Step | Hyp | Ref
| Expression |
1 | | ioccncflimc.f |
. . 3
⊢ (𝜑 → 𝐹 ∈ ((𝐴(,]𝐵)–cn→ℂ)) |
2 | | ioccncflimc.a |
. . . 4
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
3 | | ioccncflimc.b |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ ℝ) |
4 | 3 | rexrd 9968 |
. . . 4
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
5 | | ioccncflimc.altb |
. . . 4
⊢ (𝜑 → 𝐴 < 𝐵) |
6 | 3 | leidd 10473 |
. . . 4
⊢ (𝜑 → 𝐵 ≤ 𝐵) |
7 | 2, 4, 4, 5, 6 | eliocd 38577 |
. . 3
⊢ (𝜑 → 𝐵 ∈ (𝐴(,]𝐵)) |
8 | 1, 7 | cnlimci 23459 |
. 2
⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)) |
9 | | cncfrss 22502 |
. . . . . . . 8
⊢ (𝐹 ∈ ((𝐴(,]𝐵)–cn→ℂ) → (𝐴(,]𝐵) ⊆ ℂ) |
10 | 1, 9 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℂ) |
11 | | ssid 3587 |
. . . . . . 7
⊢ ℂ
⊆ ℂ |
12 | | eqid 2610 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
13 | | eqid 2610 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) = ((TopOpen‘ℂfld)
↾t (𝐴(,]𝐵)) |
14 | | eqid 2610 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t ℂ) =
((TopOpen‘ℂfld) ↾t
ℂ) |
15 | 12, 13, 14 | cncfcn 22520 |
. . . . . . 7
⊢ (((𝐴(,]𝐵) ⊆ ℂ ∧ ℂ ⊆
ℂ) → ((𝐴(,]𝐵)–cn→ℂ) =
(((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) |
16 | 10, 11, 15 | sylancl 693 |
. . . . . 6
⊢ (𝜑 → ((𝐴(,]𝐵)–cn→ℂ) =
(((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) |
17 | 1, 16 | eleqtrd 2690 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) |
18 | 12 | cnfldtopon 22396 |
. . . . . . 7
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
19 | | resttopon 20775 |
. . . . . . 7
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ (𝐴(,]𝐵) ⊆ ℂ) →
((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) ∈ (TopOn‘(𝐴(,]𝐵))) |
20 | 18, 10, 19 | sylancr 694 |
. . . . . 6
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) ∈ (TopOn‘(𝐴(,]𝐵))) |
21 | 12 | cnfldtop 22397 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) ∈ Top |
22 | | unicntop 38230 |
. . . . . . . . 9
⊢ ℂ =
∪
(TopOpen‘ℂfld) |
23 | 22 | restid 15917 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ∈ Top →
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld)) |
24 | 21, 23 | ax-mp 5 |
. . . . . . 7
⊢
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld) |
25 | 24 | cnfldtopon 22396 |
. . . . . 6
⊢
((TopOpen‘ℂfld) ↾t ℂ)
∈ (TopOn‘ℂ) |
26 | | cncnp 20894 |
. . . . . 6
⊢
((((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) ∈ (TopOn‘(𝐴(,]𝐵)) ∧
((TopOpen‘ℂfld) ↾t ℂ) ∈
(TopOn‘ℂ)) → (𝐹 ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) Cn ((TopOpen‘ℂfld)
↾t ℂ)) ↔ (𝐹:(𝐴(,]𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴(,]𝐵)𝐹 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) CnP
((TopOpen‘ℂfld) ↾t
ℂ))‘𝑥)))) |
27 | 20, 25, 26 | sylancl 693 |
. . . . 5
⊢ (𝜑 → (𝐹 ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) Cn ((TopOpen‘ℂfld)
↾t ℂ)) ↔ (𝐹:(𝐴(,]𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴(,]𝐵)𝐹 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) CnP
((TopOpen‘ℂfld) ↾t
ℂ))‘𝑥)))) |
28 | 17, 27 | mpbid 221 |
. . . 4
⊢ (𝜑 → (𝐹:(𝐴(,]𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴(,]𝐵)𝐹 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) CnP
((TopOpen‘ℂfld) ↾t
ℂ))‘𝑥))) |
29 | 28 | simpld 474 |
. . 3
⊢ (𝜑 → 𝐹:(𝐴(,]𝐵)⟶ℂ) |
30 | | ioossioc 38560 |
. . . 4
⊢ (𝐴(,)𝐵) ⊆ (𝐴(,]𝐵) |
31 | 30 | a1i 11 |
. . 3
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,]𝐵)) |
32 | | eqid 2610 |
. . 3
⊢
((TopOpen‘ℂfld) ↾t ((𝐴(,]𝐵) ∪ {𝐵})) = ((TopOpen‘ℂfld)
