Proof of Theorem dvatan
Step | Hyp | Ref
| Expression |
1 | | cnelprrecn 10029 |
. . . . 5
⊢ ℂ
∈ {ℝ, ℂ} |
2 | 1 | a1i 11 |
. . . 4
⊢ (⊤
→ ℂ ∈ {ℝ, ℂ}) |
3 | | ax-1cn 9994 |
. . . . . . 7
⊢ 1 ∈
ℂ |
4 | | ax-icn 9995 |
. . . . . . . 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 24660 |
. . . . . . . . . . 11
⊢ 𝑆 ⊆ dom
arctan |
8 | | simpr 477 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 𝑥 ∈ 𝑆) |
9 | 7, 8 | sseldi 3601 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 𝑥 ∈ dom
arctan) |
10 | | atandm2 24604 |
. . . . . . . . . 10
⊢ (𝑥 ∈ dom arctan ↔ (𝑥 ∈ ℂ ∧ (1 −
(i · 𝑥)) ≠ 0
∧ (1 + (i · 𝑥))
≠ 0)) |
11 | 9, 10 | sylib 208 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 ∈ ℂ ∧ (1 −
(i · 𝑥)) ≠ 0
∧ (1 + (i · 𝑥))
≠ 0)) |
12 | 11 | simp1d 1073 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 𝑥 ∈
ℂ) |
13 | | mulcl 10020 |
. . . . . . . 8
⊢ ((i
∈ ℂ ∧ 𝑥
∈ ℂ) → (i · 𝑥) ∈ ℂ) |
14 | 4, 12, 13 | sylancr 695 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (i
· 𝑥) ∈
ℂ) |
15 | | subcl 10280 |
. . . . . . 7
⊢ ((1
∈ ℂ ∧ (i · 𝑥) ∈ ℂ) → (1 − (i
· 𝑥)) ∈
ℂ) |
16 | 3, 14, 15 | sylancr 695 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 −
(i · 𝑥)) ∈
ℂ) |
17 | 11 | simp2d 1074 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 −
(i · 𝑥)) ≠
0) |
18 | 16, 17 | logcld 24317 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ 𝑆) →
(log‘(1 − (i · 𝑥))) ∈ ℂ) |
19 | | addcl 10018 |
. . . . . . 7
⊢ ((1
∈ ℂ ∧ (i · 𝑥) ∈ ℂ) → (1 + (i ·
𝑥)) ∈
ℂ) |
20 | 3, 14, 19 | sylancr 695 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 + (i
· 𝑥)) ∈
ℂ) |
21 | 11 | simp3d 1075 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 + (i
· 𝑥)) ≠
0) |
22 | 20, 21 | logcld 24317 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ 𝑆) →
(log‘(1 + (i · 𝑥))) ∈ ℂ) |
23 | 18, 22 | subcld 10392 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ 𝑆) →
((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥)))) ∈
ℂ) |
24 | | ovexd 6680 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((2 / i) /
(1 + (𝑥↑2))) ∈
V) |
25 | | ovexd 6680 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 /
(𝑥 + i)) ∈
V) |
26 | 5, 6 | atans2 24658 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝑆 ↔ (𝑥 ∈ ℂ ∧ (1 − (i ·
𝑥)) ∈ 𝐷 ∧ (1 + (i · 𝑥)) ∈ 𝐷)) |
27 | 26 | simp2bi 1077 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝑆 → (1 − (i · 𝑥)) ∈ 𝐷) |
28 | 27 | adantl 482 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 −
(i · 𝑥)) ∈
𝐷) |
29 | | negex 10279 |
. . . . . . . . 9
⊢ -i ∈
V |
30 | 29 | a1i 11 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → -i ∈
V) |
31 | 5 | logdmss 24388 |
. . . . . . . . . 10
⊢ 𝐷 ⊆ (ℂ ∖
{0}) |
32 | | simpr 477 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑦
∈ 𝐷) → 𝑦 ∈ 𝐷) |
33 | 31, 32 | sseldi 3601 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑦
∈ 𝐷) → 𝑦 ∈ (ℂ ∖
{0})) |
34 | | logf1o 24311 |
. . . . . . . . . . 11
⊢
log:(ℂ ∖ {0})–1-1-onto→ran
log |
35 | | f1of 6137 |
