Step | Hyp | Ref
| Expression |
1 | | dvfcn 25072 |
. . . 4
⊢ (ℂ
D (𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))):dom (ℂ D (𝑥 ∈ (ℂ ∖ {0})
↦ (𝐴 / 𝑥)))⟶ℂ |
2 | | ssidd 3944 |
. . . . . . 7
⊢ (𝐴 ∈ ℂ → ℂ
⊆ ℂ) |
3 | | eldifsn 4720 |
. . . . . . . . 9
⊢ (𝑥 ∈ (ℂ ∖ {0})
↔ (𝑥 ∈ ℂ
∧ 𝑥 ≠
0)) |
4 | | divcl 11639 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → (𝐴 / 𝑥) ∈ ℂ) |
5 | 4 | 3expb 1119 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0)) → (𝐴 / 𝑥) ∈ ℂ) |
6 | 3, 5 | sylan2b 594 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ (ℂ ∖ {0}))
→ (𝐴 / 𝑥) ∈
ℂ) |
7 | 6 | fmpttd 6989 |
. . . . . . 7
⊢ (𝐴 ∈ ℂ → (𝑥 ∈ (ℂ ∖ {0})
↦ (𝐴 / 𝑥)):(ℂ ∖
{0})⟶ℂ) |
8 | | difssd 4067 |
. . . . . . 7
⊢ (𝐴 ∈ ℂ → (ℂ
∖ {0}) ⊆ ℂ) |
9 | 2, 7, 8 | dvbss 25065 |
. . . . . 6
⊢ (𝐴 ∈ ℂ → dom
(ℂ D (𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥))) ⊆ (ℂ ∖
{0})) |
10 | | simpr 485 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ 𝑦 ∈ (ℂ
∖ {0})) |
11 | | eqid 2738 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
12 | 11 | cnfldtop 23947 |
. . . . . . . . . 10
⊢
(TopOpen‘ℂfld) ∈ Top |
13 | 11 | cnfldhaus 23948 |
. . . . . . . . . . . 12
⊢
(TopOpen‘ℂfld) ∈ Haus |
14 | | 0cn 10967 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℂ |
15 | | unicntop 23949 |
. . . . . . . . . . . . 13
⊢ ℂ =
∪
(TopOpen‘ℂfld) |
16 | 15 | sncld 22522 |
. . . . . . . . . . . 12
⊢
(((TopOpen‘ℂfld) ∈ Haus ∧ 0 ∈
ℂ) → {0} ∈
(Clsd‘(TopOpen‘ℂfld))) |
17 | 13, 14, 16 | mp2an 689 |
. . . . . . . . . . 11
⊢ {0}
∈ (Clsd‘(TopOpen‘ℂfld)) |
18 | 15 | cldopn 22182 |
. . . . . . . . . . 11
⊢ ({0}
∈ (Clsd‘(TopOpen‘ℂfld)) → (ℂ
∖ {0}) ∈ (TopOpen‘ℂfld)) |
19 | 17, 18 | ax-mp 5 |
. . . . . . . . . 10
⊢ (ℂ
∖ {0}) ∈ (TopOpen‘ℂfld) |
20 | | isopn3i 22233 |
. . . . . . . . . 10
⊢
(((TopOpen‘ℂfld) ∈ Top ∧ (ℂ
∖ {0}) ∈ (TopOpen‘ℂfld)) →
((int‘(TopOpen‘ℂfld))‘(ℂ ∖ {0}))
= (ℂ ∖ {0})) |
21 | 12, 19, 20 | mp2an 689 |
. . . . . . . . 9
⊢
((int‘(TopOpen‘ℂfld))‘(ℂ
∖ {0})) = (ℂ ∖ {0}) |
22 | 10, 21 | eleqtrrdi 2850 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ 𝑦 ∈
((int‘(TopOpen‘ℂfld))‘(ℂ ∖
{0}))) |
23 | | eldifi 4061 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ (ℂ ∖ {0})
→ 𝑦 ∈
ℂ) |
24 | 23 | adantl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ 𝑦 ∈
ℂ) |
25 | 24 | sqvald 13861 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (𝑦↑2) = (𝑦 · 𝑦)) |
26 | 25 | oveq2d 7291 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (𝐴 / (𝑦↑2)) = (𝐴 / (𝑦 · 𝑦))) |
27 | | simpl 483 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ 𝐴 ∈
ℂ) |
28 | | eldifsni 4723 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (ℂ ∖ {0})
