Step | Hyp | Ref
| Expression |
1 | | limcresioolb.d |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈
ℝ*) |
2 | | limcresioolb.cled |
. . . . . 6
⊢ (𝜑 → 𝐶 ≤ 𝐷) |
3 | | iooss2 12082 |
. . . . . 6
⊢ ((𝐷 ∈ ℝ*
∧ 𝐶 ≤ 𝐷) → (𝐵(,)𝐶) ⊆ (𝐵(,)𝐷)) |
4 | 1, 2, 3 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐵(,)𝐷)) |
5 | 4 | resabs1d 5348 |
. . . 4
⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶)) = (𝐹 ↾ (𝐵(,)𝐶))) |
6 | 5 | eqcomd 2616 |
. . 3
⊢ (𝜑 → (𝐹 ↾ (𝐵(,)𝐶)) = ((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶))) |
7 | 6 | oveq1d 6564 |
. 2
⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) limℂ 𝐵) = (((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶)) limℂ 𝐵)) |
8 | | limcresioolb.f |
. . . 4
⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
9 | | fresin 5986 |
. . . 4
⊢ (𝐹:𝐴⟶ℂ → (𝐹 ↾ (𝐵(,)𝐷)):(𝐴 ∩ (𝐵(,)𝐷))⟶ℂ) |
10 | 8, 9 | syl 17 |
. . 3
⊢ (𝜑 → (𝐹 ↾ (𝐵(,)𝐷)):(𝐴 ∩ (𝐵(,)𝐷))⟶ℂ) |
11 | | limcresioolb.bcss |
. . . 4
⊢ (𝜑 → (𝐵(,)𝐶) ⊆ 𝐴) |
12 | 11, 4 | ssind 3799 |
. . 3
⊢ (𝜑 → (𝐵(,)𝐶) ⊆ (𝐴 ∩ (𝐵(,)𝐷))) |
13 | | inss2 3796 |
. . . . 5
⊢ (𝐴 ∩ (𝐵(,)𝐷)) ⊆ (𝐵(,)𝐷) |
14 | | ioosscn 38563 |
. . . . 5
⊢ (𝐵(,)𝐷) ⊆ ℂ |
15 | 13, 14 | sstri 3577 |
. . . 4
⊢ (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℂ |
16 | 15 | a1i 11 |
. . 3
⊢ (𝜑 → (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℂ) |
17 | | eqid 2610 |
. . 3
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
18 | | eqid 2610 |
. . 3
⊢
((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) = ((TopOpen‘ℂfld)
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
19 | | limcresioolb.b |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ ℝ) |
20 | 19 | rexrd 9968 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
21 | | limcresioolb.c |
. . . . 5
⊢ (𝜑 → 𝐶 ∈
ℝ*) |
22 | | limcresioolb.bltc |
. . . . 5
⊢ (𝜑 → 𝐵 < 𝐶) |
23 | | lbico1 12099 |
. . . . 5
⊢ ((𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ* ∧ 𝐵
< 𝐶) → 𝐵 ∈ (𝐵[,)𝐶)) |
24 | 20, 21, 22, 23 | syl3anc 1318 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ (𝐵[,)𝐶)) |
25 | | snunioo1 38585 |
. . . . . . 7
⊢ ((𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ* ∧ 𝐵
< 𝐶) → ((𝐵(,)𝐶) ∪ {𝐵}) = (𝐵[,)𝐶)) |
26 | 20, 21, 22, 25 | syl3anc 1318 |
. . . . . 6
⊢ (𝜑 → ((𝐵(,)𝐶) ∪ {𝐵}) = (𝐵[,)𝐶)) |
27 | 26 | fveq2d 6107 |
. . . . 5
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘((𝐵(,)𝐶) ∪ {𝐵})) =
((int‘((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘(𝐵[,)𝐶))) |
28 | 17 | cnfldtop 22397 |
. . . . . . . 8
⊢
(TopOpen‘ℂfld) ∈ Top |
29 | | ovex 6577 |
. . . . . . . . . 10
⊢ (𝐵(,)𝐷) ∈ V |
30 | 29 | inex2 4728 |
. . . . . . . . 9
⊢ (𝐴 ∩ (𝐵(,)𝐷)) ∈ V |
31 | | snex 4835 |
. . . . . . . . 9
⊢ {𝐵} ∈ V |
32 | 30, 31 | unex 6854 |
. . . . . . . 8
⊢ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∈ V |
33 | | resttop 20774 |
. . . . . . . 8
⊢
(((TopOpen‘ℂfld) ∈ Top ∧ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∈ V) →
((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ Top) |
34 | 28, 32, 33 | mp2an 704 |
. . . . . . 7
⊢
((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ Top |
35 | 34 | a1i 11 |
. . . . . 6
⊢ (𝜑 →
((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ Top) |
36 | | mnfxr 9975 |
. . . . . . . . . . . . 13
⊢ -∞
∈ ℝ* |
37 | 36 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → -∞ ∈
ℝ*) |
