| Step |
Hyp |
Ref |
Expression |
| 1 |
|
df-tan |
⊢ tan = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) / ( cos ‘ 𝑥 ) ) ) |
| 2 |
|
cnvimass |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ dom cos |
| 3 |
|
cosf |
⊢ cos : ℂ ⟶ ℂ |
| 4 |
3
|
fdmi |
⊢ dom cos = ℂ |
| 5 |
2 4
|
sseqtri |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ ℂ |
| 6 |
5
|
sseli |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → 𝑥 ∈ ℂ ) |
| 7 |
6
|
sincld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
| 8 |
6
|
coscld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
| 9 |
|
ffn |
⊢ ( cos : ℂ ⟶ ℂ → cos Fn ℂ ) |
| 10 |
|
elpreima |
⊢ ( cos Fn ℂ → ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↔ ( 𝑥 ∈ ℂ ∧ ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) ) ) |
| 11 |
3 9 10
|
mp2b |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↔ ( 𝑥 ∈ ℂ ∧ ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) ) |
| 12 |
|
eldifsni |
⊢ ( ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) → ( cos ‘ 𝑥 ) ≠ 0 ) |
| 13 |
12
|
adantl |
⊢ ( ( 𝑥 ∈ ℂ ∧ ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) → ( cos ‘ 𝑥 ) ≠ 0 ) |
| 14 |
11 13
|
sylbi |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( cos ‘ 𝑥 ) ≠ 0 ) |
| 15 |
7 8 14
|
divrecd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( sin ‘ 𝑥 ) / ( cos ‘ 𝑥 ) ) = ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) |
| 16 |
15
|
mpteq2ia |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) / ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) |
| 17 |
1 16
|
eqtri |
⊢ tan = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) |
| 18 |
17
|
oveq2i |
⊢ ( ℂ D tan ) = ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) ) |
| 19 |
|
cnelprrecn |
⊢ ℂ ∈ { ℝ , ℂ } |
| 20 |
19
|
a1i |
⊢ ( ⊤ → ℂ ∈ { ℝ , ℂ } ) |
| 21 |
|
difss |
⊢ ( ℂ ∖ { 0 } ) ⊆ ℂ |
| 22 |
|
imass2 |
⊢ ( ( ℂ ∖ { 0 } ) ⊆ ℂ → ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ ( ◡ cos “ ℂ ) ) |
| 23 |
21 22
|
ax-mp |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ ( ◡ cos “ ℂ ) |
| 24 |
|
fimacnv |
⊢ ( cos : ℂ ⟶ ℂ → ( ◡ cos “ ℂ ) = ℂ ) |
| 25 |
3 24
|
ax-mp |
⊢ ( ◡ cos “ ℂ ) = ℂ |
| 26 |
23 25
|
sseqtri |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ ℂ |
| 27 |
26
|
sseli |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → 𝑥 ∈ ℂ ) |
| 28 |
27
|
sincld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
| 29 |
28
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
| 30 |
8
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
| 31 |
|
sincl |
⊢ ( 𝑥 ∈ ℂ → ( sin ‘ 𝑥 ) ∈ ℂ ) |
| 32 |
31
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → ( sin ‘ 𝑥 ) ∈ ℂ ) |
| 33 |
|
coscl |
⊢ ( 𝑥 ∈ ℂ → ( cos ‘ 𝑥 ) ∈ ℂ ) |
| 34 |
33
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → ( cos ‘ 𝑥 ) ∈ ℂ ) |
| 35 |
|
dvsin |
⊢ ( ℂ D sin ) = cos |
| 36 |
|
sinf |
⊢ sin : ℂ ⟶ ℂ |
| 37 |
36
|
a1i |
⊢ ( ⊤ → sin : ℂ ⟶ ℂ ) |
| 38 |
37
|
feqmptd |
⊢ ( ⊤ → sin = ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) |
| 39 |
38
|
oveq2d |
⊢ ( ⊤ → ( ℂ D sin ) = ( ℂ D ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) ) |
| 40 |
3
|
a1i |
⊢ ( ⊤ → cos : ℂ ⟶ ℂ ) |
| 41 |
40
|
feqmptd |
⊢ ( ⊤ → cos = ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) ) |
| 42 |
35 39 41
|
3eqtr3a |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ℂ ↦ ( sin ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) ) |
| 43 |
26
|
a1i |
⊢ ( ⊤ → ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ⊆ ℂ ) |