→ 𝑦 ≠
0) |
29 | 28 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ 𝑦 ≠
0) |
30 | 27, 24, 24, 29, 29 | divdiv1d 11782 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ ((𝐴 / 𝑦) / 𝑦) = (𝐴 / (𝑦 · 𝑦))) |
31 | 26, 30 | eqtr4d 2781 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (𝐴 / (𝑦↑2)) = ((𝐴 / 𝑦) / 𝑦)) |
32 | 31 | negeqd 11215 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ -(𝐴 / (𝑦↑2)) = -((𝐴 / 𝑦) / 𝑦)) |
33 | 27, 24, 29 | divcld 11751 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (𝐴 / 𝑦) ∈
ℂ) |
34 | 33, 24, 29 | divnegd 11764 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ -((𝐴 / 𝑦) / 𝑦) = (-(𝐴 / 𝑦) / 𝑦)) |
35 | 32, 34 | eqtrd 2778 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ -(𝐴 / (𝑦↑2)) = (-(𝐴 / 𝑦) / 𝑦)) |
36 | 33 | negcld 11319 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ -(𝐴 / 𝑦) ∈
ℂ) |
37 | | eqid 2738 |
. . . . . . . . . . . . 13
⊢ (𝑧 ∈ (ℂ ∖ {0})
↦ (-(𝐴 / 𝑦) / 𝑧)) = (𝑧 ∈ (ℂ ∖ {0}) ↦
(-(𝐴 / 𝑦) / 𝑧)) |
38 | 37 | cdivcncf 24084 |
. . . . . . . . . . . 12
⊢ (-(𝐴 / 𝑦) ∈ ℂ → (𝑧 ∈ (ℂ ∖ {0}) ↦
(-(𝐴 / 𝑦) / 𝑧)) ∈ ((ℂ ∖ {0})–cn→ℂ)) |
39 | 36, 38 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (𝑧 ∈ (ℂ
∖ {0}) ↦ (-(𝐴 /
𝑦) / 𝑧)) ∈ ((ℂ ∖ {0})–cn→ℂ)) |
40 | | oveq2 7283 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑦 → (-(𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑦)) |
41 | 39, 10, 40 | cnmptlimc 25054 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (-(𝐴 / 𝑦) / 𝑦) ∈ ((𝑧 ∈ (ℂ ∖ {0}) ↦
(-(𝐴 / 𝑦) / 𝑧)) limℂ 𝑦)) |
42 | 35, 41 | eqeltrd 2839 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ (ℂ ∖ {0})
↦ (-(𝐴 / 𝑦) / 𝑧)) limℂ 𝑦)) |
43 | | cncff 24056 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ (ℂ ∖ {0})
↦ (-(𝐴 / 𝑦) / 𝑧)) ∈ ((ℂ ∖ {0})–cn→ℂ) → (𝑧 ∈ (ℂ ∖ {0}) ↦
(-(𝐴 / 𝑦) / 𝑧)):(ℂ ∖
{0})⟶ℂ) |
44 | 39, 43 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (𝑧 ∈ (ℂ
∖ {0}) ↦ (-(𝐴 /
𝑦) / 𝑧)):(ℂ ∖
{0})⟶ℂ) |
45 | 44 | limcdif 25040 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ ((𝑧 ∈ (ℂ
∖ {0}) ↦ (-(𝐴 /
𝑦) / 𝑧)) limℂ 𝑦) = (((𝑧 ∈ (ℂ ∖ {0}) ↦
(-(𝐴 / 𝑦) / 𝑧)) ↾ ((ℂ ∖ {0}) ∖
{𝑦})) limℂ
𝑦)) |
46 | | eldifi 4061 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ ((ℂ ∖ {0})
∖ {𝑦}) → 𝑧 ∈ (ℂ ∖
{0})) |
47 | 46 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ 𝑧 ∈ (ℂ
∖ {0})) |
48 | 47 | eldifad 3899 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ 𝑧 ∈
ℂ) |
49 | 23 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ 𝑦 ∈
ℂ) |
50 | 48, 49 | subcld 11332 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (𝑧 − 𝑦) ∈
ℂ) |
51 | 33 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (𝐴 / 𝑦) ∈
