Step |
Hyp |
Ref |
Expression |
1 |
|
prdsxms.y |
|
2 |
|
prdsxms.s |
|
3 |
|
prdsxms.i |
|
4 |
|
prdsxms.d |
|
5 |
|
prdsxms.b |
|
6 |
|
prdsxms.r |
|
7 |
|
prdsxms.j |
|
8 |
|
prdsxms.v |
|
9 |
|
prdsxms.e |
|
10 |
|
prdsxms.k |
|
11 |
|
prdsxms.c |
|
12 |
|
topnfn |
|
13 |
6
|
ffnd |
|
14 |
|
dffn2 |
|
15 |
13 14
|
sylib |
|
16 |
|
fnfco |
|
17 |
12 15 16
|
sylancr |
|
18 |
11
|
ptval |
|
19 |
3 17 18
|
syl2anc |
|
20 |
|
eldifsn |
|
21 |
1 2 3 4 5 6
|
prdsxmslem1 |
|
22 |
|
blrn |
|
23 |
21 22
|
syl |
|
24 |
21
|
adantr |
|
25 |
|
simprl |
|
26 |
|
simprr |
|
27 |
|
xbln0 |
|
28 |
24 25 26 27
|
syl3anc |
|
29 |
3
|
3ad2ant1 |
|
30 |
29
|
mptexd |
|
31 |
|
ovex |
|
32 |
31
|
rgenw |
|
33 |
|
eqid |
|
34 |
33
|
fnmpt |
|
35 |
32 34
|
mp1i |
|
36 |
6
|
3ad2ant1 |
|
37 |
36
|
ffvelrnda |
|
38 |
8 9
|
xmsxmet |
|
39 |
37 38
|
syl |
|
40 |
|
eqid |
|
41 |
|
eqid |
|
42 |
2
|
3ad2ant1 |
|
43 |
37
|
ralrimiva |
|
44 |
|
simp2l |
|
45 |
36
|
feqmptd |
|
46 |
45
|
oveq2d |
|
47 |
1 46
|
syl5eq |
|
48 |
47
|
fveq2d |
|
49 |
5 48
|
syl5eq |
|
50 |
44 49
|
eleqtrd |
|
51 |
40 41 42 29 43 8 50
|
prdsbascl |
|
52 |
51
|
r19.21bi |
|
53 |
|
simp2r |
|
54 |
53
|
adantr |
|
55 |
|
eqid |
|
56 |
55
|
blopn |
|
57 |
39 52 54 56
|
syl3anc |
|
58 |
|
2fveq3 |
|
59 |
|
2fveq3 |
|
60 |
59 8
|
eqtr4di |
|
61 |
60
|
sqxpeqd |
|
62 |
58 61
|
reseq12d |
|
63 |
62 9
|
eqtr4di |
|
64 |
63
|
fveq2d |
|
65 |
|
fveq2 |
|
66 |
|
eqidd |
|
67 |
64 65 66
|
oveq123d |
|
68 |
|
ovex |
|
69 |
67 33 68
|
fvmpt |
|
70 |
69
|
adantl |
|
71 |
|
fvco3 |
|
72 |
71 10
|
eqtr4di |
|
73 |
36 72
|
sylan |
|
74 |
10 8 9
|
xmstopn |
|
75 |
37 74
|
syl |
|
76 |
73 75
|
eqtrd |
|
77 |
57 70 76
|
3eltr4d |
|
78 |
77
|
ralrimiva |
|
79 |
36
|
feqmptd |
|
80 |
79
|
oveq2d |
|
81 |
1 80
|
syl5eq |
|
82 |
81
|
fveq2d |
|
83 |
4 82
|
syl5eq |
|
84 |
83
|
fveq2d |
|
85 |
84
|
oveqd |
|
86 |
|
fveq2 |
|
87 |
86
|
cbvmptv |
|
88 |
87
|
oveq2i |
|
89 |
|
eqid |
|
90 |
|
eqid |
|
91 |
81
|
fveq2d |
|
92 |
5 91
|
syl5eq |
|
93 |
44 92
|
eleqtrd |
|
94 |
|
simp3 |
|
95 |
88 89 8 9 90 42 29 37 39 93 53 94
|
prdsbl |
|
96 |
85 95
|
eqtrd |
|
97 |
|
fneq1 |
|
98 |
