| Step | Hyp | Ref
| Expression |
| 1 | | icocncflimc.f |
. . 3
⊢ (𝜑 → 𝐹 ∈ ((𝐴[,)𝐵)–cn→ℂ)) |
| 2 | | icocncflimc.a |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ ℝ) |
| 3 | 2 | rexrd 11311 |
. . . 4
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| 4 | | icocncflimc.b |
. . . 4
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 5 | 2 | leidd 11829 |
. . . 4
⊢ (𝜑 → 𝐴 ≤ 𝐴) |
| 6 | | icocncflimc.altb |
. . . 4
⊢ (𝜑 → 𝐴 < 𝐵) |
| 7 | 3, 4, 3, 5, 6 | elicod 13437 |
. . 3
⊢ (𝜑 → 𝐴 ∈ (𝐴[,)𝐵)) |
| 8 | 1, 7 | cnlimci 25924 |
. 2
⊢ (𝜑 → (𝐹‘𝐴) ∈ (𝐹 limℂ 𝐴)) |
| 9 | | cncfrss 24917 |
. . . . . . . 8
⊢ (𝐹 ∈ ((𝐴[,)𝐵)–cn→ℂ) → (𝐴[,)𝐵) ⊆ ℂ) |
| 10 | 1, 9 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝐴[,)𝐵) ⊆ ℂ) |
| 11 | | ssid 4006 |
. . . . . . 7
⊢ ℂ
⊆ ℂ |
| 12 | | eqid 2737 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 13 | | eqid 2737 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵)) = ((TopOpen‘ℂfld)
↾t (𝐴[,)𝐵)) |
| 14 | | eqid 2737 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t ℂ) =
((TopOpen‘ℂfld) ↾t
ℂ) |
| 15 | 12, 13, 14 | cncfcn 24936 |
. . . . . . 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 2843 |
. . . . 5
⊢ (𝜑 → 𝐹 ∈
(((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) |
| 18 | 12 | cnfldtopon 24803 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
| 19 | 18 | a1i 11 |
. . . . . . 7
⊢ (𝜑 →
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ)) |
| 20 | | resttopon 23169 |
. . . . . . 7
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ (𝐴[,)𝐵) ⊆ ℂ) →
((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵)) ∈ (TopOn‘(𝐴[,)𝐵))) |
| 21 | 19, 10, 20 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵)) ∈ (TopOn‘(𝐴[,)𝐵))) |
| 22 | 12 | cnfldtop 24804 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) ∈ Top |
| 23 | | unicntop 24806 |
. . . . . . . . 9
⊢ ℂ =
∪
(TopOpen‘ℂfld) |
| 24 | 23 | restid 17478 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ∈ Top →
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld)) |
| 25 | 22, 24 | ax-mp 5 |
. . . . . . 7
⊢
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld) |
| 26 | 25 | cnfldtopon 24803 |
. . . . . 6
⊢
((TopOpen‘ℂfld) ↾t ℂ)
∈ (TopOn‘ℂ) |
| 27 | | cncnp 23288 |
. . . . . 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)))) |
| 28 | 21, 26, 27 | 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)))) |
| 29 | 17, 28 | mpbid 232 |
. . . 4
⊢ (𝜑 → (𝐹:(𝐴[,)𝐵)⟶ℂ ∧ ∀𝑥 ∈ (𝐴[,)𝐵)𝐹 ∈
((((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵)) CnP
((TopOpen‘ℂfld) ↾t
ℂ))‘𝑥))) |
| 30 | 29 | simpld 494 |
. . 3
⊢ (𝜑 → 𝐹:(𝐴[,)𝐵)⟶ℂ) |
| 31 | | ioossico 13478 |
. . . 4
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵) |
| 32 | 31 | a1i 11 |
. . 3
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,)𝐵)) |
| 33 | | eqid 2737 |
. . 3
⊢
((TopOpen‘ℂfld) ↾t ((𝐴[,)𝐵) ∪ {𝐴})) = ((TopOpen‘ℂfld)
↾t ((𝐴[,)𝐵) ∪ {𝐴})) |
| 34 | 2 | recnd 11289 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 35 | 23 | ntrtop 23078 |
. . . . . . . . 9
⊢
((TopOpen‘ℂfld) ∈ Top →
((int‘(TopOpen‘ℂfld))‘ℂ) =
ℂ) |
| 36 | 22, 35 | ax-mp 5 |