. . . . . . . . . . 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 6358 |
. . . . . . . . 9
⊢ (𝑦 ∈ (ℂ ∖ {0})
→ (log‘𝑦) ∈
ran log) |
38 | | logrncn 24309 |
. . . . . . . . 9
⊢
((log‘𝑦)
∈ ran log → (log‘𝑦) ∈ ℂ) |
39 | 33, 37, 38 | 3syl 18 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑦
∈ 𝐷) →
(log‘𝑦) ∈
ℂ) |
40 | | ovexd 6680 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑦
∈ 𝐷) → (1 / 𝑦) ∈ V) |
41 | 4 | a1i 11 |
. . . . . . . . . . 11
⊢ (⊤
→ i ∈ ℂ) |
42 | 41, 13 | sylan 488 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ ℂ) → (i · 𝑥) ∈ ℂ) |
43 | 3, 42, 15 | sylancr 695 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ ℂ) → (1 − (i · 𝑥)) ∈ ℂ) |
44 | 29 | a1i 11 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ ℂ) → -i ∈ V) |
45 | | 1cnd 10056 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ ℂ) → 1 ∈ ℂ) |
46 | | 0cnd 10033 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ ℂ) → 0 ∈ ℂ) |
47 | | 1cnd 10056 |
. . . . . . . . . . . 12
⊢ (⊤
→ 1 ∈ ℂ) |
48 | 2, 47 | dvmptc 23721 |
. . . . . . . . . . 11
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ 1)) = (𝑥
∈ ℂ ↦ 0)) |
49 | 4 | a1i 11 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ ℂ) → i ∈ ℂ) |
50 | | simpr 477 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ ℂ) → 𝑥
∈ ℂ) |
51 | 2 | dvmptid 23720 |
. . . . . . . . . . . . 13
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ 𝑥)) =
(𝑥 ∈ ℂ ↦
1)) |
52 | 2, 50, 45, 51, 41 | dvmptcmul 23727 |
. . . . . . . . . . . 12
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (i · 𝑥))) = (𝑥 ∈ ℂ ↦ (i ·
1))) |
53 | 4 | mulid1i 10042 |
. . . . . . . . . . . . 13
⊢ (i
· 1) = i |
54 | 53 | mpteq2i 4741 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℂ ↦ (i
· 1)) = (𝑥 ∈
ℂ ↦ i) |
55 | 52, 54 | syl6eq 2672 |
. . . . . . . . . . 11
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (i · 𝑥))) = (𝑥 ∈ ℂ ↦ i)) |
56 | 2, 45, 46, 48, 42, 49, 55 | dvmptsub 23730 |
. . . . . . . . . 10
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (1 − (i · 𝑥)))) = (𝑥 ∈ ℂ ↦ (0 −
i))) |
57 | | df-neg 10269 |
. . . . . . . . . . 11
⊢ -i = (0
− i) |
58 | 57 | mpteq2i 4741 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℂ ↦ -i) =
(𝑥 ∈ ℂ ↦
(0 − i)) |
59 | 56, 58 | syl6eqr 2674 |
. . . . . . . . 9
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (1 − (i · 𝑥)))) = (𝑥 ∈ ℂ ↦ -i)) |
60 | | eqid 2622 |
. . . . . . . . . . . 12
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
61 | 60 | cnfldtopon 22586 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
62 | 5, 6 | atansopn 24659 |
. . . . . . . . . . 11
⊢ 𝑆 ∈
(TopOpen‘ℂfld) |
63 | | toponss 20731 |
. . . . . . . . . . 11
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ 𝑆 ∈
(TopOpen‘ℂfld)) → 𝑆 ⊆ ℂ) |
64 | 61, 62, 63 | mp2an 708 |
. . . . . . . . . 10
⊢ 𝑆 ⊆
ℂ |
65 | 64 | a1i 11 |
. . . . . . . . 9
⊢ (⊤
→ 𝑆 ⊆
ℂ) |
66 | 60 | cnfldtop 22587 |
. . . . . . . . . . 11
⊢
(TopOpen‘ℂfld) ∈ Top |
67 | 61 | toponunii 20721 |
. . . . . . . . . . . 12
⊢ ℂ =
∪
(TopOpen‘ℂfld) |
68 | 67 | restid 16094 |