| 44 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
| 45 |
44
|
cnfldtopon |
⊢ ( TopOpen ‘ ℂfld ) ∈ ( TopOn ‘ ℂ ) |
| 46 |
45
|
toponrestid |
⊢ ( TopOpen ‘ ℂfld ) = ( ( TopOpen ‘ ℂfld ) ↾t ℂ ) |
| 47 |
|
dvtanlem |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ∈ ( TopOpen ‘ ℂfld ) |
| 48 |
47
|
a1i |
⊢ ( ⊤ → ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ∈ ( TopOpen ‘ ℂfld ) ) |
| 49 |
20 32 34 42 43 46 44 48
|
dvmptres |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( sin ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( cos ‘ 𝑥 ) ) ) |
| 50 |
8 14
|
reccld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( 1 / ( cos ‘ 𝑥 ) ) ∈ ℂ ) |
| 51 |
50
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → ( 1 / ( cos ‘ 𝑥 ) ) ∈ ℂ ) |
| 52 |
|
ovexd |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) ∈ V ) |
| 53 |
11
|
simprbi |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) |
| 54 |
53
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → ( cos ‘ 𝑥 ) ∈ ( ℂ ∖ { 0 } ) ) |
| 55 |
29
|
negcld |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ) → - ( sin ‘ 𝑥 ) ∈ ℂ ) |
| 56 |
|
eldifi |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 0 } ) → 𝑦 ∈ ℂ ) |
| 57 |
|
eldifsni |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 0 } ) → 𝑦 ≠ 0 ) |
| 58 |
56 57
|
reccld |
⊢ ( 𝑦 ∈ ( ℂ ∖ { 0 } ) → ( 1 / 𝑦 ) ∈ ℂ ) |
| 59 |
58
|
adantl |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → ( 1 / 𝑦 ) ∈ ℂ ) |
| 60 |
|
negex |
⊢ - ( 1 / ( 𝑦 ↑ 2 ) ) ∈ V |
| 61 |
60
|
a1i |
⊢ ( ( ⊤ ∧ 𝑦 ∈ ( ℂ ∖ { 0 } ) ) → - ( 1 / ( 𝑦 ↑ 2 ) ) ∈ V ) |
| 62 |
32
|
negcld |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℂ ) → - ( sin ‘ 𝑥 ) ∈ ℂ ) |
| 63 |
41
|
oveq2d |
⊢ ( ⊤ → ( ℂ D cos ) = ( ℂ D ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) ) ) |
| 64 |
|
dvcos |
⊢ ( ℂ D cos ) = ( 𝑥 ∈ ℂ ↦ - ( sin ‘ 𝑥 ) ) |
| 65 |
63 64
|
eqtr3di |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ℂ ↦ ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℂ ↦ - ( sin ‘ 𝑥 ) ) ) |
| 66 |
20 34 62 65 43 46 44 48
|
dvmptres |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( cos ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ - ( sin ‘ 𝑥 ) ) ) |
| 67 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
| 68 |
|
dvrec |
⊢ ( 1 ∈ ℂ → ( ℂ D ( 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( 1 / 𝑦 ) ) ) = ( 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ - ( 1 / ( 𝑦 ↑ 2 ) ) ) ) |
| 69 |
67 68
|
mp1i |
⊢ ( ⊤ → ( ℂ D ( 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ ( 1 / 𝑦 ) ) ) = ( 𝑦 ∈ ( ℂ ∖ { 0 } ) ↦ - ( 1 / ( 𝑦 ↑ 2 ) ) ) ) |
| 70 |
|
oveq2 |
⊢ ( 𝑦 = ( cos ‘ 𝑥 ) → ( 1 / 𝑦 ) = ( 1 / ( cos ‘ 𝑥 ) ) ) |
| 71 |
|
oveq1 |
⊢ ( 𝑦 = ( cos ‘ 𝑥 ) → ( 𝑦 ↑ 2 ) = ( ( cos ‘ 𝑥 ) ↑ 2 ) ) |
| 72 |
71
|
oveq2d |
⊢ ( 𝑦 = ( cos ‘ 𝑥 ) → ( 1 / ( 𝑦 ↑ 2 ) ) = ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 73 |
72
|
negeqd |
⊢ ( 𝑦 = ( cos ‘ 𝑥 ) → - ( 1 / ( 𝑦 ↑ 2 ) ) = - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 74 |
20 20 54 55 59 61 66 69 70 73
|
dvmptco |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( 1 / ( cos ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) ) ) |
| 75 |
20 29 30 49 51 52 74
|
dvmptmul |
⊢ ( ⊤ → ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) ) ) |
| 76 |
75
|
mptru |
⊢ ( ℂ D ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( sin ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) ) |
| 77 |
|
ovex |
⊢ ( ( sin ‘ 𝑥 ) / ( cos ‘ 𝑥 ) ) ∈ V |
| 78 |
77 1
|
dmmpti |
⊢ dom tan = ( ◡ cos “ ( ℂ ∖ { 0 } ) ) |
