Proof of Theorem dvatan
Step | Hyp | Ref
| Expression |
1 | | cnelprrecn 10345 |
. . . . 5
⊢ ℂ
∈ {ℝ, ℂ} |
2 | 1 | a1i 11 |
. . . 4
⊢ (⊤
→ ℂ ∈ {ℝ, ℂ}) |
3 | | ax-1cn 10310 |
. . . . . . 7
⊢ 1 ∈
ℂ |
4 | | ax-icn 10311 |
. . . . . . . 8
⊢ i ∈
ℂ |
5 | | atansopn.d |
. . . . . . . . . . . 12
⊢ 𝐷 = (ℂ ∖
(-∞(,]0)) |
6 | | atansopn.s |
. . . . . . . . . . . 12
⊢ 𝑆 = {𝑦 ∈ ℂ ∣ (1 + (𝑦↑2)) ∈ 𝐷} |
7 | 5, 6 | atansssdm 25073 |
. . . . . . . . . . 11
⊢ 𝑆 ⊆ dom
arctan |
8 | | simpr 479 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 𝑥 ∈ 𝑆) |
9 | 7, 8 | sseldi 3825 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 𝑥 ∈ dom
arctan) |
10 | | atandm2 25017 |
. . . . . . . . . 10
⊢ (𝑥 ∈ dom arctan ↔ (𝑥 ∈ ℂ ∧ (1 −
(i · 𝑥)) ≠ 0
∧ (1 + (i · 𝑥))
≠ 0)) |
11 | 9, 10 | sylib 210 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 ∈ ℂ ∧ (1 −
(i · 𝑥)) ≠ 0
∧ (1 + (i · 𝑥))
≠ 0)) |
12 | 11 | simp1d 1176 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 𝑥 ∈
ℂ) |
13 | | mulcl 10336 |
. . . . . . . 8
⊢ ((i
∈ ℂ ∧ 𝑥
∈ ℂ) → (i · 𝑥) ∈ ℂ) |
14 | 4, 12, 13 | sylancr 581 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (i
· 𝑥) ∈
ℂ) |
15 | | subcl 10600 |
. . . . . . 7
⊢ ((1
∈ ℂ ∧ (i · 𝑥) ∈ ℂ) → (1 − (i
· 𝑥)) ∈
ℂ) |
16 | 3, 14, 15 | sylancr 581 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 −
(i · 𝑥)) ∈
ℂ) |
17 | 11 | simp2d 1177 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 −
(i · 𝑥)) ≠
0) |
18 | 16, 17 | logcld 24716 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ 𝑆) →
(log‘(1 − (i · 𝑥))) ∈ ℂ) |
19 | | addcl 10334 |
. . . . . . 7
⊢ ((1
∈ ℂ ∧ (i · 𝑥) ∈ ℂ) → (1 + (i ·
𝑥)) ∈
ℂ) |
20 | 3, 14, 19 | sylancr 581 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 + (i
· 𝑥)) ∈
ℂ) |
21 | 11 | simp3d 1178 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 + (i
· 𝑥)) ≠
0) |
22 | 20, 21 | logcld 24716 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ 𝑆) →
(log‘(1 + (i · 𝑥))) ∈ ℂ) |
23 | 18, 22 | subcld 10713 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ 𝑆) →
((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))) ∈
ℂ) |
24 | | ovexd 6939 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((2 / i) /
(1 + (𝑥↑2))) ∈
V) |
25 | | ovexd 6939 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 /
(𝑥 + i)) ∈
V) |
26 | 5, 6 | atans2 25071 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℂ ∧ (1 − (i ·
𝑥)) ∈ 𝐷 ∧ (1 + (i · 𝑥)) ∈ 𝐷)) |
27 | 26 | simp2bi 1180 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝑆 → (1 − (i · 𝑥)) ∈ 𝐷) |
28 | 27 | adantl 475 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 −
(i · 𝑥)) ∈
𝐷) |
29 | | negex 10599 |
. . . . . . . . 9
⊢ -i ∈
V |
30 | 29 | a1i 11 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → -i ∈
V) |
31 | 5 | logdmss 24787 |
. . . . . . . . . 10
⊢ 𝐷 ⊆ (ℂ ∖
{0}) |
32 | | simpr 479 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑦
∈ 𝐷) → 𝑦 ∈ 𝐷) |
33 | 31, 32 | sseldi 3825 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑦
∈ 𝐷) → 𝑦 ∈ (ℂ ∖
{0})) |
34 | | logf1o 24710 |
. . . . . . . . . . 11
⊢