. . . . . . . . . . 11
⊢
((TopOpen‘ℂfld) ∈ Top →
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld)) |
69 | 66, 68 | ax-mp 5 |
. . . . . . . . . 10
⊢
((TopOpen‘ℂfld) ↾t ℂ) =
(TopOpen‘ℂfld) |
70 | 69 | eqcomi 2631 |
. . . . . . . . 9
⊢
(TopOpen‘ℂfld) =
((TopOpen‘ℂfld) ↾t
ℂ) |
71 | 62 | a1i 11 |
. . . . . . . . 9
⊢ (⊤
→ 𝑆 ∈
(TopOpen‘ℂfld)) |
72 | 2, 43, 44, 59, 65, 70, 60, 71 | dvmptres 23726 |
. . . . . . . 8
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (1 − (i
· 𝑥)))) = (𝑥 ∈ 𝑆 ↦ -i)) |
73 | | fssres 6070 |
. . . . . . . . . . . . . 14
⊢
((log:(ℂ ∖ {0})⟶ran log ∧ 𝐷 ⊆ (ℂ ∖ {0})) → (log
↾ 𝐷):𝐷⟶ran
log) |
74 | 36, 31, 73 | mp2an 708 |
. . . . . . . . . . . . 13
⊢ (log
↾ 𝐷):𝐷⟶ran log |
75 | 74 | a1i 11 |
. . . . . . . . . . . 12
⊢ (⊤
→ (log ↾ 𝐷):𝐷⟶ran log) |
76 | 75 | feqmptd 6249 |
. . . . . . . . . . 11
⊢ (⊤
→ (log ↾ 𝐷) =
(𝑦 ∈ 𝐷 ↦ ((log ↾ 𝐷)‘𝑦))) |
77 | | fvres 6207 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈ 𝐷 → ((log ↾ 𝐷)‘𝑦) = (log‘𝑦)) |
78 | 77 | mpteq2ia 4740 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ 𝐷 ↦ ((log ↾ 𝐷)‘𝑦)) = (𝑦 ∈ 𝐷 ↦ (log‘𝑦)) |
79 | 76, 78 | syl6req 2673 |
. . . . . . . . . 10
⊢ (⊤
→ (𝑦 ∈ 𝐷 ↦ (log‘𝑦)) = (log ↾ 𝐷)) |
80 | 79 | oveq2d 6666 |
. . . . . . . . 9
⊢ (⊤
→ (ℂ D (𝑦 ∈
𝐷 ↦ (log‘𝑦))) = (ℂ D (log ↾
𝐷))) |
81 | 5 | dvlog 24397 |
. . . . . . . . 9
⊢ (ℂ
D (log ↾ 𝐷)) = (𝑦 ∈ 𝐷 ↦ (1 / 𝑦)) |
82 | 80, 81 | syl6eq 2672 |
. . . . . . . 8
⊢ (⊤
→ (ℂ D (𝑦 ∈
𝐷 ↦ (log‘𝑦))) = (𝑦 ∈ 𝐷 ↦ (1 / 𝑦))) |
83 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑦 = (1 − (i · 𝑥)) → (log‘𝑦) = (log‘(1 − (i
· 𝑥)))) |
84 | | oveq2 6658 |
. . . . . . . 8
⊢ (𝑦 = (1 − (i · 𝑥)) → (1 / 𝑦) = (1 / (1 − (i ·
𝑥)))) |
85 | 2, 2, 28, 30, 39, 40, 72, 82, 83, 84 | dvmptco 23735 |
. . . . . . 7
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (log‘(1
− (i · 𝑥)))))
= (𝑥 ∈ 𝑆 ↦ ((1 / (1 − (i
· 𝑥))) ·
-i))) |
86 | | irec 12964 |
. . . . . . . . . 10
⊢ (1 / i) =
-i |
87 | 86 | oveq2i 6661 |
. . . . . . . . 9
⊢ ((1 / (1
− (i · 𝑥)))
· (1 / i)) = ((1 / (1 − (i · 𝑥))) · -i) |
88 | 4 | a1i 11 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → i ∈
ℂ) |
89 | | ine0 10465 |
. . . . . . . . . . . 12
⊢ i ≠
0 |
90 | 89 | a1i 11 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → i ≠
0) |
91 | 16, 88, 17, 90 | recdiv2d 10819 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / (1
− (i · 𝑥))) /
i) = (1 / ((1 − (i · 𝑥)) · i))) |
92 | 16, 17 | reccld 10794 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / (1
− (i · 𝑥)))
∈ ℂ) |
93 | 92, 88, 90 | divrecd 10804 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / (1
− (i · 𝑥))) /
i) = ((1 / (1 − (i · 𝑥))) · (1 / i))) |
94 | | 1cnd 10056 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 1 ∈
ℂ) |
95 | 94, 14, 88 | subdird 10487 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1
− (i · 𝑥))
· i) = ((1 · i) − ((i · 𝑥) · i))) |
96 | 4 | mulid2i 10043 |
. . . . . . . . . . . . . . 15
⊢ (1
· i) = i |
97 | 96 | a1i 11 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1