ℂ) |
52 | | eldifsni 4723 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ (ℂ ∖ {0})
→ 𝑧 ≠
0) |
53 | 47, 52 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ 𝑧 ≠
0) |
54 | 51, 48, 53 | divcld 11751 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((𝐴 / 𝑦) / 𝑧) ∈ ℂ) |
55 | | mulneg12 11413 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑧 − 𝑦) ∈ ℂ ∧ ((𝐴 / 𝑦) / 𝑧) ∈ ℂ) → (-(𝑧 − 𝑦) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑧 − 𝑦) · -((𝐴 / 𝑦) / 𝑧))) |
56 | 50, 54, 55 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (-(𝑧 − 𝑦) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑧 − 𝑦) · -((𝐴 / 𝑦) / 𝑧))) |
57 | 49, 48, 54 | subdird 11432 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((𝑦 − 𝑧) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑦 · ((𝐴 / 𝑦) / 𝑧)) − (𝑧 · ((𝐴 / 𝑦) / 𝑧)))) |
58 | 48, 49 | negsubdi2d 11348 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ -(𝑧 − 𝑦) = (𝑦 − 𝑧)) |
59 | 58 | oveq1d 7290 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (-(𝑧 − 𝑦) · ((𝐴 / 𝑦) / 𝑧)) = ((𝑦 − 𝑧) · ((𝐴 / 𝑦) / 𝑧))) |
60 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑧 → (𝐴 / 𝑥) = (𝐴 / 𝑧)) |
61 | | eqid 2738 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (ℂ ∖ {0})
↦ (𝐴 / 𝑥)) = (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥)) |
62 | | ovex 7308 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 / 𝑧) ∈ V |
63 | 60, 61, 62 | fvmpt 6875 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 ∈ (ℂ ∖ {0})
→ ((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑧) = (𝐴 / 𝑧)) |
64 | 47, 63 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑧) = (𝐴 / 𝑧)) |
65 | | simpll 764 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ 𝐴 ∈
ℂ) |
66 | 28 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ 𝑦 ≠
0) |
67 | 65, 49, 66 | divcan2d 11753 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (𝑦 · (𝐴 / 𝑦)) = 𝐴) |
68 | 67 | oveq1d 7290 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((𝑦 · (𝐴 / 𝑦)) / 𝑧) = (𝐴 / 𝑧)) |
69 | 49, 51, 48, 53 | divassd 11786 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((𝑦 · (𝐴 / 𝑦)) / 𝑧) = (𝑦 · ((𝐴 / 𝑦) / 𝑧))) |
70 | 64, 68, 69 | 3eqtr2d 2784 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑧) = (𝑦 · ((𝐴 / 𝑦) / 𝑧))) |
71 | | oveq2 7283 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑦 → (𝐴 / 𝑥) = (𝐴 / 𝑦)) |
72 | | ovex 7308 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 / 𝑦) ∈ V |
73 | 71, 61, 72 | fvmpt 6875 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (ℂ ∖ {0})
→ ((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑦) = (𝐴 / 𝑦)) |
