Step |
Hyp |
Ref |
Expression |
1 |
|
sconnpconn |
|
2 |
|
sconnpconn |
|
3 |
|
txpconn |
|
4 |
1 2 3
|
syl2an |
|
5 |
|
simpll |
|
6 |
|
simprl |
|
7 |
|
sconntop |
|
8 |
7
|
ad2antrr |
|
9 |
|
eqid |
|
10 |
9
|
toptopon |
|
11 |
8 10
|
sylib |
|
12 |
|
sconntop |
|
13 |
12
|
ad2antlr |
|
14 |
|
eqid |
|
15 |
14
|
toptopon |
|
16 |
13 15
|
sylib |
|
17 |
|
tx1cn |
|
18 |
11 16 17
|
syl2anc |
|
19 |
|
cnco |
|
20 |
6 18 19
|
syl2anc |
|
21 |
|
simprr |
|
22 |
21
|
fveq2d |
|
23 |
|
iitopon |
|
24 |
23
|
a1i |
|
25 |
|
txtopon |
|
26 |
11 16 25
|
syl2anc |
|
27 |
|
cnf2 |
|
28 |
24 26 6 27
|
syl3anc |
|
29 |
|
0elunit |
|
30 |
|
fvco3 |
|
31 |
28 29 30
|
sylancl |
|
32 |
|
1elunit |
|
33 |
|
fvco3 |
|
34 |
28 32 33
|
sylancl |
|
35 |
22 31 34
|
3eqtr4d |
|
36 |
|
sconnpht |
|
37 |
5 20 35 36
|
syl3anc |
|
38 |
|
isphtpc |
|
39 |
37 38
|
sylib |
|
40 |
39
|
simp3d |
|
41 |
|
n0 |
|
42 |
40 41
|
sylib |
|
43 |
|
simplr |
|
44 |
|
tx2cn |
|
45 |
11 16 44
|
syl2anc |
|
46 |
|
cnco |
|
47 |
6 45 46
|
syl2anc |
|
48 |
21
|
fveq2d |
|
49 |
|
fvco3 |
|
50 |
28 29 49
|
sylancl |
|
51 |
|
fvco3 |
|
52 |
28 32 51
|
sylancl |
|
53 |
48 50 52
|
3eqtr4d |
|
54 |
|
sconnpht |
|
55 |
43 47 53 54
|
syl3anc |
|
56 |
|
isphtpc |
|
57 |
55 56
|
sylib |
|
58 |
57
|
simp3d |
|
59 |
|
n0 |
|
60 |
58 59
|
sylib |
|
61 |
|
exdistrv |
|
62 |
8
|
adantr |
|
63 |
13
|
adantr |
|
64 |
6
|
adantr |
|
65 |
|
eqid |
|
66 |
|
eqid |
|
67 |
|
simprl |
|
68 |
|
simprr |
|
69 |
62 63 64 65 66 67 68
|
txsconnlem |
|
70 |
69
|
ex |
|
71 |
70
|
exlimdvv |
|
72 |
61 71
|
syl5bir |
|
73 |
42 60 72
|
mp2and |
|
74 |
73
|
expr |
|
75 |
74
|
ralrimiva |
|
76 |
|
issconn |
|
77 |
4 75 76
|
sylanbrc |
|