· i) = i) |
98 | 88, 12, 88 | mul32d 10246 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((i
· 𝑥) · i) =
((i · i) · 𝑥)) |
99 | | ixi 10656 |
. . . . . . . . . . . . . . . . 17
⊢ (i
· i) = -1 |
100 | 99 | oveq1i 6660 |
. . . . . . . . . . . . . . . 16
⊢ ((i
· i) · 𝑥) =
(-1 · 𝑥) |
101 | 12 | mulm1d 10482 |
. . . . . . . . . . . . . . . 16
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (-1
· 𝑥) = -𝑥) |
102 | 100, 101 | syl5eq 2668 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((i
· i) · 𝑥) =
-𝑥) |
103 | 98, 102 | eqtrd 2656 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((i
· 𝑥) · i) =
-𝑥) |
104 | 97, 103 | oveq12d 6668 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1
· i) − ((i · 𝑥) · i)) = (i − -𝑥)) |
105 | | subneg 10330 |
. . . . . . . . . . . . . 14
⊢ ((i
∈ ℂ ∧ 𝑥
∈ ℂ) → (i − -𝑥) = (i + 𝑥)) |
106 | 4, 12, 105 | sylancr 695 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (i −
-𝑥) = (i + 𝑥)) |
107 | 95, 104, 106 | 3eqtrd 2660 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1
− (i · 𝑥))
· i) = (i + 𝑥)) |
108 | | addcom 10222 |
. . . . . . . . . . . . 13
⊢ ((i
∈ ℂ ∧ 𝑥
∈ ℂ) → (i + 𝑥) = (𝑥 + i)) |
109 | 4, 12, 108 | sylancr 695 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (i + 𝑥) = (𝑥 + i)) |
110 | 107, 109 | eqtrd 2656 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1
− (i · 𝑥))
· i) = (𝑥 +
i)) |
111 | 110 | oveq2d 6666 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / ((1
− (i · 𝑥))
· i)) = (1 / (𝑥 +
i))) |
112 | 91, 93, 111 | 3eqtr3d 2664 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / (1
− (i · 𝑥)))
· (1 / i)) = (1 / (𝑥
+ i))) |
113 | 87, 112 | syl5eqr 2670 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / (1
− (i · 𝑥)))
· -i) = (1 / (𝑥 +
i))) |
114 | 113 | mpteq2dva 4744 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈ 𝑆 ↦ ((1 / (1 − (i
· 𝑥))) · -i))
= (𝑥 ∈ 𝑆 ↦ (1 / (𝑥 + i)))) |
115 | 85, 114 | eqtrd 2656 |
. . . . . 6
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (log‘(1
− (i · 𝑥)))))
= (𝑥 ∈ 𝑆 ↦ (1 / (𝑥 + i)))) |
116 | | ovexd 6680 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 /
(𝑥 − i)) ∈
V) |
117 | 26 | simp3bi 1078 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝑆 → (1 + (i · 𝑥)) ∈ 𝐷) |
118 | 117 | adantl 482 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 + (i
· 𝑥)) ∈ 𝐷) |
119 | 3, 42, 19 | sylancr 695 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ ℂ) → (1 + (i · 𝑥)) ∈ ℂ) |
120 | 2, 45, 46, 48, 42, 49, 55 | dvmptadd 23723 |
. . . . . . . . . 10
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (1 + (i · 𝑥)))) = (𝑥 ∈ ℂ ↦ (0 +
i))) |
121 | 4 | addid2i 10224 |
. . . . . . . . . . 11
⊢ (0 + i) =
i |
122 | 121 | mpteq2i 4741 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℂ ↦ (0 + i))
= (𝑥 ∈ ℂ ↦
i) |
123 | 120, 122 | syl6eq 2672 |
. . . . . . . . 9
⊢ (⊤
→ (ℂ D (𝑥 ∈
ℂ ↦ (1 + (i · 𝑥)))) = (𝑥 ∈ ℂ ↦ i)) |
124 | 2, 119, 49, 123, 65, 70, 60, 71 | dvmptres 23726 |
. . . . . . . 8