74 | 73 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑦) = (𝐴 / 𝑦)) |
75 | 51, 48, 53 | divcan2d 11753 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (𝑧 · ((𝐴 / 𝑦) / 𝑧)) = (𝐴 / 𝑦)) |
76 | 74, 75 | eqtr4d 2781 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑦) = (𝑧 · ((𝐴 / 𝑦) / 𝑧))) |
77 | 70, 76 | oveq12d 7293 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) = ((𝑦 · ((𝐴 / 𝑦) / 𝑧)) − (𝑧 · ((𝐴 / 𝑦) / 𝑧)))) |
78 | 57, 59, 77 | 3eqtr4d 2788 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (-(𝑧 − 𝑦) · ((𝐴 / 𝑦) / 𝑧)) = (((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦))) |
79 | 51, 48, 53 | divnegd 11764 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ -((𝐴 / 𝑦) / 𝑧) = (-(𝐴 / 𝑦) / 𝑧)) |
80 | 79 | oveq2d 7291 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((𝑧 − 𝑦) · -((𝐴 / 𝑦) / 𝑧)) = ((𝑧 − 𝑦) · (-(𝐴 / 𝑦) / 𝑧))) |
81 | 56, 78, 80 | 3eqtr3d 2786 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) = ((𝑧 − 𝑦) · (-(𝐴 / 𝑦) / 𝑧))) |
82 | 81 | oveq1d 7290 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((((𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦)) = (((𝑧 − 𝑦) · (-(𝐴 / 𝑦) / 𝑧)) / (𝑧 − 𝑦))) |
83 | 51 | negcld 11319 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ -(𝐴 / 𝑦) ∈
ℂ) |
84 | 83, 48, 53 | divcld 11751 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (-(𝐴 / 𝑦) / 𝑧) ∈ ℂ) |
85 | | eldifsni 4723 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑧 ∈ ((ℂ ∖ {0})
∖ {𝑦}) → 𝑧 ≠ 𝑦) |
86 | 85 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ 𝑧 ≠ 𝑦) |
87 | 48, 49, 86 | subne0d 11341 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (𝑧 − 𝑦) ≠ 0) |
88 | 84, 50, 87 | divcan3d 11756 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ (((𝑧 − 𝑦) · (-(𝐴 / 𝑦) / 𝑧)) / (𝑧 − 𝑦)) = (-(𝐴 / 𝑦) / 𝑧)) |
89 | 82, 88 | eqtrd 2778 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
∧ 𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦}))
→ ((((𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦)) = (-(𝐴 / 𝑦) / 𝑧)) |
90 | 89 | mpteq2dva 5174 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦})
↦ ((((𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) = (𝑧 ∈ ((ℂ ∖ {0}) ∖ {𝑦}) ↦ (-(𝐴 / 𝑦) / 𝑧))) |
91 | | difss 4066 |
. . . . . . . . . . . . 13
⊢ ((ℂ
∖ {0}) ∖ {𝑦})
⊆ (ℂ ∖ {0}) |
92 | | resmpt 5945 |
. . . . . . . . . . . . 13
⊢
(((ℂ ∖ {0}) ∖ {𝑦}) ⊆ (ℂ ∖ {0}) →
((𝑧 ∈ (ℂ ∖
{0}) ↦ (-(𝐴 / 𝑦) / 𝑧)) ↾ ((ℂ ∖ {0}) ∖
{𝑦})) = (𝑧 ∈ ((ℂ ∖ {0}) ∖ {𝑦}) ↦ (-(𝐴 / 𝑦) / 𝑧))) |
93 | 91, 92 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ (ℂ ∖ {0})
↦ (-(𝐴 / 𝑦) / 𝑧)) ↾ ((ℂ ∖ {0}) ∖
