 |
61 67 69
|
ne0gt0d |
|
71 |
61 70
|
elrpd |
|
72 |
71 5
|
syldan |
|
73 |
57 72
|
mpbid |
|
74 |
|
rpcn |
|
75 |
|
rpne0 |
|
76 |
74 75
|
recrecd |
|
77 |
71 76
|
syl |
|
78 |
77
|
eqcomd |
|
79 |
|
oveq2 |
|
80 |
79
|
rspceeqv |
|
81 |
73 78 80
|
syl2anc |
|
82 |
|
breq1 |
|
83 |
7
|
fvoveq1d |
|
84 |
83
|
breq1d |
|
85 |
82 84
|
imbi12d |
|
86 |
85
|
adantl |
|
87 |
55 81 86
|
ralxfrd |
|
88 |
87
|
ad2antrr |
|
89 |
37 88
|
bitr4d |
|
90 |
|
elsni |
|
91 |
90
|
adantl |
|
92 |
91 6
|
syl |
|
93 |
92
|
oveq1d |
|
94 |
6
|
eleq1d |
|
95 |
4
|
ralrimiva |
|
96 |
94 95 2
|
rspcdva |
|
97 |
96
|
subidd |
|
98 |
97
|
ad2antrr |
|
99 |
93 98
|
eqtrd |
|
100 |
99
|
abs00bd |
|
101 |
|
rpgt0 |
|
102 |
101
|
ad2antlr |
|
103 |
100 102
|
eqbrtrd |
|
104 |
103
|
a1d |
|
105 |
104
|
ralrimiva |
|
106 |
105
|
adantr |
|
107 |
106
|
biantrud |
|
108 |
|
ralunb |
|
109 |
107 108
|
bitr4di |
|
110 |
|
undif1 |
|
111 |
2
|
ad2antrr |
|
112 |
111
|
snssd |
|
113 |
|
ssequn2 |
|
114 |
112 113
|
sylib |
|
115 |
110 114
|
syl5eq |
|
116 |
115
|
raleqdv |
|
117 |
89 109 116
|
3bitrd |
|
118 |
117
|
rexbidva |
|
119 |
26 118
|
bitrd |
|
120 |
119
|
ralbidva |
|
121 |
|
nfv |
|
122 |
|
nffvmpt1 |
|
123 |
|
nfcv |
|
124 |
|
nffvmpt1 |
|
125 |
122 123 124
|
nfov |
|
126 |
|
nfcv |
|
127 |
|
nfcv |
|
128 |
125 126 127
|
nfbr |
|
129 |
121 128
|
nfim |
|
130 |
|
nfv |
|
131 |
|
oveq1 |
|
132 |
131
|
breq1d |
|
133 |
|
fveq2 |
|
134 |
133
|
oveq1d |
|
135 |
134
|
breq1d |
|
136 |
132 135
|
imbi12d |
|
137 |
129 130 136
|
cbvralw |
|
138 |
|
simpr |
|
139 |
2
|
adantr |
|
140 |
138 139
|
ovresd |
|
141 |
1 58
|
sstrdi |
|
142 |
|
ax-resscn |
|
143 |
141 142
|
sstrdi |
|
144 |
143
|
sselda |
|
145 |
|
0cnd |
|
146 |
|
eqid |
|
147 |
146
|
cnmetdval |
|
148 |
144 145 147
|
syl2anc |
|
149 |
144
|
subid1d |
|
150 |
149
|
fveq2d |
|
151 |
140 148 150
|
3eqtrd |
|
152 |
141
|
sselda |
|
153 |
1
|
sselda |
|
154 |
153 66
|
syl |
|
155 |
152 154
|
absidd |
|
156 |
151 155
|
eqtrd |
|
157 |
156
|
breq1d |
|
158 |
|
eqid |
|
159 |
158
|
fvmpt2 |
|
160 |
138 4 159
|
syl2anc |
|
161 |
96
|
adantr |
|
162 |
158 6 139 161
|
fvmptd3 |
|
163 |
160 162
|
oveq12d |
|
164 |
146
|
cnmetdval |
|
165 |
4 161 164
|
syl2anc |
|
166 |
163 165
|
eqtrd |
|
167 |
166
|
breq1d |
|
168 |
157 167
|
imbi12d |
|
169 |
168
|
ralbidva |
|
170 |
137 169
|
syl5bb |
|
171 |
170
|
rexbidv |
|
172 |
171
|
ralbidv |
|
173 |
4
|
fmpttd |
|
174 |
173
|
biantrurd |
|
175 |
120 172 174
|
3bitr2d |
|
176 |
7
|
eleq1d |
|
177 |
95
|
adantr |
|
178 |
176 177 52
|
rspcdva |
|
179 |
178
|
ralrimiva |
|
180 |
|
rpssre |
|
181 |
3 180
|
sstrdi |
|
182 |
|
1red |
|
183 |
179 181 96 182
|
rlim3 |
|
184 |
|
0xr |
|
185 |
|
0lt1 |
|
186 |
|
df-ioo |
|
187 |
|
df-ico |
|
188 |
|
xrltletr |
|
189 |
186 187 188
|
ixxss1 |
|
190 |
184 185 189
|
mp2an |
|
191 |
|
ioorp |
|
192 |
190 191
|
sseqtri |
|
193 |
|
ssrexv |
|
194 |
192 193
|
ax-mp |
|
195 |
|
simplr |
|
196 |
180 195
|
sseldi |
|
197 |
181
|
adantr |
|
198 |
197
|
sselda |
|
199 |
|
ltle |
|
200 |
196 198 199
|
syl2anc |
|
201 |
200
|
imim1d |
|
202 |
201
|
ralimdva |
|
203 |
202
|
reximdva |
|
204 |
194 203
|
syl5 |
|
205 |
204
|
ralimdv |
|
206 |
183 205
|
sylbid |
|
207 |
|
ssrexv |
|
208 |
180 207
|
ax-mp |
|
209 |
208
|
ralimi |
|
210 |
179 181 96
|
rlim2lt |
|
211 |
209 210
|
syl5ibr |
|
212 |
206 211
|
impbid |
|
213 |
|
cnxmet |
|
214 |
|
xmetres2 |
|
215 |
213 143 214
|
sylancr |
|
216 |
213
|
a1i |
|
217 |
|
eqid |
|
218 |
8
|
cnfldtopn |
|
219 |
217 218
|
metcnp2 |
|
220 |
215 216 2 219
|
syl3anc |
|
221 |
175 212 220
|
3bitr4d |
|
222 |
|
eqid |
|
223 |
222 218 217
|
metrest |
|
224 |
213 143 223
|
sylancr |
|
225 |
9 224
|
syl5eq |
|
226 |
225
|
oveq1d |
|
227 |
226
|
fveq1d |
|
228 |
227
|
eleq2d |
|
229 |
221 228
|
bitr4d |
|