Metamath Proof Explorer


Theorem sepfsepc

Description: If two sets are separated by a continuous function, then they are separated by closed neighborhoods. (Contributed by Zhi Wang, 9-Sep-2024)

Ref Expression
Hypothesis sepfsepc.1 φ f J Cn II S f -1 0 T f -1 1
Assertion sepfsepc φ n nei J S m nei J T n Clsd J m Clsd J n m =

Proof

Step Hyp Ref Expression
1 sepfsepc.1 φ f J Cn II S f -1 0 T f -1 1
2 simpl f J Cn II S f -1 0 T f -1 1 f J Cn II
3 0re 0
4 1re 1
5 0le0 0 0
6 3re 3
7 3ne0 3 0
8 6 7 rereccli 1 3
9 1lt3 1 < 3
10 recgt1i 3 1 < 3 0 < 1 3 1 3 < 1
11 6 9 10 mp2an 0 < 1 3 1 3 < 1
12 11 simpri 1 3 < 1
13 8 4 12 ltleii 1 3 1
14 iccss 0 1 0 0 1 3 1 0 1 3 0 1
15 3 4 5 13 14 mp4an 0 1 3 0 1
16 i0oii 1 3 1 0 1 3 II
17 13 16 ax-mp 0 1 3 II
18 11 simpli 0 < 1 3
19 8 rexri 1 3 *
20 elico2 0 1 3 * 0 0 1 3 0 0 0 0 < 1 3
21 3 19 20 mp2an 0 0 1 3 0 0 0 0 < 1 3
22 21 biimpri 0 0 0 0 < 1 3 0 0 1 3
23 22 snssd 0 0 0 0 < 1 3 0 0 1 3
24 3 5 18 23 mp3an 0 0 1 3
25 icossicc 0 1 3 0 1 3
26 24 25 pm3.2i 0 0 1 3 0 1 3 0 1 3
27 sseq2 g = 0 1 3 0 g 0 0 1 3
28 sseq1 g = 0 1 3 g 0 1 3 0 1 3 0 1 3
29 27 28 anbi12d g = 0 1 3 0 g g 0 1 3 0 0 1 3 0 1 3 0 1 3
30 29 rspcev 0 1 3 II 0 0 1 3 0 1 3 0 1 3 g II 0 g g 0 1 3
31 17 26 30 mp2an g II 0 g g 0 1 3
32 iitop II Top
33 24 25 sstri 0 0 1 3
34 33 15 sstri 0 0 1
35 iiuni 0 1 = II
36 35 isnei II Top 0 0 1 0 1 3 nei II 0 0 1 3 0 1 g II 0 g g 0 1 3
37 32 34 36 mp2an 0 1 3 nei II 0 0 1 3 0 1 g II 0 g g 0 1 3
38 15 31 37 mpbir2an 0 1 3 nei II 0
39 38 a1i f J Cn II S f -1 0 T f -1 1 0 1 3 nei II 0
40 simprl f J Cn II S f -1 0 T f -1 1 S f -1 0
41 2 39 40 cnneiima f J Cn II S f -1 0 T f -1 1 f -1 0 1 3 nei J S
42 halfge0 0 1 2
43 1le1 1 1
44 iccss 0 1 0 1 2 1 1 1 2 1 0 1
45 3 4 42 43 44 mp4an 1 2 1 0 1
46 io1ii 0 1 2 1 2 1 II
47 42 46 ax-mp 1 2 1 II
48 halflt1 1 2 < 1
49 halfre 1 2
50 49 rexri 1 2 *
51 elioc2 1 2 * 1 1 1 2 1 1 1 2 < 1 1 1
52 50 4 51 mp2an 1 1 2 1 1 1 2 < 1 1 1
53 52 biimpri 1 1 2 < 1 1 1 1 1 2 1
54 53 snssd 1 1 2 < 1 1 1 1 1 2 1
55 4 48 43 54 mp3an 1 1 2 1
56 iocssicc 1 2 1 1 2 1
57 55 56 pm3.2i 1 1 2 1 1 2 1 1 2 1
58 sseq2 h = 1 2 1 1 h 1 1 2 1
59 sseq1 h = 1 2 1 h 1 2 1 1 2 1 1 2 1
60 58 59 anbi12d h = 1 2 1 1 h h 1 2 1 1 1 2 1 1 2 1 1 2 1
61 60 rspcev 1 2 1 II 1 1 2 1 1 2 1 1 2 1 h II 1 h h 1 2 1
62 47 57 61 mp2an h II 1 h h 1 2 1