38 | 21 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝐶 ∈
ℝ*) |
39 | | icossre 12125 |
. . . . . . . . . . . . . 14
⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ*)
→ (𝐵[,)𝐶) ⊆
ℝ) |
40 | 19, 21, 39 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐵[,)𝐶) ⊆ ℝ) |
41 | 40 | sselda 3568 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ ℝ) |
42 | 41 | mnfltd 11834 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → -∞ < 𝑥) |
43 | 20 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝐵 ∈
ℝ*) |
44 | | simpr 476 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ (𝐵[,)𝐶)) |
45 | | icoltub 38579 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ* ∧ 𝑥
∈ (𝐵[,)𝐶)) → 𝑥 < 𝐶) |
46 | 43, 38, 44, 45 | syl3anc 1318 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 < 𝐶) |
47 | 37, 38, 41, 42, 46 | eliood 38567 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ (-∞(,)𝐶)) |
48 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) |
49 | | snidg 4153 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ∈ ℝ → 𝐵 ∈ {𝐵}) |
50 | | elun2 3743 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ∈ {𝐵} → 𝐵 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
51 | 19, 49, 50 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐵 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
52 | 51 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐵 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
53 | 48, 52 | eqeltrd 2688 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
54 | 53 | adantlr 747 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
55 | | simpll 786 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝜑) |
56 | 43 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈
ℝ*) |
57 | 38 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐶 ∈
ℝ*) |
58 | 41 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ) |
59 | 19 | ad2antrr 758 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈ ℝ) |
60 | | icogelb 12096 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐵 ∈ ℝ*
∧ 𝐶 ∈
ℝ* ∧ 𝑥
∈ (𝐵[,)𝐶)) → 𝐵 ≤ 𝑥) |
61 | 43, 38, 44, 60 | syl3anc 1318 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝐵 ≤ 𝑥) |
62 | 61 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ≤ 𝑥) |
63 | | neqne 2790 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑥 = 𝐵 → 𝑥 ≠ 𝐵) |
64 | 63 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ≠ 𝐵) |
65 | 59, 58, 62, 64 | leneltd 10070 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝐵 < 𝑥) |
66 | 46 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐶) |
67 | 56, 57, 58, 65, 66 | eliood 38567 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐶)) |
68 | 12 | sselda 3568 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷))) |
69 | | elun1 3742 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷)) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
70 | 68, 69 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵(,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
71 | 55, 67, 70 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
72 | 54, 71 | pm2.61dan 828 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
73 | 47, 72 | elind 3760 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐵[,)𝐶)) → 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) |
74 | 24 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐵 ∈ (𝐵[,)𝐶)) |