. . . . . . . 8
⊢
((int‘(TopOpen‘ℂfld))‘ℂ) =
ℂ |
| 37 | | undif 4482 |
. . . . . . . . . . 11
⊢ ((𝐴[,)𝐵) ⊆ ℂ ↔ ((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵))) = ℂ) |
| 38 | 10, 37 | sylib 218 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵))) = ℂ) |
| 39 | 38 | eqcomd 2743 |
. . . . . . . . 9
⊢ (𝜑 → ℂ = ((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))) |
| 40 | 39 | fveq2d 6910 |
. . . . . . . 8
⊢ (𝜑 →
((int‘(TopOpen‘ℂfld))‘ℂ) =
((int‘(TopOpen‘ℂfld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵))))) |
| 41 | 36, 40 | eqtr3id 2791 |
. . . . . . 7
⊢ (𝜑 → ℂ =
((int‘(TopOpen‘ℂfld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵))))) |
| 42 | 34, 41 | eleqtrd 2843 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈
((int‘(TopOpen‘ℂfld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵))))) |
| 43 | 42, 7 | elind 4200 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈
(((int‘(TopOpen‘ℂfld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))) ∩ (𝐴[,)𝐵))) |
| 44 | 22 | a1i 11 |
. . . . . 6
⊢ (𝜑 →
(TopOpen‘ℂfld) ∈ Top) |
| 45 | | ssid 4006 |
. . . . . . 7
⊢ (𝐴[,)𝐵) ⊆ (𝐴[,)𝐵) |
| 46 | 45 | a1i 11 |
. . . . . 6
⊢ (𝜑 → (𝐴[,)𝐵) ⊆ (𝐴[,)𝐵)) |
| 47 | 23, 13 | restntr 23190 |
. . . . . 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))) |
| 48 | 44, 10, 46, 47 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵)))‘(𝐴[,)𝐵)) =
(((int‘(TopOpen‘ℂfld))‘((𝐴[,)𝐵) ∪ (ℂ ∖ (𝐴[,)𝐵)))) ∩ (𝐴[,)𝐵))) |
| 49 | 43, 48 | eleqtrrd 2844 |
. . . 4
⊢ (𝜑 → 𝐴 ∈
((int‘((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵)))‘(𝐴[,)𝐵))) |
| 50 | 7 | snssd 4809 |
. . . . . . . . 9
⊢ (𝜑 → {𝐴} ⊆ (𝐴[,)𝐵)) |
| 51 | | ssequn2 4189 |
. . . . . . . . 9
⊢ ({𝐴} ⊆ (𝐴[,)𝐵) ↔ ((𝐴[,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵)) |
| 52 | 50, 51 | sylib 218 |
. . . . . . . 8
⊢ (𝜑 → ((𝐴[,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵)) |
| 53 | 52 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → (𝐴[,)𝐵) = ((𝐴[,)𝐵) ∪ {𝐴})) |
| 54 | 53 | oveq2d 7447 |
. . . . . 6
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵)) = ((TopOpen‘ℂfld)
↾t ((𝐴[,)𝐵) ∪ {𝐴}))) |
| 55 | 54 | fveq2d 6910 |
. . . . 5
⊢ (𝜑 →
(int‘((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵))) =
(int‘((TopOpen‘ℂfld) ↾t ((𝐴[,)𝐵) ∪ {𝐴})))) |
| 56 | | snunioo1 45525 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
< 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵)) |
| 57 | 3, 4, 6, 56 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → ((𝐴(,)𝐵) ∪ {𝐴}) = (𝐴[,)𝐵)) |
| 58 | 57 | eqcomd 2743 |
. . . . 5
⊢ (𝜑 → (𝐴[,)𝐵) = ((𝐴(,)𝐵) ∪ {𝐴})) |
| 59 | 55, 58 | fveq12d 6913 |
. . . 4
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t (𝐴[,)𝐵)))‘(𝐴[,)𝐵)) =
((int‘((TopOpen‘ℂfld) ↾t ((𝐴[,)𝐵) ∪ {𝐴})))‘((𝐴(,)𝐵) ∪ {𝐴}))) |
| 60 | 49, 59 | eleqtrd 2843 |
. . 3
⊢ (𝜑 → 𝐴 ∈
((int‘((TopOpen‘ℂfld) ↾t ((𝐴[,)𝐵) ∪ {𝐴})))‘((𝐴(,)𝐵) ∪ {𝐴}))) |
| 61 | 30, 32, 10, 12, 33, 60 | limcres 25921 |
. 2
⊢ (𝜑 → ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐴) = (𝐹 limℂ 𝐴)) |
| 62 | 8, 61 | eleqtrrd 2844 |
1
⊢ (𝜑 → (𝐹‘𝐴) ∈ ((𝐹 ↾ (𝐴(,)𝐵)) limℂ 𝐴)) |