{𝑦})) = (𝑧 ∈ ((ℂ ∖ {0}) ∖ {𝑦}) ↦ (-(𝐴 / 𝑦) / 𝑧)) |
94 | 90, 93 | eqtr4di 2796 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦})
↦ ((((𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) = ((𝑧 ∈ (ℂ ∖ {0}) ↦
(-(𝐴 / 𝑦) / 𝑧)) ↾ ((ℂ ∖ {0}) ∖
{𝑦}))) |
95 | 94 | oveq1d 7290 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ ((𝑧 ∈ ((ℂ
∖ {0}) ∖ {𝑦})
↦ ((((𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) limℂ 𝑦) = (((𝑧 ∈ (ℂ ∖ {0}) ↦
(-(𝐴 / 𝑦) / 𝑧)) ↾ ((ℂ ∖ {0}) ∖
{𝑦})) limℂ
𝑦)) |
96 | 45, 95 | eqtr4d 2781 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ ((𝑧 ∈ (ℂ
∖ {0}) ↦ (-(𝐴 /
𝑦) / 𝑧)) limℂ 𝑦) = ((𝑧 ∈ ((ℂ ∖ {0}) ∖ {𝑦}) ↦ ((((𝑥 ∈ (ℂ ∖ {0})
↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) limℂ 𝑦)) |
97 | 42, 96 | eleqtrd 2841 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ ((ℂ ∖ {0})
∖ {𝑦}) ↦
((((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) limℂ 𝑦)) |
98 | 11 | cnfldtopon 23946 |
. . . . . . . . . 10
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
99 | 98 | toponrestid 22070 |
. . . . . . . . 9
⊢
(TopOpen‘ℂfld) =
((TopOpen‘ℂfld) ↾t
ℂ) |
100 | | eqid 2738 |
. . . . . . . . 9
⊢ (𝑧 ∈ ((ℂ ∖ {0})
∖ {𝑦}) ↦
((((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) = (𝑧 ∈ ((ℂ ∖ {0}) ∖ {𝑦}) ↦ ((((𝑥 ∈ (ℂ ∖ {0})
↦ (𝐴 / 𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) |
101 | | ssidd 3944 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ ℂ ⊆ ℂ) |
102 | 7 | adantr 481 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥)):(ℂ ∖
{0})⟶ℂ) |
103 | | difssd 4067 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (ℂ ∖ {0}) ⊆ ℂ) |
104 | 99, 11, 100, 101, 102, 103 | eldv 25062 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ (𝑦(ℂ D (𝑥 ∈ (ℂ ∖ {0})
↦ (𝐴 / 𝑥)))-(𝐴 / (𝑦↑2)) ↔ (𝑦 ∈
((int‘(TopOpen‘ℂfld))‘(ℂ ∖ {0}))
∧ -(𝐴 / (𝑦↑2)) ∈ ((𝑧 ∈ ((ℂ ∖ {0})
∖ {𝑦}) ↦
((((𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))‘𝑧) − ((𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥))‘𝑦)) / (𝑧 − 𝑦))) limℂ 𝑦)))) |
105 | 22, 97, 104 | mpbir2and 710 |
. . . . . . 7
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ 𝑦(ℂ D (𝑥 ∈ (ℂ ∖ {0})
↦ (𝐴 / 𝑥)))-(𝐴 / (𝑦↑2))) |
106 | | vex 3436 |
. . . . . . . 8
⊢ 𝑦 ∈ V |
107 | | negex 11219 |
. . . . . . . 8
⊢ -(𝐴 / (𝑦↑2)) ∈ V |
108 | 106, 107 | breldm 5817 |
. . . . . . 7
⊢ (𝑦(ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥)))-(𝐴 / (𝑦↑2)) → 𝑦 ∈ dom (ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥)))) |
109 | 105, 108 | syl 17 |
. . . . . 6
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ 𝑦 ∈ dom
(ℂ D (𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥)))) |