|
fveq1 |
|
99 |
98
|
eleq1d |
|
100 |
99
|
ralbidv |
|
101 |
97 100
|
anbi12d |
|
102 |
98 69
|
sylan9eq |
|
103 |
102
|
ixpeq2dva |
|
104 |
103
|
eqeq2d |
|
105 |
101 104
|
anbi12d |
|
106 |
105
|
spcegv |
|
107 |
106
|
3impib |
|
108 |
30 35 78 96 107
|
syl121anc |
|
109 |
108
|
3expia |
|
110 |
28 109
|
sylbid |
|
111 |
110
|
adantr |
|
112 |
|
simpr |
|
113 |
112
|
neeq1d |
|
114 |
|
df-3an |
|
115 |
|
ral0 |
|
116 |
|
difeq2 |
|
117 |
|
difid |
|
118 |
116 117
|
eqtrdi |
|
119 |
118
|
raleqdv |
|
120 |
119
|
rspcev |
|
121 |
3 115 120
|
sylancl |
|
122 |
121
|
adantr |
|
123 |
122
|
biantrud |
|
124 |
114 123
|
bitr4id |
|
125 |
|
eqeq1 |
|
126 |
124 125
|
bi2anan9 |
|
127 |
126
|
exbidv |
|
128 |
111 113 127
|
3imtr4d |
|
129 |
128
|
ex |
|
130 |
129
|
rexlimdvva |
|
131 |
23 130
|
sylbid |
|
132 |
131
|
impd |
|
133 |
20 132
|
syl5bi |
|
134 |
133
|
alrimiv |
|
135 |
|
ssab |
|
136 |
134 135
|
sylibr |
|
137 |
136 11
|
sseqtrrdi |
|
138 |
|
ssv |
|
139 |
|
fnssres |
|
140 |
12 138 139
|
mp2an |
|
141 |
|
fvres |
|
142 |
|
xmstps |
|
143 |
|
eqid |
|
144 |
143
|
tpstop |
|
145 |
142 144
|
syl |
|
146 |
141 145
|
eqeltrd |
|
147 |
146
|
rgen |
|
148 |
|
ffnfv |
|
149 |
140 147 148
|
mpbir2an |
|
150 |
|
fco2 |
|
151 |
149 6 150
|
sylancr |
|
152 |
|
eqid |
|
153 |
11 152
|
ptbasfi |
|
154 |
3 151 153
|
syl2anc |
|
155 |
|
eqid |
|
156 |
155
|
mopntop |
|
157 |
21 156
|
syl |
|
158 |
1 5 2 3 13
|
prdsbas2 |
|
159 |
6 72
|
sylan |
|
160 |
6
|
ffvelrnda |
|
161 |
|
xmstps |
|
162 |
160 161
|
syl |
|
163 |
8 10
|
istps |
|
164 |
162 163
|
sylib |
|
165 |
159 164
|
eqeltrd |
|
166 |
|
toponuni |
|
167 |
165 166
|
syl |
|
168 |
8 167
|
eqtr3id |
|
169 |
168
|
ixpeq2dva |
|
170 |
158 169
|
eqtrd |
|
171 |
|
fveq2 |
|
172 |
171
|
unieqd |
|
173 |
172
|
cbvixpv |
|
174 |
170 173
|
eqtrdi |
|
175 |
155
|
mopntopon |
|
176 |
21 175
|
syl |
|
177 |
|
toponmax |
|
178 |
176 177
|
syl |
|
179 |
174 178
|
eqeltrrd |
|
180 |
179
|
snssd |
|
181 |
174
|
mpteq1d |
|
182 |
181
|
ad2antrr |
|
183 |
182
|
cnveqd |
|
184 |
183
|
imaeq1d |
|
185 |
|
fveq1 |
|
186 |
185
|
eleq1d |
|
187 |
|
eqid |
|
188 |
187
|
mptpreima |