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (1 + (i ·
𝑥)))) = (𝑥 ∈ 𝑆 ↦ i)) |
125 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑦 = (1 + (i · 𝑥)) → (log‘𝑦) = (log‘(1 + (i ·
𝑥)))) |
126 | | oveq2 6658 |
. . . . . . . 8
⊢ (𝑦 = (1 + (i · 𝑥)) → (1 / 𝑦) = (1 / (1 + (i · 𝑥)))) |
127 | 2, 2, 118, 88, 39, 40, 124, 82, 125, 126 | dvmptco 23735 |
. . . . . . 7
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (log‘(1 +
(i · 𝑥))))) = (𝑥 ∈ 𝑆 ↦ ((1 / (1 + (i · 𝑥))) ·
i))) |
128 | 94, 20, 88, 21, 90 | divdiv2d 10833 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / ((1 +
(i · 𝑥)) / i)) = ((1
· i) / (1 + (i · 𝑥)))) |
129 | 94, 14, 88, 90 | divdird 10839 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 + (i
· 𝑥)) / i) = ((1 /
i) + ((i · 𝑥) /
i))) |
130 | 86 | a1i 11 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / i) =
-i) |
131 | 12, 88, 90 | divcan3d 10806 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((i
· 𝑥) / i) = 𝑥) |
132 | 130, 131 | oveq12d 6668 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / i) +
((i · 𝑥) / i)) = (-i
+ 𝑥)) |
133 | | negicn 10282 |
. . . . . . . . . . . . 13
⊢ -i ∈
ℂ |
134 | | addcom 10222 |
. . . . . . . . . . . . 13
⊢ ((-i
∈ ℂ ∧ 𝑥
∈ ℂ) → (-i + 𝑥) = (𝑥 + -i)) |
135 | 133, 12, 134 | sylancr 695 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (-i +
𝑥) = (𝑥 + -i)) |
136 | | negsub 10329 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → (𝑥 + -i) =
(𝑥 −
i)) |
137 | 12, 4, 136 | sylancl 694 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 + -i) = (𝑥 − i)) |
138 | 135, 137 | eqtrd 2656 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (-i +
𝑥) = (𝑥 − i)) |
139 | 129, 132,
138 | 3eqtrd 2660 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 + (i
· 𝑥)) / i) = (𝑥 − i)) |
140 | 139 | oveq2d 6666 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / ((1 +
(i · 𝑥)) / i)) = (1
/ (𝑥 −
i))) |
141 | 94, 88, 20, 21 | div23d 10838 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1
· i) / (1 + (i · 𝑥))) = ((1 / (1 + (i · 𝑥))) ·
i)) |
142 | 128, 140,
141 | 3eqtr3rd 2665 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 / (1 +
(i · 𝑥))) ·
i) = (1 / (𝑥 −
i))) |
143 | 142 | mpteq2dva 4744 |
. . . . . . 7
⊢ (⊤
→ (𝑥 ∈ 𝑆 ↦ ((1 / (1 + (i ·
𝑥))) · i)) = (𝑥 ∈ 𝑆 ↦ (1 / (𝑥 − i)))) |
144 | 127, 143 | eqtrd 2656 |
. . . . . 6
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ (log‘(1 +
(i · 𝑥))))) = (𝑥 ∈ 𝑆 ↦ (1 / (𝑥 − i)))) |
145 | 2, 18, 25, 115, 22, 116, 144 | dvmptsub 23730 |
. . . . 5
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥)))))) = (𝑥 ∈ 𝑆 ↦ ((1 / (𝑥 + i)) − (1 / (𝑥 − i))))) |
146 | | subcl 10280 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → (𝑥 −
i) ∈ ℂ) |
147 | 12, 4, 146 | sylancl 694 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 − i) ∈
ℂ) |
148 | | addcl 10018 |
. . . . . . . . 9
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → (𝑥 + i)
∈ ℂ) |
149 | 12, 4, 148 | sylancl 694 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 + i) ∈
ℂ) |
