| Step |
Hyp |
Ref |
Expression |
| 1 |
|
df-csc |
⊢ csc = ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ ( 1 / ( sin ‘ 𝑥 ) ) ) |
| 2 |
1
|
oveq2i |
⊢ ( ℂ D csc ) = ( ℂ D ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ ( 1 / ( sin ‘ 𝑥 ) ) ) ) |
| 3 |
|
cnelprrecn |
⊢ ℂ ∈ { ℝ , ℂ } |
| 4 |
3
|
a1i |
⊢ ( ⊤ → ℂ ∈ { ℝ , ℂ } ) |
| 5 |
|
1cnd |
⊢ ( ⊤ → 1 ∈ ℂ ) |
| 6 |
|
elrabi |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → 𝑥 ∈ ℂ ) |
| 7 |
|
sincl |
⊢ ( 𝑥 ∈ ℂ → ( sin ‘ 𝑥 ) ∈ ℂ ) |
| 8 |
6 7
|
syl |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( sin ‘ 𝑥 ) ∈ ℂ ) |
| 9 |
|
fveq2 |
⊢ ( 𝑦 = 𝑥 → ( sin ‘ 𝑦 ) = ( sin ‘ 𝑥 ) ) |
| 10 |
9
|
neeq1d |
⊢ ( 𝑦 = 𝑥 → ( ( sin ‘ 𝑦 ) ≠ 0 ↔ ( sin ‘ 𝑥 ) ≠ 0 ) ) |
| 11 |
10
|
elrab |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↔ ( 𝑥 ∈ ℂ ∧ ( sin ‘ 𝑥 ) ≠ 0 ) ) |
| 12 |
11
|
simprbi |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( sin ‘ 𝑥 ) ≠ 0 ) |
| 13 |
8 12
|
jca |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( ( sin ‘ 𝑥 ) ∈ ℂ ∧ ( sin ‘ 𝑥 ) ≠ 0 ) ) |
| 14 |
|
eldifsn |
⊢ ( ( sin ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ↔ ( ( sin ‘ 𝑥 ) ∈ ℂ ∧ ( sin ‘ 𝑥 ) ≠ 0 ) ) |
| 15 |
13 14
|
sylibr |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( sin ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) |
| 16 |
15
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ) → ( sin ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) |
| 17 |
|
coscl |
⊢ ( 𝑥 ∈ ℂ → ( cos ‘ 𝑥 ) ∈ ℂ ) |
| 18 |
6 17
|
syl |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( cos ‘ 𝑥 ) ∈ ℂ ) |
| 19 |
18
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
| 20 |
7
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
| 21 |
17
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
| 22 |
|
sinf |
⊢ sin : ℂ ⟶ ℂ |
| 23 |
22
|
a1i |
⊢ ( ⊤ → sin : ℂ ⟶ ℂ ) |
| 24 |
23
|
feqmptd |
⊢ ( ⊤ → sin = ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) |
| 25 |
24
|
mptru |
⊢ sin = ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) |
| 26 |
25
|
oveq2i |
⊢ ( ℂ D sin ) = ( ℂ D ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) |
| 27 |
|
dvsin |
⊢ ( ℂ D sin ) = cos |
| 28 |
26 27
|
eqtr3i |
⊢ ( ℂ D ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) = cos |
| 29 |
|
cosf |
⊢ cos : ℂ ⟶ ℂ |
| 30 |
29
|
a1i |
⊢ ( ⊤ → cos : ℂ ⟶ ℂ ) |
| 31 |
30
|
feqmptd |
⊢ ( ⊤ → cos = ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) ) |
| 32 |
31
|
mptru |
⊢ cos = ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) |
| 33 |
28 32
|
eqtri |
⊢ ( ℂ D ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) |
| 34 |
33
|
a1i |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) ) |
| 35 |
|
ssrab2 |
⊢ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ⊆ ℂ |
| 36 |
35
|
a1i |
⊢ ( ⊤ → { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ⊆ ℂ ) |
| 37 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
| 38 |
37
|
cnfldtopon |
⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) |
| 39 |
38
|
toponrestid |
⊢ ( TopOpen ‘ ℂfld ) = ( ( TopOpen ‘ ℂfld ) ↾t ℂ ) |
| 40 |
|
sincn |
⊢ sin ∈ ( ℂ –cn→ ℂ ) |
| 41 |
|
ssid |
⊢ ℂ ⊆ ℂ |
| 42 |
37 39 39
|
cncfcn |
⊢ ( ( ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ℂ –cn→ ℂ ) = ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ) |
| 43 |
41 41 42
|
mp2an |
⊢ ( ℂ –cn→ ℂ ) = ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) |
| 44 |
40 43
|
eleqtri |
⊢ sin ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) |
| 45 |
|
cnn0opn |
⊢ ( ℂ ∖ { 0 } ) ∈ ( TopOpen ‘ ℂfld ) |
| 46 |