|
189 |
186 188
|
elrab2 |
|
190 |
160 38
|
syl |
|
191 |
190
|
adantr |
|
192 |
|
simprl |
|
193 |
160 74
|
syl |
|
194 |
193
|
adantr |
|
195 |
192 194
|
eleqtrd |
|
196 |
|
simprrr |
|
197 |
55
|
mopni2 |
|
198 |
191 195 196 197
|
syl3anc |
|
199 |
21
|
ad3antrrr |
|
200 |
|
simprrl |
|
201 |
200
|
adantr |
|
202 |
|
rpxr |
|
203 |
202
|
ad2antrl |
|
204 |
155
|
blopn |
|
205 |
199 201 203 204
|
syl3anc |
|
206 |
|
simprl |
|
207 |
|
blcntr |
|
208 |
199 201 206 207
|
syl3anc |
|
209 |
|
blssm |
|
210 |
199 201 203 209
|
syl3anc |
|
211 |
|
simplrr |
|
212 |
|
simplll |
|
213 |
|
rpgt0 |
|
214 |
213
|
ad2antrl |
|
215 |
212 201 203 214 96
|
syl121anc |
|
216 |
215
|
eleq2d |
|
217 |
216
|
biimpa |
|
218 |
|
vex |
|
219 |
218
|
elixp |
|
220 |
219
|
simprbi |
|
221 |
217 220
|
syl |
|
222 |
|
simp-4r |
|
223 |
|
rsp |
|
224 |
221 222 223
|
sylc |
|
225 |
211 224
|
sseldd |
|
226 |
210 225
|
ssrabdv |
|
227 |
226 188
|
sseqtrrdi |
|
228 |
|
eleq2 |
|
229 |
|
sseq1 |
|
230 |
228 229
|
anbi12d |
|
231 |
230
|
rspcev |
|
232 |
205 208 227 231
|
syl12anc |
|
233 |
198 232
|
rexlimddv |
|
234 |
233
|
expr |
|
235 |
189 234
|
syl5bi |
|
236 |
235
|
ralrimiv |
|
237 |
157
|
ad2antrr |
|
238 |
|
eltop2 |
|
239 |
237 238
|
syl |
|
240 |
236 239
|
mpbird |
|
241 |
184 240
|
eqeltrrd |
|
242 |
241
|
ralrimiva |
|
243 |
159
|
raleqdv |
|
244 |
242 243
|
mpbird |
|
245 |
244
|
ralrimiva |
|
246 |
|
fveq2 |
|
247 |
|
fveq2 |
|
248 |
247
|
mpteq2dv |
|
249 |
248
|
cnveqd |
|
250 |
249
|
imaeq1d |
|
251 |
250
|
eleq1d |
|
252 |
246 251
|
raleqbidv |
|
253 |
252
|
cbvralvw |
|
254 |
245 253
|
sylib |
|
255 |
|
eqid |
|
256 |
255
|
fmpox |
|
257 |
254 256
|
sylib |
|
258 |
257
|
frnd |
|
259 |
180 258
|
unssd |
|
260 |
|
fiss |
|
261 |
157 259 260
|
syl2anc |
|
262 |
154 261
|
eqsstrd |
|
263 |
|
fitop |
|
264 |
157 263
|
syl |
|
265 |
155
|
mopnval |
|
266 |
21 265
|
syl |
|
267 |
|
tgdif0 |
|
268 |
266 267
|
eqtr4di |
|
269 |
264 268
|
eqtrd |
|
270 |
262 269
|
sseqtrd |
|
271 |
|
2basgen |
|
272 |
137 270 271
|
syl2anc |
|
273 |
19 272
|
eqtr4d |
|
274 |
1 2 3 13 7
|
prdstopn |
|
275 |
273 274 268
|
3eqtr4d |
|