150 | 12 | sqcld 13006 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥↑2) ∈
ℂ) |
151 | | addcl 10018 |
. . . . . . . . 9
⊢ ((1
∈ ℂ ∧ (𝑥↑2) ∈ ℂ) → (1 + (𝑥↑2)) ∈
ℂ) |
152 | 3, 150, 151 | sylancr 695 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 +
(𝑥↑2)) ∈
ℂ) |
153 | | atandm4 24606 |
. . . . . . . . . 10
⊢ (𝑥 ∈ dom arctan ↔ (𝑥 ∈ ℂ ∧ (1 +
(𝑥↑2)) ≠
0)) |
154 | 153 | simprbi 480 |
. . . . . . . . 9
⊢ (𝑥 ∈ dom arctan → (1 +
(𝑥↑2)) ≠
0) |
155 | 9, 154 | syl 17 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 +
(𝑥↑2)) ≠
0) |
156 | 147, 149,
152, 155 | divsubdird 10840 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 − i) − (𝑥 + i)) / (1 + (𝑥↑2))) = (((𝑥 − i) / (1 + (𝑥↑2))) − ((𝑥 + i) / (1 + (𝑥↑2))))) |
157 | 137 | oveq1d 6665 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 + -i) − (𝑥 + i)) = ((𝑥 − i) − (𝑥 + i))) |
158 | 133 | a1i 11 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → -i ∈
ℂ) |
159 | 12, 158, 88 | pnpcand 10429 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 + -i) − (𝑥 + i)) = (-i −
i)) |
160 | 157, 159 | eqtr3d 2658 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) − (𝑥 + i)) = (-i −
i)) |
161 | | 2cn 11091 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℂ |
162 | 161, 4, 89 | divreci 10770 |
. . . . . . . . . . 11
⊢ (2 / i) =
(2 · (1 / i)) |
163 | 86 | oveq2i 6661 |
. . . . . . . . . . 11
⊢ (2
· (1 / i)) = (2 · -i) |
164 | 162, 163 | eqtri 2644 |
. . . . . . . . . 10
⊢ (2 / i) =
(2 · -i) |
165 | 133 | 2timesi 11147 |
. . . . . . . . . 10
⊢ (2
· -i) = (-i + -i) |
166 | 133, 4 | negsubi 10359 |
. . . . . . . . . 10
⊢ (-i + -i)
= (-i − i) |
167 | 164, 165,
166 | 3eqtri 2648 |
. . . . . . . . 9
⊢ (2 / i) =
(-i − i) |
168 | 160, 167 | syl6eqr 2674 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) − (𝑥 + i)) = (2 /
i)) |
169 | 168 | oveq1d 6665 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 − i) − (𝑥 + i)) / (1 + (𝑥↑2))) = ((2 / i) / (1 +
(𝑥↑2)))) |
170 | 147 | mulid1d 10057 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) · 1) = (𝑥 − i)) |
171 | 147, 149 | mulcomd 10061 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) · (𝑥 + i)) = ((𝑥 + i) · (𝑥 − i))) |
172 | | i2 12965 |
. . . . . . . . . . . . . 14
⊢
(i↑2) = -1 |
173 | 172 | oveq2i 6661 |
. . . . . . . . . . . . 13
⊢ ((𝑥↑2) − (i↑2)) =
((𝑥↑2) −
-1) |
174 | | subneg 10330 |
. . . . . . . . . . . . . 14
⊢ (((𝑥↑2) ∈ ℂ ∧ 1
∈ ℂ) → ((𝑥↑2) − -1) = ((𝑥↑2) + 1)) |
175 | 150, 3, 174 | sylancl 694 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥↑2) − -1) = ((𝑥↑2) + 1)) |
176 | 173, 175 | syl5eq 2668 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥↑2) − (i↑2)) =
((𝑥↑2) +
1)) |
177 | | subsq 12972 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → ((𝑥↑2)
− (i↑2)) = ((𝑥 +
i) · (𝑥 −
i))) |
178 | 12, 4, 177 | sylancl 694 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥↑2) − (i↑2)) =
((𝑥 + i) · (𝑥 − i))) |
179 | | addcom 10222 |
. . . . . . . . . . . . 13
⊢ (((𝑥↑2) ∈ ℂ ∧ 1
∈ ℂ) → ((𝑥↑2) + 1) = (1 + (𝑥↑2))) |