log:(ℂ ∖ {0})–1-1-onto→ran
log |
35 | | f1of 6378 |
. . . . . . . . . . 11
⊢
(log:(ℂ ∖ {0})–1-1-onto→ran
log → log:(ℂ ∖ {0})⟶ran log) |
36 | 34, 35 | ax-mp 5 |
. . . . . . . . . 10
⊢
log:(ℂ ∖ {0})⟶ran log |
37 | 36 | ffvelrni 6607 |
. . . . . . . . 9
⊢ (𝑦 ∈ (ℂ ∖ {0})
→ (log‘𝑦) ∈
ran log) |
38 | | logrncn 24708 |
. . . . . . . . 9
⊢
((log‘𝑦)
∈ ran log → (log‘𝑦) ∈ ℂ) |
39 | 33, 37, 38 | 3syl 18 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑦
∈ 𝐷) →
(log‘𝑦) ∈
ℂ) |
40 | | ovexd 6939 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑦
∈ 𝐷) → (1 / 𝑦) ∈ V) |
41 | 4 | a1i 11 |
. . . . . . . . . . 11
⊢ (⊤
→ i ∈ ℂ) |
42 | 41, 13 | sylan 575 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ ℂ) → (i · 𝑥) ∈ ℂ) |
43 | 3, 42, 15 | sylancr 581 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ ℂ) → (1 − (i · 𝑥)) ∈ ℂ) |
44 | 29 | a1i 11 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ ℂ) → -i ∈ V) |
45 | | 1cnd 10351 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ ℂ) → 1 ∈ ℂ) |
46 | | 0cnd 10349 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ ℂ) → 0 ∈ ℂ) |
47 | | 1cnd 10351 |
. . . . . . . . . . . 12
⊢ (⊤
→ 1 ∈ ℂ) |
48 | 2, 47 | dvmptc 24120 |
. . . . . . . . . . 11
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ 1)) = (𝑥
∈ ℂ ↦ 0)) |
49 | 4 | a1i 11 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ ℂ) → i ∈ ℂ) |
50 | | simpr 479 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ ℂ) → 𝑥
∈ ℂ) |
51 | 2 | dvmptid 24119 |
. . . . . . . . . . . . 13
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ 𝑥)) =
(𝑥 ∈ ℂ ↦
1)) |
52 | 2, 50, 45, 51, 41 | dvmptcmul 24126 |
. . . . . . . . . . . 12
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (i · 𝑥))) = (𝑥 ∈ ℂ ↦ (i ·
1))) |
53 | 4 | mulid1i 10361 |
. . . . . . . . . . . . 13
⊢ (i
· 1) = i |
54 | 53 | mpteq2i 4964 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℂ ↦ (i
· 1)) = (𝑥 ∈
ℂ ↦ i) |
55 | 52, 54 | syl6eq 2877 |
. . . . . . . . . . 11
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (i · 𝑥))) = (𝑥 ∈ ℂ ↦ i)) |
56 | 2, 45, 46, 48, 42, 49, 55 | dvmptsub 24129 |
. . . . . . . . . 10
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (1 − (i · 𝑥)))) = (𝑥 ∈ ℂ ↦ (0 −
i))) |
57 | | df-neg 10588 |
. . . . . . . . . . 11
⊢ -i = (0
− i) |
58 | 57 | mpteq2i 4964 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℂ ↦ -i) =
(𝑥 ∈ ℂ ↦
(0 − i)) |
59 | 56, 58 | syl6eqr 2879 |
. . . . . . . . 9
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (1 − (i · 𝑥)))) = (𝑥 ∈ ℂ ↦ -i)) |
60 | | eqid 2825 |
. . . . . . . . . . . 12
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
61 | 60 | cnfldtopon 22956 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
62 | 5, 6 | atansopn 25072 |
. . . . . . . . . . 11
⊢ 𝑆 ∈
(TopOpen‘ℂfld) |
63 | | toponss 21102 |
. . . . . . . . . . 11
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ 𝑆 ∈
(TopOpen‘ℂfld)) → 𝑆 ⊆ ℂ) |
64 | 61, 62, 63 | mp2an 683 |
. . . . . . . . . 10
⊢ 𝑆 ⊆
ℂ |
65 | 64 | a1i 11 |
. . . . . . . . 9
⊢ (⊤
→ 𝑆 ⊆
ℂ) |
66 | 61 | toponrestid 21096 |