110 | 9, 109 | eqelssd 3942 |
. . . . 5
⊢ (𝐴 ∈ ℂ → dom
(ℂ D (𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥))) = (ℂ ∖ {0})) |
111 | 110 | feq2d 6586 |
. . . 4
⊢ (𝐴 ∈ ℂ → ((ℂ
D (𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))):dom (ℂ D (𝑥 ∈ (ℂ ∖ {0})
↦ (𝐴 / 𝑥)))⟶ℂ ↔
(ℂ D (𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥))):(ℂ ∖
{0})⟶ℂ)) |
112 | 1, 111 | mpbii 232 |
. . 3
⊢ (𝐴 ∈ ℂ → (ℂ
D (𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))):(ℂ ∖
{0})⟶ℂ) |
113 | 112 | ffnd 6601 |
. 2
⊢ (𝐴 ∈ ℂ → (ℂ
D (𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))) Fn (ℂ ∖
{0})) |
114 | | negex 11219 |
. . . 4
⊢ -(𝐴 / (𝑥↑2)) ∈ V |
115 | 114 | rgenw 3076 |
. . 3
⊢
∀𝑥 ∈
(ℂ ∖ {0})-(𝐴 /
(𝑥↑2)) ∈
V |
116 | | eqid 2738 |
. . . 4
⊢ (𝑥 ∈ (ℂ ∖ {0})
↦ -(𝐴 / (𝑥↑2))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ -(𝐴 / (𝑥↑2))) |
117 | 116 | fnmpt 6573 |
. . 3
⊢
(∀𝑥 ∈
(ℂ ∖ {0})-(𝐴 /
(𝑥↑2)) ∈ V →
(𝑥 ∈ (ℂ ∖
{0}) ↦ -(𝐴 / (𝑥↑2))) Fn (ℂ ∖
{0})) |
118 | 115, 117 | mp1i 13 |
. 2
⊢ (𝐴 ∈ ℂ → (𝑥 ∈ (ℂ ∖ {0})
↦ -(𝐴 / (𝑥↑2))) Fn (ℂ ∖
{0})) |
119 | | ffun 6603 |
. . . . 5
⊢ ((ℂ
D (𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))):dom (ℂ D (𝑥 ∈ (ℂ ∖ {0})
↦ (𝐴 / 𝑥)))⟶ℂ → Fun
(ℂ D (𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥)))) |
120 | 1, 119 | mp1i 13 |
. . . 4
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ Fun (ℂ D (𝑥
∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥)))) |
121 | | funbrfv 6820 |
. . . 4
⊢ (Fun
(ℂ D (𝑥 ∈
(ℂ ∖ {0}) ↦ (𝐴 / 𝑥))) → (𝑦(ℂ D (𝑥 ∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥)))-(𝐴 / (𝑦↑2)) → ((ℂ D (𝑥 ∈ (ℂ ∖ {0})
↦ (𝐴 / 𝑥)))‘𝑦) = -(𝐴 / (𝑦↑2)))) |
122 | 120, 105,
121 | sylc 65 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ ((ℂ D (𝑥
∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥)))‘𝑦) = -(𝐴 / (𝑦↑2))) |
123 | | oveq1 7282 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑥↑2) = (𝑦↑2)) |
124 | 123 | oveq2d 7291 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝐴 / (𝑥↑2)) = (𝐴 / (𝑦↑2))) |
125 | 124 | negeqd 11215 |
. . . . 5
⊢ (𝑥 = 𝑦 → -(𝐴 / (𝑥↑2)) = -(𝐴 / (𝑦↑2))) |
126 | 125, 116,
107 | fvmpt 6875 |
. . . 4
⊢ (𝑦 ∈ (ℂ ∖ {0})
→ ((𝑥 ∈ (ℂ
∖ {0}) ↦ -(𝐴 /
(𝑥↑2)))‘𝑦) = -(𝐴 / (𝑦↑2))) |
127 | 126 | adantl 482 |
. . 3
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ ((𝑥 ∈ (ℂ
∖ {0}) ↦ -(𝐴 /
(𝑥↑2)))‘𝑦) = -(𝐴 / (𝑦↑2))) |
128 | 122, 127 | eqtr4d 2781 |
. 2
⊢ ((𝐴 ∈ ℂ ∧ 𝑦 ∈ (ℂ ∖ {0}))
→ ((ℂ D (𝑥
∈ (ℂ ∖ {0}) ↦ (𝐴 / 𝑥)))‘𝑦) = ((𝑥 ∈ (ℂ ∖ {0}) ↦ -(𝐴 / (𝑥↑2)))‘𝑦)) |
129 | 113, 118,
128 | eqfnfvd 6912 |
1
⊢ (𝐴 ∈ ℂ → (ℂ
D (𝑥 ∈ (ℂ
∖ {0}) ↦ (𝐴 /
𝑥))) = (𝑥 ∈ (ℂ ∖ {0}) ↦ -(𝐴 / (𝑥↑2)))) |