180 | 150, 3, 179 | sylancl 694 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥↑2) + 1) = (1 + (𝑥↑2))) |
181 | 176, 178,
180 | 3eqtr3d 2664 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 + i) · (𝑥 − i)) = (1 + (𝑥↑2))) |
182 | 171, 181 | eqtrd 2656 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) · (𝑥 + i)) = (1 + (𝑥↑2))) |
183 | 170, 182 | oveq12d 6668 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 − i) · 1) /
((𝑥 − i) ·
(𝑥 + i))) = ((𝑥 − i) / (1 + (𝑥↑2)))) |
184 | | subneg 10330 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → (𝑥 −
-i) = (𝑥 +
i)) |
185 | 12, 4, 184 | sylancl 694 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 − -i) = (𝑥 + i)) |
186 | | atandm 24603 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ dom arctan ↔ (𝑥 ∈ ℂ ∧ 𝑥 ≠ -i ∧ 𝑥 ≠ i)) |
187 | 9, 186 | sylib 208 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 ∈ ℂ ∧ 𝑥 ≠ -i ∧ 𝑥 ≠ i)) |
188 | 187 | simp2d 1074 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 𝑥 ≠ -i) |
189 | | subeq0 10307 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℂ ∧ -i ∈
ℂ) → ((𝑥 −
-i) = 0 ↔ 𝑥 =
-i)) |
190 | 189 | necon3bid 2838 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ -i ∈
ℂ) → ((𝑥 −
-i) ≠ 0 ↔ 𝑥 ≠
-i)) |
191 | 12, 133, 190 | sylancl 694 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − -i) ≠ 0 ↔ 𝑥 ≠ -i)) |
192 | 188, 191 | mpbird 247 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 − -i) ≠
0) |
193 | 185, 192 | eqnetrrd 2862 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 + i) ≠ 0) |
194 | 187 | simp3d 1075 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → 𝑥 ≠ i) |
195 | | subeq0 10307 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → ((𝑥 −
i) = 0 ↔ 𝑥 =
i)) |
196 | 195 | necon3bid 2838 |
. . . . . . . . . . . 12
⊢ ((𝑥 ∈ ℂ ∧ i ∈
ℂ) → ((𝑥 −
i) ≠ 0 ↔ 𝑥 ≠
i)) |
197 | 12, 4, 196 | sylancl 694 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) ≠ 0 ↔ 𝑥 ≠ i)) |
198 | 194, 197 | mpbird 247 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (𝑥 − i) ≠
0) |
199 | 94, 149, 147, 193, 198 | divcan5d 10827 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 − i) · 1) /
((𝑥 − i) ·
(𝑥 + i))) = (1 / (𝑥 + i))) |
200 | 183, 199 | eqtr3d 2658 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 − i) / (1 + (𝑥↑2))) = (1 / (𝑥 + i))) |
201 | 149 | mulid1d 10057 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 + i) · 1) = (𝑥 + i)) |
202 | 201, 181 | oveq12d 6668 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 + i) · 1) / ((𝑥 + i) · (𝑥 − i))) = ((𝑥 + i) / (1 + (𝑥↑2)))) |
203 | 94, 147, 149, 198, 193 | divcan5d 10827 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 + i) · 1) / ((𝑥 + i) · (𝑥 − i))) = (1 / (𝑥 − i))) |
204 | 202, 203 | eqtr3d 2658 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((𝑥 + i) / (1 + (𝑥↑2))) = (1 / (𝑥 − i))) |
205 | 200, 204 | oveq12d 6668 |
. . . . . . 7
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((𝑥 − i) / (1 + (𝑥↑2))) − ((𝑥 + i) / (1 + (𝑥↑2)))) = ((1 / (𝑥 + i)) − (1 / (𝑥 − i)))) |
