Step |
Hyp |
Ref |
Expression |
1 |
|
conntop |
|
2 |
1
|
adantr |
|
3 |
|
eqid |
|
4 |
|
simpll |
|
5 |
|
inss1 |
|
6 |
|
simplr |
|
7 |
1
|
ad2antrr |
|
8 |
3
|
topopn |
|
9 |
7 8
|
syl |
|
10 |
|
simprr |
|
11 |
|
nlly2i |
|
12 |
6 9 10 11
|
syl3anc |
|
13 |
|
simprr1 |
|
14 |
|
eqeq2 |
|
15 |
14
|
anbi2d |
|
16 |
15
|
rexbidv |
|
17 |
16
|
elrab |
|
18 |
17
|
simprbi |
|
19 |
|
simprr3 |
|
20 |
19
|
adantr |
|
21 |
|
simprr2 |
|
22 |
21
|
adantr |
|
23 |
|
simprll |
|
24 |
22 23
|
sseldd |
|
25 |
7
|
ad2antrr |
|
26 |
|
elpwi |
|
27 |
26
|
ad2antrl |
|
28 |
27
|
adantr |
|
29 |
3
|
restuni |
|
30 |
25 28 29
|
syl2anc |
|
31 |
24 30
|
eleqtrd |
|
32 |
|
simprr |
|
33 |
22 32
|
sseldd |
|
34 |
33 30
|
eleqtrd |
|
35 |
|
eqid |
|
36 |
35
|
pconncn |
|
37 |
20 31 34 36
|
syl3anc |
|
38 |
|
simplrl |
|
39 |
38
|
ad2antlr |
|
40 |
25
|
adantr |
|
41 |
|
cnrest2r |
|
42 |
40 41
|
syl |
|
43 |
|
simprl |
|
44 |
42 43
|
sseldd |
|
45 |
|
simplrr |
|
46 |
45
|
ad2antlr |
|
47 |
46
|
simprd |
|
48 |
|
simprrl |
|
49 |
47 48
|
eqtr4d |
|
50 |
39 44 49
|
pcocn |
|
51 |
39 44
|
pco0 |
|
52 |
46
|
simpld |
|
53 |
51 52
|
eqtrd |
|
54 |
39 44
|
pco1 |
|
55 |
|
simprrr |
|
56 |
54 55
|
eqtrd |
|
57 |
|
fveq1 |
|
58 |
57
|
eqeq1d |
|
59 |
|
fveq1 |
|
60 |
59
|
eqeq1d |
|
61 |
58 60
|
anbi12d |
|
62 |
61
|
rspcev |
|
63 |
50 53 56 62
|
syl12anc |
|
64 |
37 63
|
rexlimddv |
|
65 |
64
|
anassrs |
|
66 |
65
|
ralrimiva |
|
67 |
66
|
anassrs |
|
68 |
67
|
rexlimdvaa |
|
69 |
21
|
adantr |
|
70 |
|
simplrl |
|
71 |
70 26
|
syl |
|
72 |
69 71
|
sstrd |
|
73 |
68 72
|
jctild |
|
74 |
|
fveq1 |
|
75 |
74
|
eqeq1d |
|
76 |
|
fveq1 |
|
77 |
76
|
eqeq1d |
|
78 |
75 77
|
anbi12d |
|
79 |
78
|
cbvrexvw |
|
80 |
|
ssrab |
|
81 |
73 79 80
|
3imtr4g |
|
82 |
18 81
|
syl5 |
|
83 |
82
|
ralrimiva |
|
84 |
13 83
|
jca |
|
85 |
84
|
expr |
|
86 |
85
|
reximdv |
|
87 |
86
|
rexlimdva |
|
88 |
12 87
|
mpd |
|
89 |
88
|
anassrs |
|
90 |
89
|
ralrimiva |
|
91 |
1
|
ad2antrr |
|
92 |
|
ssrab2 |
|
93 |
3
|
isclo2 |
|
94 |
91 92 93
|
sylancl |
|
95 |
90 94
|
mpbird |
|
96 |
5 95
|
sselid |
|
97 |
|
simpr |
|
98 |
|
iitopon |
|
99 |
98
|
a1i |
|
100 |
3
|
toptopon |
|
101 |
91 100
|
sylib |
|
102 |
|
cnconst2 |
|
103 |
99 101 97 102
|
syl3anc |
|
104 |
|
0elunit |
|
105 |
|
vex |
|
106 |
105
|
fvconst2 |
|
107 |
104 106
|
mp1i |
|
108 |
|
1elunit |
|
109 |
105
|
fvconst2 |
|
110 |
108 109
|
mp1i |
|
111 |
|
eqeq2 |
|
112 |
111
|
anbi2d |
|
113 |
|
fveq1 |
|
114 |
113
|
eqeq1d |
|
115 |
|
fveq1 |
|
116 |
115
|
eqeq1d |
|
117 |
114 116
|
anbi12d |
|
118 |
112 117
|
rspc2ev |
|
119 |
97 103 107 110 118
|
syl112anc |
|
120 |
|
rabn0 |
|
121 |
119 120
|
sylibr |
|
122 |
|
inss2 |
|
123 |
122 95
|
sselid |
|
124 |
3 4 96 121 123
|
connclo |
|
125 |
124
|
eqcomd |
|
126 |
|
rabid2 |
|
127 |
125 126
|
sylib |
|
128 |
127
|
ralrimiva |
|
129 |
3
|
ispconn |
|
130 |
2 128 129
|
sylanbrc |
|