| 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 11312 | . . . 4
⊢ (𝜑 → 𝐵 ∈
ℝ*) | 
| 5 |  | ioccncflimc.altb | . . . 4
⊢ (𝜑 → 𝐴 < 𝐵) | 
| 6 | 3 | leidd 11830 | . . . 4
⊢ (𝜑 → 𝐵 ≤ 𝐵) | 
| 7 | 2, 4, 4, 5, 6 | eliocd 45525 | . . 3
⊢ (𝜑 → 𝐵 ∈ (𝐴(,]𝐵)) | 
| 8 | 1, 7 | cnlimci 25925 | . 2
⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 limℂ 𝐵)) | 
| 9 |  | cncfrss 24918 | . . . . . . . 8
⊢ (𝐹 ∈ ((𝐴(,]𝐵)–cn→ℂ) → (𝐴(,]𝐵) ⊆ ℂ) | 
| 10 | 1, 9 | syl 17 | . . . . . . 7
⊢ (𝜑 → (𝐴(,]𝐵) ⊆ ℂ) | 
| 11 |  | ssid 4005 | . . . . . . 7
⊢ ℂ
⊆ ℂ | 
| 12 |  | eqid 2736 | . . . . . . . 8
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) | 
| 13 |  | eqid 2736 | . . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) = ((TopOpen‘ℂfld)
↾t (𝐴(,]𝐵)) | 
| 14 |  | eqid 2736 | . . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t ℂ) =
((TopOpen‘ℂfld) ↾t
ℂ) | 
| 15 | 12, 13, 14 | cncfcn 24937 | . . . . . . 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 586 | . . . . . 6
⊢ (𝜑 → ((𝐴(,]𝐵)–cn→ℂ) =
(((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) | 
| 17 | 1, 16 | eleqtrd 2842 | . . . . 5
⊢ (𝜑 → 𝐹 ∈
(((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) | 
| 18 | 12 | cnfldtopon 24804 | . . . . . . 7
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) | 
| 19 |  | resttopon 23170 | . . . . . . 7
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ (𝐴(,]𝐵) ⊆ ℂ) →
((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) ∈ (TopOn‘(𝐴(,]𝐵))) | 
| 20 | 18, 10, 19 | sylancr 587 | . . . . . 6
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) ∈ (TopOn‘(𝐴(,]𝐵))) | 
| 21 | 12 | cnfldtop 24805 | . . . . . . . 8
⊢
(TopOpen‘ℂfld) ∈ Top | 
| 22 |  | unicntop 24807 | . . . . . . . . 9
⊢ ℂ =
∪
(TopOpen‘ℂfld) | 
| 23 | 22 | restid 17479 | . . . . . . . 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 24804 | . . . . . 6
⊢
((TopOpen‘ℂfld) ↾t ℂ)
∈ (TopOn‘ℂ) | 
| 26 |  | cncnp 23289 | . . . . . 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 586 | . . . . 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 232 | . . . 4
⊢ (𝜑 → (𝐹:(𝐴(,]𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴(,]𝐵)𝐹 ∈
((((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) CnP
((TopOpen‘ℂfld) ↾t
ℂ))‘𝑥))) | 
| 29 | 28 | simpld 494 | . . 3
⊢ (𝜑 → 𝐹:(𝐴(,]𝐵)⟶ℂ) | 
| 30 |  | ioossioc 45510 | . . . 4
⊢ (𝐴(,)𝐵) ⊆ (𝐴(,]𝐵) | 
| 31 | 30 | a1i 11 | . . 3
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,]𝐵)) | 
| 32 |  | eqid 2736 | . . 3
⊢
((TopOpen‘ℂfld) ↾t ((𝐴(,]𝐵) ∪ {𝐵})) = ((TopOpen‘ℂfld)
↾t ((𝐴(,]𝐵) ∪ {𝐵})) | 
| 33 | 3 | recnd 11290 | . . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℂ) | 
| 34 | 22 | ntrtop 23079 | . . . . . . . . 9
⊢
((TopOpen‘ℂfld) ∈ Top →
((int‘(TopOpen‘ℂfld))‘ℂ) =
ℂ) | 
| 35 | 21, 34 | ax-mp 5 | . . . . . . . 8
⊢
((int‘(TopOpen‘ℂfld))‘ℂ) =
ℂ | 
| 36 |  | undif 4481 | . . . . . . . . . . 11