. . . . . . . . 9
⊢
(TopOpen‘ℂfld) =
((TopOpen‘ℂfld) ↾t
ℂ) |
67 | 62 | a1i 11 |
. . . . . . . . 9
⊢ (⊤
→ 𝑆 ∈
(TopOpen‘ℂfld)) |
68 | 2, 43, 44, 59, 65, 66, 60, 67 | dvmptres 24125 |
. . . . . . . 8
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (1 − (i
· 𝑥)))) = (𝑥 ∈ 𝑆 ↦ -i)) |
69 | | fssres 6307 |
. . . . . . . . . . . . . 14
⊢
((log:(ℂ ∖ {0})⟶ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log
↾ 𝐷):𝐷⟶ran
log) |
70 | 36, 31, 69 | mp2an 683 |
. . . . . . . . . . . . 13
⊢ (log
↾ 𝐷):𝐷⟶ran log |
71 | 70 | a1i 11 |
. . . . . . . . . . . 12
⊢ (⊤
→ (log ↾ 𝐷):𝐷⟶ran log) |
72 | 71 | feqmptd 6496 |
. . . . . . . . . . 11
⊢ (⊤
→ (log ↾ 𝐷) =
(𝑦 ∈ 𝐷 ↦ ((log ↾ 𝐷)‘𝑦))) |
73 | | fvres 6452 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ 𝐷 → ((log ↾ 𝐷)‘𝑦) = (log‘𝑦)) |
74 | 73 | mpteq2ia 4963 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ 𝐷 ↦ ((log ↾ 𝐷)‘𝑦)) = (𝑦 ∈ 𝐷 ↦ (log‘𝑦)) |
75 | 72, 74 | syl6req 2878 |
. . . . . . . . . 10
⊢ (⊤
→ (𝑦 ∈ 𝐷 ↦ (log‘𝑦)) = (log ↾ 𝐷)) |
76 | 75 | oveq2d 6921 |
. . . . . . . . 9
⊢ (⊤
→ (ℂ D (𝑦 ∈
𝐷 ↦ (log‘𝑦))) = (ℂ D (log ↾
𝐷))) |
77 | 5 | dvlog 24796 |
. . . . . . . . 9
⊢ (ℂ
D (log ↾ 𝐷)) = (𝑦 ∈ 𝐷 ↦ (1 / 𝑦)) |
78 | 76, 77 | syl6eq 2877 |
. . . . . . . 8
⊢ (⊤
→ (ℂ D (𝑦 ∈
𝐷 ↦ (log‘𝑦))) = (𝑦 ∈ 𝐷 ↦ (1 / 𝑦))) |
79 | | fveq2 6433 |
. . . . . . . 8
⊢ (𝑦 = (1 − (i · 𝑥)) → (log‘𝑦) = (log‘(1 − (i
· 𝑥)))) |
80 | | oveq2 6913 |
. . . . . . . 8
⊢ (𝑦 = (1 − (i · 𝑥)) → (1 / 𝑦) = (1 / (1 − (i ·
𝑥)))) |
81 | 2, 2, 28, 30, 39, 40, 68, 78, 79, 80 | dvmptco 24134 |
. . . . . . 7
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (log‘(1
− (i · 𝑥)))))
= (𝑥 ∈ 𝑆 ↦ ((1 / (1 − (i
· 𝑥))) ·
-i))) |
82 | | irec 13258 |
. . . . . . . . . 10
⊢ (1 / i) =
-i |
83 | 82 | oveq2i 6916 |
. . . . . . . . 9
⊢ ((1 / (1
− (i · 𝑥)))
· (1 / i)) = ((1 / (1 − (i · 𝑥))) · -i) |
84 | 4 | a1i 11 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → i ∈
ℂ) |
85 | | ine0 10789 |
. . . . . . . . . . . 12
⊢ i ≠
0 |
86 | 85 | a1i 11 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → i ≠
0) |
87 | 16, 84, 17, 86 | recdiv2d 11145 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / (1
− (i · 𝑥))) /
i) = (1 / ((1 − (i · 𝑥)) · i))) |
88 | 16, 17 | reccld 11120 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / (1
− (i · 𝑥)))
∈ ℂ) |
89 | 88, 84, 86 | divrecd 11130 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / (1
− (i · 𝑥))) /
i) = ((1 / (1 − (i · 𝑥))) · (1 / i))) |
90 | | 1cnd 10351 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 1 ∈
ℂ) |
91 | 90, 14, 84 | subdird 10811 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1
− (i · 𝑥))
· i) = ((1 · i) − ((i · 𝑥) · i))) |
92 | 4 | mulid2i 10362 |
. . . . . . . . . . . . . . 15
⊢ (1
· i) = i |
93 | 92 | a1i 11 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1
· i) = i) |
94 | 84, 12, 84 | mul32d 10565 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((i
· 𝑥) · i) =
((i · i) · 𝑥)) |
95 | | ixi 10981 |
. . . . . . . . . . . . . . . . 17
⊢ (i
· i) = -1 |
96 | 95 | oveq1i 6915 |
. . . . . . . . . . . . . . . 16
⊢ ((i
