Step |
Hyp |
Ref |
Expression |
1 |
|
lptioo2cn.1 |
⊢ 𝐽 = ( TopOpen ‘ ℂfld ) |
2 |
|
lptioo2cn.2 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
3 |
|
lptioo2cn.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
4 |
|
lptioo2cn.4 |
⊢ ( 𝜑 → 𝐴 < 𝐵 ) |
5 |
|
eqid |
⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) ) |
6 |
5 2 3 4
|
lptioo2 |
⊢ ( 𝜑 → 𝐵 ∈ ( ( limPt ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ) |
7 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
8 |
7
|
cnfldtop |
⊢ ( TopOpen ‘ ℂfld ) ∈ Top |
9 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
10 |
|
unicntop |
⊢ ℂ = ∪ ( TopOpen ‘ ℂfld ) |
11 |
9 10
|
sseqtri |
⊢ ℝ ⊆ ∪ ( TopOpen ‘ ℂfld ) |
12 |
|
ioossre |
⊢ ( 𝐴 (,) 𝐵 ) ⊆ ℝ |
13 |
|
eqid |
⊢ ∪ ( TopOpen ‘ ℂfld ) = ∪ ( TopOpen ‘ ℂfld ) |
14 |
7
|
tgioo2 |
⊢ ( topGen ‘ ran (,) ) = ( ( TopOpen ‘ ℂfld ) ↾t ℝ ) |
15 |
13 14
|
restlp |
⊢ ( ( ( TopOpen ‘ ℂfld ) ∈ Top ∧ ℝ ⊆ ∪ ( TopOpen ‘ ℂfld ) ∧ ( 𝐴 (,) 𝐵 ) ⊆ ℝ ) → ( ( limPt ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) = ( ( ( limPt ‘ ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ∩ ℝ ) ) |
16 |
8 11 12 15
|
mp3an |
⊢ ( ( limPt ‘ ( topGen ‘ ran (,) ) ) ‘ ( 𝐴 (,) 𝐵 ) ) = ( ( ( limPt ‘ ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ∩ ℝ ) |
17 |
6 16
|
eleqtrdi |
⊢ ( 𝜑 → 𝐵 ∈ ( ( ( limPt ‘ ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ∩ ℝ ) ) |
18 |
|
elin |
⊢ ( 𝐵 ∈ ( ( ( limPt ‘ ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ∩ ℝ ) ↔ ( 𝐵 ∈ ( ( limPt ‘ ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ∧ 𝐵 ∈ ℝ ) ) |
19 |
17 18
|
sylib |
⊢ ( 𝜑 → ( 𝐵 ∈ ( ( limPt ‘ ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ∧ 𝐵 ∈ ℝ ) ) |
20 |
19
|
simpld |
⊢ ( 𝜑 → 𝐵 ∈ ( ( limPt ‘ ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐴 (,) 𝐵 ) ) ) |
21 |
1
|
eqcomi |
⊢ ( TopOpen ‘ ℂfld ) = 𝐽 |
22 |
21
|
fveq2i |
⊢ ( limPt ‘ ( TopOpen ‘ ℂfld ) ) = ( limPt ‘ 𝐽 ) |
23 |
22
|
fveq1i |
⊢ ( ( limPt ‘ ( TopOpen ‘ ℂfld ) ) ‘ ( 𝐴 (,) 𝐵 ) ) = ( ( limPt ‘ 𝐽 ) ‘ ( 𝐴 (,) 𝐵 ) ) |
24 |
20 23
|
eleqtrdi |
⊢ ( 𝜑 → 𝐵 ∈ ( ( limPt ‘ 𝐽 ) ‘ ( 𝐴 (,) 𝐵 ) ) ) |