⊢ ((𝐴(,]𝐵) ⊆ ℂ ↔ ((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵))) = ℂ) | 
| 37 | 10, 36 | sylib 218 | . . . . . . . . . 10
⊢ (𝜑 → ((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵))) = ℂ) | 
| 38 | 37 | eqcomd 2742 | . . . . . . . . 9
⊢ (𝜑 → ℂ = ((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵)))) | 
| 39 | 38 | fveq2d 6909 | . . . . . . . 8
⊢ (𝜑 →
((int‘(TopOpen‘ℂfld))‘ℂ) =
((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵))))) | 
| 40 | 35, 39 | eqtr3id 2790 | . . . . . . 7
⊢ (𝜑 → ℂ =
((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵))))) | 
| 41 | 33, 40 | eleqtrd 2842 | . . . . . 6
⊢ (𝜑 → 𝐵 ∈
((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵))))) | 
| 42 | 41, 7 | elind 4199 | . . . . 5
⊢ (𝜑 → 𝐵 ∈
(((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵)))) ∩ (𝐴(,]𝐵))) | 
| 43 | 21 | a1i 11 | . . . . . 6
⊢ (𝜑 →
(TopOpen‘ℂfld) ∈ Top) | 
| 44 |  | ssid 4005 | . . . . . . 7
⊢ (𝐴(,]𝐵) ⊆ (𝐴(,]𝐵) | 
| 45 | 44 | a1i 11 | . . . . . 6
⊢ (𝜑 → (𝐴(,]𝐵) ⊆ (𝐴(,]𝐵)) | 
| 46 | 22, 13 | restntr 23191 | . . . . . 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 1372 | . . . . 5
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)))‘(𝐴(,]𝐵)) =
(((int‘(TopOpen‘ℂfld))‘((𝐴(,]𝐵) ∪ (ℂ ∖ (𝐴(,]𝐵)))) ∩ (𝐴(,]𝐵))) | 
| 48 | 42, 47 | eleqtrrd 2843 | . . . 4
⊢ (𝜑 → 𝐵 ∈
((int‘((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)))‘(𝐴(,]𝐵))) | 
| 49 | 7 | snssd 4808 | . . . . . . . . 9
⊢ (𝜑 → {𝐵} ⊆ (𝐴(,]𝐵)) | 
| 50 |  | ssequn2 4188 | . . . . . . . . 9
⊢ ({𝐵} ⊆ (𝐴(,]𝐵) ↔ ((𝐴(,]𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) | 
| 51 | 49, 50 | sylib 218 | . . . . . . . 8
⊢ (𝜑 → ((𝐴(,]𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) | 
| 52 | 51 | eqcomd 2742 | . . . . . . 7
⊢ (𝜑 → (𝐴(,]𝐵) = ((𝐴(,]𝐵) ∪ {𝐵})) | 
| 53 | 52 | oveq2d 7448 | . . . . . 6
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)) = ((TopOpen‘ℂfld)
↾t ((𝐴(,]𝐵) ∪ {𝐵}))) | 
| 54 | 53 | fveq2d 6909 | . . . . 5
⊢ (𝜑 →
(int‘((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵))) =
(int‘((TopOpen‘ℂfld) ↾t ((𝐴(,]𝐵) ∪ {𝐵})))) | 
| 55 |  | ioounsn 13518 | . . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
< 𝐵) → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) | 
| 56 | 2, 4, 5, 55 | syl3anc 1372 | . . . . . 6
⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐵}) = (𝐴(,]𝐵)) | 
| 57 | 56 | eqcomd 2742 | . . . . 5
⊢ (𝜑 → (𝐴(,]𝐵) = ((𝐴(,)𝐵) ∪ {𝐵})) | 
| 58 | 54, 57 | fveq12d 6912 | . . . 4
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t (𝐴(,]𝐵)))‘(𝐴(,]𝐵)) =
((int‘((TopOpen‘ℂfld) ↾t ((𝐴(,]𝐵) ∪ {𝐵})))‘((𝐴(,)𝐵) ∪ {𝐵}))) | 
| 59 | 48, 58 | eleqtrd 2842 | . . 3
⊢ (𝜑 → 𝐵 ∈
((int‘((TopOpen‘ℂfld) ↾t ((𝐴(,]𝐵) ∪ {𝐵})))‘((𝐴(,)𝐵) ∪ {𝐵}))) | 
| 60 | 29, 31, 10, 12, 32, 59 | limcres 25922 | . 2
⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵) = (𝐹 limℂ 𝐵)) | 
| 61 | 8, 60 | eleqtrrd 2843 | 1
⊢ (𝜑 → (𝐹‘𝐵) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐵)) |