· i) · 𝑥) =
(-1 · 𝑥) |
97 | 12 | mulm1d 10806 |
. . . . . . . . . . . . . . . 16
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (-1
· 𝑥) = -𝑥) |
98 | 96, 97 | syl5eq 2873 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((i
· i) · 𝑥) =
-𝑥) |
99 | 94, 98 | eqtrd 2861 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((i
· 𝑥) · i) =
-𝑥) |
100 | 93, 99 | oveq12d 6923 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1
· i) − ((i · 𝑥) · i)) = (i − -𝑥)) |
101 | | subneg 10651 |
. . . . . . . . . . . . . 14
⊢ ((i
∈ ℂ ∧ 𝑥
∈ ℂ) → (i − -𝑥) = (i + 𝑥)) |
102 | 4, 12, 101 | sylancr 581 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (i −
-𝑥) = (i + 𝑥)) |
103 | 91, 100, 102 | 3eqtrd 2865 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1
− (i · 𝑥))
· i) = (i + 𝑥)) |
104 | | addcom 10541 |
. . . . . . . . . . . . 13
⊢ ((i
∈ ℂ ∧ 𝑥
∈ ℂ) → (i + 𝑥) = (𝑥 + i)) |
105 | 4, 12, 104 | sylancr 581 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (i + 𝑥) = (𝑥 + i)) |
106 | 103, 105 | eqtrd 2861 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1
− (i · 𝑥))
· i) = (𝑥 +
i)) |
107 | 106 | oveq2d 6921 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / ((1
− (i · 𝑥))
· i)) = (1 / (𝑥 +
i))) |
108 | 87, 89, 107 | 3eqtr3d 2869 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / (1
− (i · 𝑥)))
· (1 / i)) = (1 / (𝑥
+ i))) |
109 | 83, 108 | syl5eqr 2875 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / (1
− (i · 𝑥)))
· -i) = (1 / (𝑥 +
i))) |
110 | 109 | mpteq2dva 4967 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈ 𝑆 ↦ ((1 / (1 − (i
· 𝑥))) · -i))
= (𝑥 ∈ 𝑆 ↦ (1 / (𝑥 + i)))) |
111 | 81, 110 | eqtrd 2861 |
. . . . . 6
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (log‘(1
− (i · 𝑥)))))
= (𝑥 ∈ 𝑆 ↦ (1 / (𝑥 + i)))) |
112 | | ovexd 6939 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 /
(𝑥 − i)) ∈
V) |
113 | 26 | simp3bi 1181 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝑆 → (1 + (i · 𝑥)) ∈ 𝐷) |
114 | 113 | adantl 475 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 + (i
· 𝑥)) ∈ 𝐷) |
115 | 3, 42, 19 | sylancr 581 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ ℂ) → (1 + (i · 𝑥)) ∈ ℂ) |
116 | 2, 45, 46, 48, 42, 49, 55 | dvmptadd 24122 |
. . . . . . . . . 10
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (1 + (i · 𝑥)))) = (𝑥 ∈ ℂ ↦ (0 +
i))) |
117 | 4 | addid2i 10543 |
. . . . . . . . . . 11
⊢ (0 + i) =
i |
118 | 117 | mpteq2i 4964 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℂ ↦ (0 + i))
= (𝑥 ∈ ℂ ↦
i) |
119 | 116, 118 | syl6eq 2877 |
. . . . . . . . 9
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (1 + (i · 𝑥)))) = (𝑥 ∈ ℂ ↦ i)) |
120 | 2, 115, 49, 119, 65, 66, 60, 67 | dvmptres 24125 |
. . . . . . . 8
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (1 + (i ·
𝑥)))) = (𝑥 ∈ 𝑆 ↦ i)) |
121 | | fveq2 6433 |
. . . . . . . 8
⊢ (𝑦 = (1 + (i · 𝑥)) → (log‘𝑦) = (log‘(1 + (i ·
𝑥)))) |
122 | | oveq2 6913 |
. . . . . . . 8
⊢ (𝑦 = (1 + (i · 𝑥)) → (1 / 𝑦) = (1 / (1 + (i · 𝑥)))) |
123 | 2, 2, 114, 84, 39, 40, 120, 78, 121, 122 | dvmptco 24134 |