206 | 156, 169,
205 | 3eqtr3rd 2665 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → ((1 /
(𝑥 + i)) − (1 /
(𝑥 − i))) = ((2 / i)
/ (1 + (𝑥↑2)))) |
207 | 206 | mpteq2dva 4744 |
. . . . 5
⊢ (⊤
→ (𝑥 ∈ 𝑆 ↦ ((1 / (𝑥 + i)) − (1 / (𝑥 − i)))) = (𝑥 ∈ 𝑆 ↦ ((2 / i) / (1 + (𝑥↑2))))) |
208 | 145, 207 | eqtrd 2656 |
. . . 4
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥)))))) = (𝑥 ∈ 𝑆 ↦ ((2 / i) / (1 + (𝑥↑2))))) |
209 | | halfcl 11257 |
. . . . 5
⊢ (i ∈
ℂ → (i / 2) ∈ ℂ) |
210 | 4, 209 | mp1i 13 |
. . . 4
⊢ (⊤
→ (i / 2) ∈ ℂ) |
211 | 2, 23, 24, 208, 210 | dvmptcmul 23727 |
. . 3
⊢ (⊤
→ (ℂ D (𝑥 ∈
𝑆 ↦ ((i / 2) ·
((log‘(1 − (i · 𝑥))) − (log‘(1 + (i · 𝑥))))))) = (𝑥 ∈ 𝑆 ↦ ((i / 2) · ((2 / i) / (1 +
(𝑥↑2)))))) |
212 | | df-atan 24594 |
. . . . . . 7
⊢ arctan =
(𝑥 ∈ (ℂ ∖
{-i, i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i
· 𝑥)))))) |
213 | 212 | reseq1i 5392 |
. . . . . 6
⊢ (arctan
↾ 𝑆) = ((𝑥 ∈ (ℂ ∖ {-i,
i}) ↦ ((i / 2) · ((log‘(1 − (i · 𝑥))) − (log‘(1 + (i
· 𝑥)))))) ↾
𝑆) |
214 | | atanf 24607 |
. . . . . . . . 9
⊢
arctan:(ℂ ∖ {-i, i})⟶ℂ |
215 | 214 | fdmi 6052 |
. . . . . . . 8
⊢ dom
arctan = (ℂ ∖ {-i, i}) |
216 | 7, 215 | sseqtri 3637 |
. . . . . . 7
⊢ 𝑆 ⊆ (ℂ ∖ {-i,
i}) |
217 | | resmpt 5449 |
. . . . . . 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))))))) |
218 | 216, 217 | 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)))))) |
219 | 213, 218 | eqtri 2644 |
. . . . 5
⊢ (arctan
↾ 𝑆) = (𝑥 ∈ 𝑆 ↦ ((i / 2) · ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥)))))) |
220 | 219 | a1i 11 |
. . . 4
⊢ (⊤
→ (arctan ↾ 𝑆) =
(𝑥 ∈ 𝑆 ↦ ((i / 2) · ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥))))))) |
221 | 220 | oveq2d 6666 |
. . 3
⊢ (⊤
→ (ℂ D (arctan ↾ 𝑆)) = (ℂ D (𝑥 ∈ 𝑆 ↦ ((i / 2) · ((log‘(1
− (i · 𝑥)))
− (log‘(1 + (i · 𝑥)))))))) |
222 | | 2ne0 11113 |
. . . . . . 7
⊢ 2 ≠
0 |
223 | | divcan6 10732 |
. . . . . . 7
⊢ (((i
∈ ℂ ∧ i ≠ 0) ∧ (2 ∈ ℂ ∧ 2 ≠ 0)) →
((i / 2) · (2 / i)) = 1) |
224 | 4, 89, 161, 222, 223 | mp4an 709 |
. . . . . 6
⊢ ((i / 2)
· (2 / i)) = 1 |
225 | 224 | oveq1i 6660 |
. . . . 5
⊢ (((i / 2)
· (2 / i)) / (1 + (𝑥↑2))) = (1 / (1 + (𝑥↑2))) |
226 | 4, 209 | mp1i 13 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (i / 2)
∈ ℂ) |
227 | 161, 4, 89 | divcli 10767 |
. . . . . . 7
⊢ (2 / i)
∈ ℂ |
228 | 227 | a1i 11 |
. . . . . 6
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (2 / i)
∈ ℂ) |
229 | 226, 228,
152, 155 | divassd 10836 |
. . . . 5
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (((i / 2)
· (2 / i)) / (1 + (𝑥↑2))) = ((i / 2) · ((2 / i) / (1
+ (𝑥↑2))))) |
230 | 225, 229 | syl5eqr 2670 |
. . . 4
⊢
((⊤ ∧ 𝑥
∈ 𝑆) → (1 / (1 +
(𝑥↑2))) = ((i / 2)
· ((2 / i) / (1 + (𝑥↑2))))) |
231 | 230 | mpteq2dva 4744 |
. . 3
⊢ (⊤
→ (𝑥 ∈ 𝑆 ↦ (1 / (1 + (𝑥↑2)))) = (𝑥 ∈ 𝑆 ↦ ((i / 2) · ((2 / i) / (1 +
(𝑥↑2)))))) |
232 | 211, 221,
231 | 3eqtr4d 2666 |
. 2
⊢ (⊤
→ (ℂ D (arctan ↾ 𝑆)) = (𝑥 ∈ 𝑆 ↦ (1 / (1 + (𝑥↑2))))) |
233 | 232 | trud 1493 |
1
⊢ (ℂ
D (arctan ↾ 𝑆)) =
(𝑥 ∈ 𝑆 ↦ (1 / (1 + (𝑥↑2)))) |