75 | 48, 74 | eqeltrd 2688 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,)𝐶)) |
76 | 75 | adantlr 747 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,)𝐶)) |
77 | | ioossico 12133 |
. . . . . . . . . . . 12
⊢ (𝐵(,)𝐶) ⊆ (𝐵[,)𝐶) |
78 | 20 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐵 ∈
ℝ*) |
79 | 21 | ad2antrr 758 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐶 ∈
ℝ*) |
80 | | elinel1 3761 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) → 𝑥 ∈ (-∞(,)𝐶)) |
81 | | elioore 12076 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ (-∞(,)𝐶) → 𝑥 ∈ ℝ) |
82 | 80, 81 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) → 𝑥 ∈ ℝ) |
83 | 82 | ad2antlr 759 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ℝ) |
84 | 1 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐷 ∈
ℝ*) |
85 | | elinel2 3762 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) → 𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
86 | | id 22 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
𝑥 = 𝐵 → ¬ 𝑥 = 𝐵) |
87 | | velsn 4141 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) |
88 | 86, 87 | sylnibr 318 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑥 = 𝐵 → ¬ 𝑥 ∈ {𝐵}) |
89 | | elunnel2 38221 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ∈ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∧ ¬ 𝑥 ∈ {𝐵}) → 𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷))) |
90 | 85, 88, 89 | syl2an 493 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴 ∩ (𝐵(,)𝐷))) |
91 | 13, 90 | sseldi 3566 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐷)) |
92 | 91 | adantll 746 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐷)) |
93 | | ioogtlb 38564 |
. . . . . . . . . . . . . 14
⊢ ((𝐵 ∈ ℝ*
∧ 𝐷 ∈
ℝ* ∧ 𝑥
∈ (𝐵(,)𝐷)) → 𝐵 < 𝑥) |
94 | 78, 84, 92, 93 | syl3anc 1318 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝐵 < 𝑥) |
95 | 36 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → -∞ ∈
ℝ*) |
96 | 21 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → 𝐶 ∈
ℝ*) |
97 | 80 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → 𝑥 ∈ (-∞(,)𝐶)) |
98 | | iooltub 38582 |
. . . . . . . . . . . . . . 15
⊢
((-∞ ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑥 ∈ (-∞(,)𝐶)) → 𝑥 < 𝐶) |
99 | 95, 96, 97, 98 | syl3anc 1318 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → 𝑥 < 𝐶) |
100 | 99 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 < 𝐶) |
101 | 78, 79, 83, 94, 100 | eliood 38567 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵(,)𝐶)) |
102 | 77, 101 | sseldi 3566 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐵[,)𝐶)) |
103 | 76, 102 | pm2.61dan 828 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) → 𝑥 ∈ (𝐵[,)𝐶)) |
104 | 73, 103 | impbida 873 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ (𝐵[,)𝐶) ↔ 𝑥 ∈ ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))) |
105 | 104 | eqrdv 2608 |
. . . . . . . 8
⊢ (𝜑 → (𝐵[,)𝐶) = ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) |
106 | | retop 22375 |
. . . . . . . . . 10
⊢
(topGen‘ran (,)) ∈ Top |
107 | 106 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (topGen‘ran (,))
∈ Top) |
108 | 32 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∈ V) |
109 | | iooretop 22379 |
. . . . . . . . . 10
⊢
(-∞(,)𝐶)
∈ (topGen‘ran (,)) |