63 55 56 sstri 1 1 2 1
64 63 45 sstri 1 0 1
65 35 isnei II Top 1 0 1 1 2 1 nei II 1 1 2 1 0 1 h II 1 h h 1 2 1
66 32 64 65 mp2an 1 2 1 nei II 1 1 2 1 0 1 h II 1 h h 1 2 1
67 45 62 66 mpbir2an 1 2 1 nei II 1
68 67 a1i f J Cn II S f -1 0 T f -1 1 1 2 1 nei II 1
69 simprr f J Cn II S f -1 0 T f -1 1 T f -1 1
70 2 68 69 cnneiima f J Cn II S f -1 0 T f -1 1 f -1 1 2 1 nei J T
71 icccldii 0 0 1 3 1 0 1 3 Clsd II
72 5 13 71 mp2an 0 1 3 Clsd II
73 cnclima f J Cn II 0 1 3 Clsd II f -1 0 1 3 Clsd J
74 2 72 73 sylancl f J Cn II S f -1 0 T f -1 1 f -1 0 1 3 Clsd J
75 icccldii 0 1 2 1 1 1 2 1 Clsd II
76 42 43 75 mp2an 1 2 1 Clsd II
77 cnclima f J Cn II 1 2 1 Clsd II f -1 1 2 1 Clsd J
78 2 76 77 sylancl f J Cn II S f -1 0 T f -1 1 f -1 1 2 1 Clsd J
79 eqid J = J
80 79 35 cnf f J Cn II f : J 0 1
81 80 ffund f J Cn II Fun f
82 2 81 syl f J Cn II S f -1 0 T f -1 1 Fun f
83 0xr 0 *
84 1xr 1 *
85 2lt3 2 < 3
86 2re 2
87 2pos 0 < 2
88 3pos 0 < 3
89 86 6 87 88 ltrecii 2 < 3 1 3 < 1 2
90 85 89 mpbi 1 3 < 1 2
91 iccdisj2 0 * 1 * 1 3 < 1 2 0 1 3 1 2 1 =
92 83 84 90 91 mp3an 0 1 3 1 2 1 =
93 92 a1i f J Cn II S f -1 0 T f -1 1 0 1 3 1 2 1 =
94 ssidd f J Cn II S f -1 0 T f -1 1 f -1 0 1 3 f -1 0 1 3
95 ssidd f J Cn II S f -1 0 T f -1 1 f -1 1 2 1 f -1 1 2 1
96 82 93 94 95 predisj f J Cn II S f -1 0 T f -1 1 f -1 0 1 3 f -1 1 2 1 =
97 eleq1 n = f -1 0 1 3 n Clsd J f -1 0 1 3 Clsd J
98 ineq1 n = f -1 0 1 3 n m = f -1 0 1 3 m
99 98 eqeq1d n = f -1 0 1 3 n m = f -1 0 1 3 m =
100 97 99 3anbi13d n = f -1 0 1 3 n Clsd J m Clsd J n m = f -1 0 1 3 Clsd J m Clsd J f -1 0 1 3 m =
101 eleq1 m = f -1 1 2 1 m Clsd J f -1 1 2 1 Clsd J
102 ineq2 m = f -1 1 2 1 f -1 0 1 3 m = f -1 0 1 3 f -1 1 2 1
103 102 eqeq1d m = f -1 1 2 1 f -1 0 1 3 m = f -1 0 1 3 f -1 1 2 1 =
104 101 103 3anbi23d m = f -1 1 2 1 f -1 0 1 3 Clsd J m Clsd J f -1 0 1 3 m = f -1 0 1 3 Clsd J f -1 1 2 1 Clsd J f -1 0 1 3 f -1 1 2 1 =
105 100 104 rspc2ev f -1 0 1 3 nei J S f -1 1 2 1 nei J T f -1 0 1 3 Clsd J f -1 1 2 1 Clsd J f -1 0 1 3 f -1 1 2 1 = n nei J S m nei J T n Clsd J m Clsd J n m =
106 41 70 74 78 96 105 syl113anc f J Cn II S f -1 0 T f -1 1 n nei J S m nei J T n Clsd J m Clsd J n m =
107 106 rexlimiva f J Cn II S f -1 0 T f -1 1 n nei J S m nei J T n Clsd J m Clsd J n m =
108 1 107 syl φ n nei J S m nei J T n Clsd J m Clsd J n m =