Step | Hyp | Ref
| Expression |
1 | | limccl.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
2 | 1 | fdmd 6509 |
. . . . . . 7
⊢ (𝜑 → dom 𝐹 = 𝐴) |
3 | 2 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → dom 𝐹 = 𝐴) |
4 | | limcrcl 24618 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
5 | 4 | adantl 485 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
6 | 5 | simp2d 1144 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → dom 𝐹 ⊆ ℂ) |
7 | 3, 6 | eqsstrrd 3914 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → 𝐴 ⊆ ℂ) |
8 | 5 | simp3d 1145 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → 𝐵 ∈ ℂ) |
9 | 7, 8 | jca 515 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐹 limℂ 𝐵)) → (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
10 | 9 | ex 416 |
. . 3
⊢ (𝜑 → (𝑥 ∈ (𝐹 limℂ 𝐵) → (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ))) |
11 | | undif1 4362 |
. . . . . . 7
⊢ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = (𝐴 ∪ {𝐵}) |
12 | | difss 4020 |
. . . . . . . . . . . 12
⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴 |
13 | | fssres 6538 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝐴⟶ℂ ∧ (𝐴 ∖ {𝐵}) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) |
14 | 1, 12, 13 | sylancl 589 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) |
15 | 14 | fdmd 6509 |
. . . . . . . . . 10
⊢ (𝜑 → dom (𝐹 ↾ (𝐴 ∖ {𝐵})) = (𝐴 ∖ {𝐵})) |
16 | 15 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → dom (𝐹 ↾ (𝐴 ∖ {𝐵})) = (𝐴 ∖ {𝐵})) |
17 | | limcrcl 24618 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵) → ((𝐹 ↾ (𝐴 ∖ {𝐵})):dom (𝐹 ↾ (𝐴 ∖ {𝐵}))⟶ℂ ∧ dom (𝐹 ↾ (𝐴 ∖ {𝐵})) ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
18 | 17 | adantl 485 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → ((𝐹 ↾ (𝐴 ∖ {𝐵})):dom (𝐹 ↾ (𝐴 ∖ {𝐵}))⟶ℂ ∧ dom (𝐹 ↾ (𝐴 ∖ {𝐵})) ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
19 | 18 | simp2d 1144 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → dom (𝐹 ↾ (𝐴 ∖ {𝐵})) ⊆ ℂ) |
20 | 16, 19 | eqsstrrd 3914 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → (𝐴 ∖ {𝐵}) ⊆ ℂ) |
21 | 18 | simp3d 1145 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → 𝐵 ∈ ℂ) |
22 | 21 | snssd 4694 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → {𝐵} ⊆ ℂ) |
23 | 20, 22 | unssd 4074 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ ℂ) |
24 | 11, 23 | eqsstrrid 3924 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → (𝐴 ∪ {𝐵}) ⊆ ℂ) |
25 | 24 | unssad 4075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → 𝐴 ⊆ ℂ) |
26 | 25, 21 | jca 515 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) → (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) |
27 | 26 | ex 416 |
. . 3
⊢ (𝜑 → (𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵) → (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ))) |
28 | | eqid 2738 |
. . . . . 6
⊢
((TopOpen‘ℂfld) ↾t (𝐴 ∪ {𝐵})) = ((TopOpen‘ℂfld)
↾t (𝐴
∪ {𝐵})) |
29 | | eqid 2738 |
. . . . . 6
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
30 | | eqid 2738 |
. . . . . 6
⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) = (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) |
31 | 1 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐹:𝐴⟶ℂ) |
32 | | simprl 771 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐴 ⊆ ℂ) |
33 | | simprr 773 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → 𝐵 ∈ ℂ) |
34 | 28, 29, 30, 31, 32, 33 | ellimc 24617 |
. . . . 5
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) ∈
((((TopOpen‘ℂfld) ↾t (𝐴 ∪ {𝐵})) CnP
(TopOpen‘ℂfld))‘𝐵))) |
35 | 11 | eqcomi 2747 |
. . . . . . 7
⊢ (𝐴 ∪ {𝐵}) = ((𝐴 ∖ {𝐵}) ∪ {𝐵}) |
36 | 35 | oveq2i 7175 |
. . . . . 6
⊢
((TopOpen‘ℂfld) ↾t (𝐴 ∪ {𝐵})) = ((TopOpen‘ℂfld)
↾t ((𝐴
∖ {𝐵}) ∪ {𝐵})) |
37 | 35 | mpteq1i 5117 |
. . . . . . 7
⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) = (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) |
38 | | elun 4037 |
. . . . . . . . 9
⊢ (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 ∈ {𝐵})) |
39 | | velsn 4529 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵) |
40 | 39 | orbi2i 912 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 ∈ {𝐵}) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 = 𝐵)) |
41 | | pm5.61 1000 |
. . . . . . . . . . . 12
⊢ (((𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) ↔ (𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ ¬ 𝑧 = 𝐵)) |
42 | | fvres 6687 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) → ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧) = (𝐹‘𝑧)) |
43 | 42 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ ¬ 𝑧 = 𝐵) → ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧) = (𝐹‘𝑧)) |
44 | 41, 43 | sylbi 220 |
. . . . . . . . . . 11
⊢ (((𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 = 𝐵) ∧ ¬ 𝑧 = 𝐵) → ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧) = (𝐹‘𝑧)) |
45 | 44 | ifeq2da 4443 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 = 𝐵) → if(𝑧 = 𝐵, 𝑥, ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧)) = if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) |
46 | 40, 45 | sylbi 220 |
. . . . . . . . 9
⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∨ 𝑧 ∈ {𝐵}) → if(𝑧 = 𝐵, 𝑥, ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧)) = if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) |
47 | 38, 46 | sylbi 220 |
. . . . . . . 8
⊢ (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) → if(𝑧 = 𝐵, 𝑥, ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧)) = if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) |
48 | 47 | mpteq2ia 5118 |
. . . . . . 7
⊢ (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧))) = (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) |
49 | 37, 48 | eqtr4i 2764 |
. . . . . 6
⊢ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) = (𝑧 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, ((𝐹 ↾ (𝐴 ∖ {𝐵}))‘𝑧))) |
50 | 14 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → (𝐹 ↾ (𝐴 ∖ {𝐵})):(𝐴 ∖ {𝐵})⟶ℂ) |
51 | 32 | ssdifssd 4031 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → (𝐴 ∖ {𝐵}) ⊆ ℂ) |
52 | 36, 29, 49, 50, 51, 33 | ellimc 24617 |
. . . . 5
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → (𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵) ↔ (𝑧 ∈ (𝐴 ∪ {𝐵}) ↦ if(𝑧 = 𝐵, 𝑥, (𝐹‘𝑧))) ∈
((((TopOpen‘ℂfld) ↾t (𝐴 ∪ {𝐵})) CnP
(TopOpen‘ℂfld))‘𝐵))) |
53 | 34, 52 | bitr4d 285 |
. . . 4
⊢ ((𝜑 ∧ (𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ)) → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵))) |
54 | 53 | ex 416 |
. . 3
⊢ (𝜑 → ((𝐴 ⊆ ℂ ∧ 𝐵 ∈ ℂ) → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)))) |
55 | 10, 27, 54 | pm5.21ndd 384 |
. 2
⊢ (𝜑 → (𝑥 ∈ (𝐹 limℂ 𝐵) ↔ 𝑥 ∈ ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵))) |
56 | 55 | eqrdv 2736 |
1
⊢ (𝜑 → (𝐹 limℂ 𝐵) = ((𝐹 ↾ (𝐴 ∖ {𝐵})) limℂ 𝐵)) |