. . . . . . 7
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (log‘(1 +
(i · 𝑥))))) = (𝑥 ∈ 𝑆 ↦ ((1 / (1 + (i · 𝑥))) ·
i))) |
124 | 90, 20, 84, 21, 86 | divdiv2d 11159 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / ((1 +
(i · 𝑥)) / i)) = ((1
· i) / (1 + (i · 𝑥)))) |
125 | 90, 14, 84, 86 | divdird 11165 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 + (i
· 𝑥)) / i) = ((1 /
i) + ((i · 𝑥) /
i))) |
126 | 82 | a1i 11 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / i) =
-i) |
127 | 12, 84, 86 | divcan3d 11132 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((i
· 𝑥) / i) = 𝑥) |
128 | 126, 127 | oveq12d 6923 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / i) +
((i · 𝑥) / i)) = (-i
+ 𝑥)) |
129 | | negicn 10602 |
. . . . . . . . . . . . 13
⊢ -i ∈
ℂ |
130 | | addcom 10541 |
. . . . . . . . . . . . 13
⊢ ((-i
∈ ℂ ∧ 𝑥
∈ ℂ) → (-i + 𝑥) = (𝑥 + -i)) |
131 | 129, 12, 130 | sylancr 581 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (-i +
𝑥) = (𝑥 + -i)) |
132 | | negsub 10650 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → (𝑥 + -i) =
(𝑥 −
i)) |
133 | 12, 4, 132 | sylancl 580 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 + -i) = (𝑥 − i)) |
134 | 131, 133 | eqtrd 2861 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (-i +
𝑥) = (𝑥 − i)) |
135 | 125, 128,
134 | 3eqtrd 2865 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 + (i
· 𝑥)) / i) = (𝑥 − i)) |
136 | 135 | oveq2d 6921 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / ((1 +
(i · 𝑥)) / i)) = (1
/ (𝑥 −
i))) |
137 | 90, 84, 20, 21 | div23d 11164 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1
· i) / (1 + (i · 𝑥))) = ((1 / (1 + (i · 𝑥))) ·
i)) |
138 | 124, 136,
137 | 3eqtr3rd 2870 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / (1 +
(i · 𝑥))) ·
i) = (1 / (𝑥 −
i))) |
139 | 138 | mpteq2dva 4967 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈ 𝑆 ↦ ((1 / (1 + (i ·
𝑥))) · i)) = (𝑥 ∈ 𝑆 ↦ (1 / (𝑥 − i)))) |
140 | 123, 139 | eqtrd 2861 |
. . . . . 6
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (log‘(1 +
(i · 𝑥))))) = (𝑥 ∈ 𝑆 ↦ (1 / (𝑥 − i)))) |
141 | 2, 18, 25, 111, 22, 112, 140 | dvmptsub 24129 |
. . . . 5
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥)))))) = (𝑥 ∈ 𝑆 ↦ ((1 / (𝑥 + i)) − (1 / (𝑥 − i))))) |
142 | | subcl 10600 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → (𝑥 −
i) ∈ ℂ) |
143 | 12, 4, 142 | sylancl 580 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 − i) ∈
ℂ) |
144 | | addcl 10334 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → (𝑥 + i)
∈ ℂ) |
145 | 12, 4, 144 | sylancl 580 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 + i) ∈
ℂ) |
146 | 12 | sqcld 13300 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥↑2) ∈
ℂ) |
147 | | addcl 10334 |
. . . . . . . . 9
⊢ ((1
∈ ℂ ∧ (𝑥↑2) ∈ ℂ) → (1 + (𝑥↑2)) ∈
ℂ) |
148 | 3, 146, 147 | sylancr 581 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 +
(𝑥↑2)) ∈
ℂ) |
149 | | atandm4 25019 |
. . . . . . . . . 10
⊢ (𝑥 ∈ dom arctan ↔ (𝑥 ∈ ℂ ∧ (1 +
(𝑥↑2)) ≠