|
cnima |
⊢ ( ( sin ∈ ( ( TopOpen ‘ ℂfld ) Cn ( TopOpen ‘ ℂfld ) ) ∧ ( ℂ ∖ { 0 } ) ∈ ( TopOpen ‘ ℂfld ) ) → ( ◡ sin “ ( ℂ ∖ { 0 } ) ) ∈ ( TopOpen ‘ ℂfld ) ) |
| 47 |
44 45 46
|
mp2an |
⊢ ( ◡ sin “ ( ℂ ∖ { 0 } ) ) ∈ ( TopOpen ‘ ℂfld ) |
| 48 |
|
sincl |
⊢ ( 𝑦 ∈ ℂ → ( sin ‘ 𝑦 ) ∈ ℂ ) |
| 49 |
|
eldifsn |
⊢ ( ( sin ‘ 𝑦 ) ∈ ( ℂ ∖ { 0 } ) ↔ ( ( sin ‘ 𝑦 ) ∈ ℂ ∧ ( sin ‘ 𝑦 ) ≠ 0 ) ) |
| 50 |
49
|
baib |
⊢ ( ( sin ‘ 𝑦 ) ∈ ℂ → ( ( sin ‘ 𝑦 ) ∈ ( ℂ ∖ { 0 } ) ↔ ( sin ‘ 𝑦 ) ≠ 0 ) ) |
| 51 |
48 50
|
syl |
⊢ ( 𝑦 ∈ ℂ → ( ( sin ‘ 𝑦 ) ∈ ( ℂ ∖ { 0 } ) ↔ ( sin ‘ 𝑦 ) ≠ 0 ) ) |
| 52 |
51
|
bicomd |
⊢ ( 𝑦 ∈ ℂ → ( ( sin ‘ 𝑦 ) ≠ 0 ↔ ( sin ‘ 𝑦 ) ∈ ( ℂ ∖ { 0 } ) ) ) |
| 53 |
52
|
rabbiia |
⊢ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } = { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ∈ ( ℂ ∖ { 0 } ) } |
| 54 |
23
|
feqmptd |
⊢ ( ⊤ → sin = ( 𝑦 ∈ ℂ ↦ ( sin ‘ 𝑦 ) ) ) |
| 55 |
54
|
mptru |
⊢ sin = ( 𝑦 ∈ ℂ ↦ ( sin ‘ 𝑦 ) ) |
| 56 |
55
|
mptpreima |
⊢ ( ◡ sin “ ( ℂ ∖ { 0 } ) ) = { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ∈ ( ℂ ∖ { 0 } ) } |
| 57 |
53 56
|
eqtr4i |
⊢ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } = ( ◡ sin “ ( ℂ ∖ { 0 } ) ) |
| 58 |
57
|
eleq1i |
⊢ ( { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ∈ ( TopOpen ‘ ℂfld ) ↔ ( ◡ sin “ ( ℂ ∖ { 0 } ) ) ∈ ( TopOpen ‘ ℂfld ) ) |
| 59 |
47 58
|
mpbir |
⊢ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ∈ ( TopOpen ‘ ℂfld ) |
| 60 |
59
|
a1i |
⊢ ( ⊤ → { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ∈ ( TopOpen ‘ ℂfld ) ) |
| 61 |
4 20 21 34 36 39 37 60
|
dvmptres |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ ( sin ‘ 𝑥 ) ) ) = ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ ( cos ‘ 𝑥 ) ) ) |
| 62 |
4 5 16 19 61
|
dvrecg |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ ( 1 / ( sin ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ - ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) ↑ 2 ) ) ) ) |
| 63 |
62
|
mptru |
⊢ ( ℂ D ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ ( 1 / ( sin ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ - ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) ↑ 2 ) ) ) |
| 64 |
18
|
mullidd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( 1 · ( cos ‘ 𝑥 ) ) = ( cos ‘ 𝑥 ) ) |
| 65 |
|
sqval |
⊢ ( ( sin ‘ 𝑥 ) ∈ ℂ → ( ( sin ‘ 𝑥 ) ↑ 2 ) = ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) |
| 66 |
8 65
|
syl |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( ( sin ‘ 𝑥 ) ↑ 2 ) = ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) |
| 67 |
64 66
|
oveq12d |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) ↑ 2 ) ) = ( ( cos ‘ 𝑥 ) / ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) ) |
| 68 |
67
|
negeqd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → - ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) ↑ 2 ) ) = - ( ( cos ‘ 𝑥 ) / ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) ) |
| 69 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 70 |
69
|
a1i |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → 1 ∈ ℂ ) |
| 71 |
70 8 18 8 12 12
|
divmuldivd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) = ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) ) |
| 72 |
64
|
oveq1d |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) = ( ( cos ‘ 𝑥 ) / ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) ) |
| 73 |
71 72
|
eqtrd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) = ( ( cos ‘ 𝑥 ) / ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) ) |
| 74 |
73
|