110 | 109 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (-∞(,)𝐶) ∈ (topGen‘ran
(,))) |
111 | | elrestr 15912 |
. . . . . . . . 9
⊢
(((topGen‘ran (,)) ∈ Top ∧ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ∈ V ∧ (-∞(,)𝐶) ∈ (topGen‘ran (,)))
→ ((-∞(,)𝐶)
∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ ((topGen‘ran (,))
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵}))) |
112 | 107, 108,
110, 111 | syl3anc 1318 |
. . . . . . . 8
⊢ (𝜑 → ((-∞(,)𝐶) ∩ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ ((topGen‘ran (,))
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵}))) |
113 | 105, 112 | eqeltrd 2688 |
. . . . . . 7
⊢ (𝜑 → (𝐵[,)𝐶) ∈ ((topGen‘ran (,))
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵}))) |
114 | 17 | tgioo2 22414 |
. . . . . . . . 9
⊢
(topGen‘ran (,)) = ((TopOpen‘ℂfld)
↾t ℝ) |
115 | 114 | oveq1i 6559 |
. . . . . . . 8
⊢
((topGen‘ran (,)) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) =
(((TopOpen‘ℂfld) ↾t ℝ)
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵})) |
116 | 28 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 →
(TopOpen‘ℂfld) ∈ Top) |
117 | | ioossre 12106 |
. . . . . . . . . . . 12
⊢ (𝐵(,)𝐷) ⊆ ℝ |
118 | 13, 117 | sstri 3577 |
. . . . . . . . . . 11
⊢ (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℝ |
119 | 118 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ∩ (𝐵(,)𝐷)) ⊆ ℝ) |
120 | 19 | snssd 4281 |
. . . . . . . . . 10
⊢ (𝜑 → {𝐵} ⊆ ℝ) |
121 | 119, 120 | unssd 3751 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ⊆ ℝ) |
122 | | reex 9906 |
. . . . . . . . . 10
⊢ ℝ
∈ V |
123 | 122 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → ℝ ∈
V) |
124 | | restabs 20779 |
. . . . . . . . 9
⊢
(((TopOpen‘ℂfld) ∈ Top ∧ ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}) ⊆ ℝ ∧ ℝ ∈ V)
→ (((TopOpen‘ℂfld) ↾t ℝ)
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵})) = ((TopOpen‘ℂfld)
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵}))) |
125 | 116, 121,
123, 124 | syl3anc 1318 |
. . . . . . . 8
⊢ (𝜑 →
(((TopOpen‘ℂfld) ↾t ℝ)
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵})) = ((TopOpen‘ℂfld)
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵}))) |
126 | 115, 125 | syl5eq 2656 |
. . . . . . 7
⊢ (𝜑 → ((topGen‘ran (,))
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵})) = ((TopOpen‘ℂfld)
↾t ((𝐴
∩ (𝐵(,)𝐷)) ∪ {𝐵}))) |
127 | 113, 126 | eleqtrd 2690 |
. . . . . 6
⊢ (𝜑 → (𝐵[,)𝐶) ∈
((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) |
128 | | isopn3i 20696 |
. . . . . 6
⊢
((((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})) ∈ Top ∧ (𝐵[,)𝐶) ∈
((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵}))) →
((int‘((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘(𝐵[,)𝐶)) = (𝐵[,)𝐶)) |
129 | 35, 127, 128 | syl2anc 691 |
. . . . 5
⊢ (𝜑 →
((int‘((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘(𝐵[,)𝐶)) = (𝐵[,)𝐶)) |
130 | 27, 129 | eqtr2d 2645 |
. . . 4
⊢ (𝜑 → (𝐵[,)𝐶) =
((int‘((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘((𝐵(,)𝐶) ∪ {𝐵}))) |
131 | 24, 130 | eleqtrd 2690 |
. . 3
⊢ (𝜑 → 𝐵 ∈
((int‘((TopOpen‘ℂfld) ↾t ((𝐴 ∩ (𝐵(,)𝐷)) ∪ {𝐵})))‘((𝐵(,)𝐶) ∪ {𝐵}))) |
132 | 10, 12, 16, 17, 18, 131 | limcres 23456 |
. 2
⊢ (𝜑 → (((𝐹 ↾ (𝐵(,)𝐷)) ↾ (𝐵(,)𝐶)) limℂ 𝐵) = ((𝐹 ↾ (𝐵(,)𝐷)) limℂ 𝐵)) |
133 | 7, 132 | eqtrd 2644 |
1
⊢ (𝜑 → ((𝐹 ↾ (𝐵(,)𝐶)) limℂ 𝐵) = ((𝐹 ↾ (𝐵(,)𝐷)) limℂ 𝐵)) |