0)) |
150 | 149 | simprbi 492 |
. . . . . . . . 9
⊢ (𝑥 ∈ dom arctan → (1 +
(𝑥↑2)) ≠
0) |
151 | 9, 150 | syl 17 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 +
(𝑥↑2)) ≠
0) |
152 | 143, 145,
148, 151 | divsubdird 11166 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 − i) − (𝑥 + i)) / (1 + (𝑥↑2))) = (((𝑥 − i) / (1 + (𝑥↑2))) − ((𝑥 + i) / (1 + (𝑥↑2))))) |
153 | 133 | oveq1d 6920 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 + -i) − (𝑥 + i)) = ((𝑥 − i) − (𝑥 + i))) |
154 | 129 | a1i 11 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → -i ∈
ℂ) |
155 | 12, 154, 84 | pnpcand 10750 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 + -i) − (𝑥 + i)) = (-i −
i)) |
156 | 153, 155 | eqtr3d 2863 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) − (𝑥 + i)) = (-i −
i)) |
157 | | 2cn 11426 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℂ |
158 | 157, 4, 85 | divreci 11096 |
. . . . . . . . . . 11
⊢ (2 / i) =
(2 · (1 / i)) |
159 | 82 | oveq2i 6916 |
. . . . . . . . . . 11
⊢ (2
· (1 / i)) = (2 · -i) |
160 | 158, 159 | eqtri 2849 |
. . . . . . . . . 10
⊢ (2 / i) =
(2 · -i) |
161 | 129 | 2timesi 11496 |
. . . . . . . . . 10
⊢ (2
· -i) = (-i + -i) |
162 | 129, 4 | negsubi 10680 |
. . . . . . . . . 10
⊢ (-i + -i)
= (-i − i) |
163 | 160, 161,
162 | 3eqtri 2853 |
. . . . . . . . 9
⊢ (2 / i) =
(-i − i) |
164 | 156, 163 | syl6eqr 2879 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) − (𝑥 + i)) = (2 /
i)) |
165 | 164 | oveq1d 6920 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 − i) − (𝑥 + i)) / (1 + (𝑥↑2))) = ((2 / i) / (1 +
(𝑥↑2)))) |
166 | 143 | mulid1d 10374 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) · 1) = (𝑥 − i)) |
167 | 143, 145 | mulcomd 10378 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) · (𝑥 + i)) = ((𝑥 + i) · (𝑥 − i))) |
168 | | i2 13259 |
. . . . . . . . . . . . . 14
⊢
(i↑2) = -1 |
169 | 168 | oveq2i 6916 |
. . . . . . . . . . . . 13
⊢ ((𝑥↑2) − (i↑2)) =
((𝑥↑2) −
-1) |
170 | | subneg 10651 |
. . . . . . . . . . . . . 14
⊢ (((𝑥↑2) ∈ ℂ ∧ 1
∈ ℂ) → ((𝑥↑2) − -1) = ((𝑥↑2) + 1)) |
171 | 146, 3, 170 | sylancl 580 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥↑2) − -1) = ((𝑥↑2) + 1)) |
172 | 169, 171 | syl5eq 2873 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥↑2) − (i↑2)) =
((𝑥↑2) +
1)) |
173 | | subsq 13266 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → ((𝑥↑2)
− (i↑2)) = ((𝑥 +
i) · (𝑥 −
i))) |
174 | 12, 4, 173 | sylancl 580 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥↑2) − (i↑2)) =
((𝑥 + i) · (𝑥 − i))) |
175 | | addcom 10541 |
. . . . . . . . . . . . 13
⊢ (((𝑥↑2) ∈ ℂ ∧ 1
∈ ℂ) → ((𝑥↑2) + 1) = (1 + (𝑥↑2))) |
176 | 146, 3, 175 | sylancl 580 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥↑2) + 1) = (1 + (𝑥↑2))) |
177 | 172, 174,
176 | 3eqtr3d 2869 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 + i) · (𝑥 − i)) = (1 + (𝑥↑2))) |
178 | 167, 177 | eqtrd 2861 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) · (𝑥 + i)) = (1 + (𝑥↑2))) |