↾t ((𝐴(,]𝐵) ∪ {𝐵})) |
33 | 3 | recnd 9947 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℂ) |
34 | 22 | ntrtop 20684 |
. . . . . . . . 9
⊢
((TopOpen‘ℂfld) ∈ Top →
((int‘(TopOpen‘ℂfld))‘ℂ) =
ℂ) |
35 | 21, 34 | ax-mp 5 |
. . . . . . . 8
⊢
((int‘(TopOpen‘ℂfld))‘ℂ) =
ℂ |
36 | | undif 4001 |
. . . . . . . . . . 11
⊢ ((𝐴(,]𝐵) ⊆ ℂ ↔ ((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵))) = ℂ) |
37 | 10, 36 | sylib 207 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵))) = ℂ) |
38 | 37 | eqcomd 2616 |
. . . . . . . . 9
⊢ (𝜑 → ℂ = ((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵)))) |
39 | 38 | fveq2d 6107 |
. . . . . . . 8
⊢ (𝜑 →
((int‘(TopOpen‘ℂfld))‘ℂ) =
((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵))))) |
40 | 35, 39 | syl5eqr 2658 |
. . . . . . 7
⊢ (𝜑 → ℂ =
((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵))))) |
41 | 33, 40 | eleqtrd 2690 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈
((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵))))) |
42 | 41, 7 | elind 3760 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈
(((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵)))) ∩ (𝐴(,]𝐵))) |
43 | 21 | a1i 11 |
. . . . . 6
⊢ (𝜑 →
(TopOpen‘ℂfld) ∈ Top) |
44 | | ssid 3587 |
. . . . . . 7
⊢ (𝐴(,]𝐵) ⊆ (𝐴(,]𝐵) |
45 | 44 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝐴(,]𝐵) ⊆ (𝐴(,]𝐵)) |
46 | 22, 13 | restntr 20796 |
. . . . . 6
⊢
(((TopOpen‘ℂfld) ∈ Top ∧ (𝐴(,]𝐵) ⊆ ℂ ∧ (𝐴(,]𝐵) ⊆ (𝐴(,]𝐵)) →
((int‘((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)))‘(𝐴(,]𝐵)) =
(((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵)))) ∩ (𝐴(,]𝐵))) |
47 | 43, 10, 45, 46 | syl3anc 1318 |
. . . . 5
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)))‘(𝐴(,]𝐵)) =
(((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵)))) ∩ (𝐴(,]𝐵))) |
48 | 42, 47 | eleqtrrd 2691 |
. . . 4
⊢ (𝜑 → 𝐵 ∈
((int‘((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)))‘(𝐴(,]𝐵))) |
49 | 7 | snssd 4281 |
. . . . . . . . 9
⊢ (𝜑 → {𝐵} ⊆ (𝐴(,]𝐵)) |
50 | | ssequn2 3748 |
. . . . . . . . 9
⊢ ({𝐵} ⊆ (𝐴(,]𝐵) ↔ ((𝐴(,]𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) |
51 | 49, 50 | sylib 207 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴(,]𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) |
52 | 51 | eqcomd 2616 |
. . . . . . 7
⊢ (𝜑 → (𝐴(,]𝐵) = ((𝐴(,]𝐵) ∪ {𝐵})) |
53 | 52 | oveq2d 6565 |
. . . . . 6
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) = ((TopOpen‘ℂfld)
↾t ((𝐴(,]𝐵) ∪ {𝐵}))) |
54 | 53 | fveq2d 6107 |
. . . . 5
⊢ (𝜑 →
(int‘((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵))) =
(int‘((TopOpen‘ℂfld) ↾t ((𝐴(,]𝐵) ∪ {𝐵})))) |
55 | | snunioo2 38578 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
< 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) |
56 | 2, 4, 5, 55 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) |
57 | 56 | eqcomd 2616 |
. . . . 5
⊢ (𝜑 → (𝐴(,]𝐵) = ((𝐴(,)𝐵) ∪ {𝐵})) |
58 | 54, 57 | fveq12d 6109 |
. . . 4
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)))‘(𝐴(,]𝐵)) =
((int‘((TopOpen‘ℂfld) ↾t ((𝐴(,]𝐵) ∪ {𝐵})))‘((𝐴(,)𝐵) ∪ {𝐵}))) |
59 | 48, 58 | eleqtrd 2690 |
. . 3
⊢ (𝜑 → 𝐵 ∈
((int‘((TopOpen‘ℂfld) ↾t ((𝐴(,]𝐵) ∪ {𝐵})))‘((𝐴(,)𝐵) ∪ {𝐵}))) |
60 | 29, 31, 10, 12, 32, 59 | limcres 23456 |
. 2
⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵) = (𝐹 limℂ 𝐵)) |
61 | 8, 60 | eleqtrrd 2691 |
1
⊢ (𝜑 → (𝐹‘𝐵) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵)) |