eqcomd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( ( cos ‘ 𝑥 ) / ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) = ( ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) ) |
| 75 |
74
|
negeqd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → - ( ( cos ‘ 𝑥 ) / ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) = - ( ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) ) |
| 76 |
68 75
|
eqtrd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → - ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) ↑ 2 ) ) = - ( ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) ) |
| 77 |
70 8 12
|
divcld |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( 1 / ( sin ‘ 𝑥 ) ) ∈ ℂ ) |
| 78 |
18 8 12
|
divcld |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ∈ ℂ ) |
| 79 |
77 78
|
mulneg1d |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( - ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) = - ( ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) ) |
| 80 |
79
|
eqcomd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → - ( ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) = ( - ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) ) |
| 81 |
76 80
|
eqtrd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → - ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) ↑ 2 ) ) = ( - ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) ) |
| 82 |
|
cscval |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( sin ‘ 𝑥 ) ≠ 0 ) → ( csc ‘ 𝑥 ) = ( 1 / ( sin ‘ 𝑥 ) ) ) |
| 83 |
11 82
|
sylbi |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( csc ‘ 𝑥 ) = ( 1 / ( sin ‘ 𝑥 ) ) ) |
| 84 |
83
|
eqcomd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( 1 / ( sin ‘ 𝑥 ) ) = ( csc ‘ 𝑥 ) ) |
| 85 |
84
|
negeqd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → - ( 1 / ( sin ‘ 𝑥 ) ) = - ( csc ‘ 𝑥 ) ) |
| 86 |
|
cotval |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( sin ‘ 𝑥 ) ≠ 0 ) → ( cot ‘ 𝑥 ) = ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) |
| 87 |
11 86
|
sylbi |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( cot ‘ 𝑥 ) = ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) |
| 88 |
87
|
eqcomd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) = ( cot ‘ 𝑥 ) ) |
| 89 |
85 88
|
oveq12d |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → ( - ( 1 / ( sin ‘ 𝑥 ) ) · ( ( cos ‘ 𝑥 ) / ( sin ‘ 𝑥 ) ) ) = ( - ( csc ‘ 𝑥 ) · ( cot ‘ 𝑥 ) ) ) |
| 90 |
81 89
|
eqtrd |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } → - ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) ↑ 2 ) ) = ( - ( csc ‘ 𝑥 ) · ( cot ‘ 𝑥 ) ) ) |
| 91 |
90
|
mpteq2ia |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ - ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) ↑ 2 ) ) ) = ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ ( - ( csc ‘ 𝑥 ) · ( cot ‘ 𝑥 ) ) ) |
| 92 |
1 77
|
fmpti |
⊢ csc : { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ⟶ ℂ |
| 93 |
92
|
fdmi |
⊢ dom csc = { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } |
| 94 |
93
|
eqcomi |
⊢ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } = dom csc |
| 95 |
94
|
mpteq1i |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ ( - ( csc ‘ 𝑥 ) · ( cot ‘ 𝑥 ) ) ) = ( 𝑥 ∈ dom csc ↦ ( - ( csc ‘ 𝑥 ) · ( cot ‘ 𝑥 ) ) ) |
| 96 |
91 95
|
eqtri |
⊢ ( 𝑥 ∈ { 𝑦 ∈ ℂ ∣ ( sin ‘ 𝑦 ) ≠ 0 } ↦ - ( ( 1 · ( cos ‘ 𝑥 ) ) / ( ( sin ‘ 𝑥 ) ↑ 2 ) ) ) = ( 𝑥 ∈ dom csc ↦ ( - ( csc ‘ 𝑥 ) · ( cot ‘ 𝑥 ) ) ) |
| 97 |
2 63 96
|
3eqtri |
⊢ ( ℂ D csc ) = ( 𝑥 ∈ dom csc ↦ ( - ( csc ‘ 𝑥 ) · ( cot ‘ 𝑥 ) ) ) |