179 | 166, 178 | oveq12d 6923 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 − i) · 1) /
((𝑥 − i) ·
(𝑥 + i))) = ((𝑥 − i) / (1 + (𝑥↑2)))) |
180 | | subneg 10651 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → (𝑥 −
-i) = (𝑥 +
i)) |
181 | 12, 4, 180 | sylancl 580 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 − -i) = (𝑥 + i)) |
182 | | atandm 25016 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ dom arctan ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ -i ∧ 𝑥 ≠ i)) |
183 | 9, 182 | sylib 210 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ -i ∧ 𝑥 ≠ i)) |
184 | 183 | simp2d 1177 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 𝑥 ≠ -i) |
185 | | subeq0 10628 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℂ ∧ -i ∈
ℂ) → ((𝑥 −
-i) = 0 ↔ 𝑥 =
-i)) |
186 | 185 | necon3bid 3043 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ -i ∈
ℂ) → ((𝑥 −
-i) ≠ 0 ↔ 𝑥 ≠
-i)) |
187 | 12, 129, 186 | sylancl 580 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − -i) ≠ 0 ↔ 𝑥 ≠ -i)) |
188 | 184, 187 | mpbird 249 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 − -i) ≠
0) |
189 | 181, 188 | eqnetrrd 3067 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 + i) ≠ 0) |
190 | 183 | simp3d 1178 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 𝑥 ≠ i) |
191 | | subeq0 10628 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → ((𝑥 −
i) = 0 ↔ 𝑥 =
i)) |
192 | 191 | necon3bid 3043 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → ((𝑥 −
i) ≠ 0 ↔ 𝑥 ≠
i)) |
193 | 12, 4, 192 | sylancl 580 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) ≠ 0 ↔ 𝑥 ≠ i)) |
194 | 190, 193 | mpbird 249 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 − i) ≠
0) |
195 | 90, 145, 143, 189, 194 | divcan5d 11153 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 − i) · 1) /
((𝑥 − i) ·
(𝑥 + i))) = (1 / (𝑥 + i))) |
196 | 179, 195 | eqtr3d 2863 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) / (1 + (𝑥↑2))) = (1 / (𝑥 + i))) |
197 | 145 | mulid1d 10374 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 + i) · 1) = (𝑥 + i)) |
198 | 197, 177 | oveq12d 6923 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 + i) · 1) / ((𝑥 + i) · (𝑥 − i))) = ((𝑥 + i) / (1 + (𝑥↑2)))) |
199 | 90, 143, 145, 194, 189 | divcan5d 11153 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 + i) · 1) / ((𝑥 + i) · (𝑥 − i))) = (1 / (𝑥 − i))) |
200 | 198, 199 | eqtr3d 2863 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 + i) / (1 + (𝑥↑2))) = (1 / (𝑥 − i))) |
201 | 196, 200 | oveq12d 6923 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 − i) / (1 + (𝑥↑2))) − ((𝑥 + i) / (1 + (𝑥↑2)))) = ((1 / (𝑥 + i)) − (1 / (𝑥 − i)))) |
202 | 152, 165,
201 | 3eqtr3rd 2870 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 /
(𝑥 + i)) − (1 /
(𝑥 − i))) = ((2 / i)
/ (1 + (𝑥↑2)))) |
203 | 202 | mpteq2dva 4967 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈ 𝑆 ↦ ((1 / (𝑥 + i)) − (1 / (𝑥 − i)))) = (𝑥 ∈ 𝑆 ↦ ((2 / i) / (1 + (𝑥↑2))))) |
204 | 141, 203 | eqtrd 2861 |
. . . 4
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥)))))) = (𝑥 ∈ 𝑆 ↦ ((2 / i) / (1 + (𝑥↑2))))) |
205 | | halfcl 11583 |
. . . . 5
⊢ (i ∈
ℂ → (i / 2) ∈ ℂ) |
206 | 4, 205 | mp1i 13 |
. . . 4
⊢ (⊤
→ (i / 2) ∈ ℂ) |
207 | 2, 23, 24, 204, 206 | dvmptcmul 24126 |
. . 3
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ ((i / 2) ·
((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))) = (𝑥 ∈ 𝑆 ↦ ((i / 2) · ((2 / i) / (1 +
(𝑥↑2)))))) |
208 | | df-atan 25007 |
. . . . . . 7
⊢ arctan =
(𝑥 ∈ (ℂ ∖
{-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i
· 𝑥)))))) |
209 | 208 | reseq1i 5625 |
. . . . . 6
⊢ (arctan
↾ 𝑆) = ((𝑥 ∈ (ℂ ∖ {-i,
i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i
· 𝑥)))))) ↾
𝑆) |
210 | | atanf 25020 |
. . . . . . . . 9
⊢
arctan:(ℂ ∖ {-i, i})⟶ℂ |
211 | 210 | fdmi 6288 |
. . . . . . . 8
⊢ dom
arctan = (ℂ ∖ {-i, i}) |
212 | 7, 211 | sseqtri 3862 |
. . . . . . 7
⊢ 𝑆 ⊆ (ℂ ∖ {-i,
i}) |
213 | | resmpt 5686 |
. . . . . . 7
⊢ (𝑆 ⊆ (ℂ ∖ {-i,
i}) → ((𝑥 ∈
(ℂ ∖ {-i, i}) ↦ ((i / 2) · ((log‘(1 − (i
· 𝑥))) −
(log‘(1 + (i · 𝑥)))))) ↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ ((i / 2) · ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥))))))) |
214 | 212, 213 | ax-mp 5 |
. . . . . 6
⊢ ((𝑥 ∈ (ℂ ∖ {-i,
i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i
· 𝑥)))))) ↾
𝑆) = (𝑥 ∈ 𝑆 ↦ ((i / 2) · ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥)))))) |
215 | 209, 214 | eqtri 2849 |
. . . . 5
⊢ (arctan
↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ ((i / 2) · ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥)))))) |
216 | 215 | a1i 11 |
. . . 4
⊢ (⊤
→ (arctan ↾ 𝑆) =
(𝑥 ∈ 𝑆 ↦ ((i / 2) · ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥))))))) |
217 | 216 | oveq2d 6921 |
. . 3
⊢ (⊤
→ (ℂ D (arctan ↾ 𝑆)) = (ℂ D (𝑥 ∈ 𝑆 ↦ ((i / 2) · ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥)))))))) |
218 | | 2ne0 11462 |
. . . . . . 7
⊢ 2 ≠
0 |
219 | | divcan6 11058 |
. . . . . . 7
⊢ (((i
∈ ℂ ∧ i ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) →
((i / 2) · (2 / i)) = 1) |
220 | 4, 85, 157, 218, 219 | mp4an 684 |
. . . . . 6
⊢ ((i / 2)
· (2 / i)) = 1 |
221 | 220 | oveq1i 6915 |
. . . . 5
⊢ (((i / 2)
· (2 / i)) / (1 + (𝑥↑2))) = (1 / (1 + (𝑥↑2))) |
222 | 4, 205 | mp1i 13 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (i / 2)
∈ ℂ) |
223 | 157, 4, 85 | divcli 11093 |
. . . . . . 7
⊢ (2 / i)
∈ ℂ |
224 | 223 | a1i 11 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (2 / i)
∈ ℂ) |
225 | 222, 224,
148, 151 | divassd 11162 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((i / 2)
· (2 / i)) / (1 + (𝑥↑2))) = ((i / 2) · ((2 / i) / (1
+ (𝑥↑2))))) |
226 | 221, 225 | syl5eqr 2875 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / (1 +
(𝑥↑2))) = ((i / 2)
· ((2 / i) / (1 + (𝑥↑2))))) |
227 | 226 | mpteq2dva 4967 |
. . 3
⊢ (⊤
→ (𝑥 ∈ 𝑆 ↦ (1 / (1 + (𝑥↑2)))) = (𝑥 ∈ 𝑆 ↦ ((i / 2) · ((2 / i) / (1 +
(𝑥↑2)))))) |
228 | 207, 217,
227 | 3eqtr4d 2871 |
. 2
⊢ (⊤
→ (ℂ D (arctan ↾ 𝑆)) = (𝑥 ∈ 𝑆 ↦ (1 / (1 + (𝑥↑2))))) |
229 | 228 | mptru 1664 |
1
⊢ (ℂ
D (arctan ↾ 𝑆)) =
(𝑥 ∈ 𝑆 ↦ (1 / (1 + (𝑥↑2)))) |