| 79 |
78
|
eqcomi |
⊢ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) = dom tan |
| 80 |
8
|
sqcld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( cos ‘ 𝑥 ) ↑ 2 ) ∈ ℂ ) |
| 81 |
7
|
sqcld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( sin ‘ 𝑥 ) ↑ 2 ) ∈ ℂ ) |
| 82 |
|
sqne0 |
⊢ ( ( cos ‘ 𝑥 ) ∈ ℂ → ( ( ( cos ‘ 𝑥 ) ↑ 2 ) ≠ 0 ↔ ( cos ‘ 𝑥 ) ≠ 0 ) ) |
| 83 |
8 82
|
syl |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) ↑ 2 ) ≠ 0 ↔ ( cos ‘ 𝑥 ) ≠ 0 ) ) |
| 84 |
14 83
|
mpbird |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( cos ‘ 𝑥 ) ↑ 2 ) ≠ 0 ) |
| 85 |
80 81 80 84
|
divdird |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( ( cos ‘ 𝑥 ) ↑ 2 ) + ( ( sin ‘ 𝑥 ) ↑ 2 ) ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = ( ( ( ( cos ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) + ( ( ( sin ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) ) |
| 86 |
80 81
|
addcomd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) ↑ 2 ) + ( ( sin ‘ 𝑥 ) ↑ 2 ) ) = ( ( ( sin ‘ 𝑥 ) ↑ 2 ) + ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 87 |
|
sincossq |
⊢ ( 𝑥 ∈ ℂ → ( ( ( sin ‘ 𝑥 ) ↑ 2 ) + ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = 1 ) |
| 88 |
6 87
|
syl |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( sin ‘ 𝑥 ) ↑ 2 ) + ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = 1 ) |
| 89 |
86 88
|
eqtrd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) ↑ 2 ) + ( ( sin ‘ 𝑥 ) ↑ 2 ) ) = 1 ) |
| 90 |
89
|
oveq1d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( ( cos ‘ 𝑥 ) ↑ 2 ) + ( ( sin ‘ 𝑥 ) ↑ 2 ) ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 91 |
85 90
|
eqtr3d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( ( cos ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) + ( ( ( sin ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) = ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 92 |
8 14
|
recidd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) = 1 ) |
| 93 |
80 84
|
dividd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = 1 ) |
| 94 |
92 93
|
eqtr4d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) = ( ( ( cos ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 95 |
7 7 80 84
|
div23d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = ( ( ( sin ‘ 𝑥 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · ( sin ‘ 𝑥 ) ) ) |
| 96 |
7
|
sqvald |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( sin ‘ 𝑥 ) ↑ 2 ) = ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) ) |
| 97 |
96
|
oveq1d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( sin ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = ( ( ( sin ‘ 𝑥 ) · ( sin ‘ 𝑥 ) ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 98 |
80 84
|
reccld |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ∈ ℂ ) |
| 99 |
98 7
|
mul2negd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) = ( ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · ( sin ‘ 𝑥 ) ) ) |
| 100 |
7 80 84
|
divrec2d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( sin ‘ 𝑥 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) = ( ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · ( sin ‘ 𝑥 ) ) ) |
| 101 |
99 100
|
eqtr4d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) = ( ( sin ‘ 𝑥 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 102 |
101
|
oveq1d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) = ( ( ( sin ‘ 𝑥 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · ( sin ‘ 𝑥 ) ) ) |
| 103 |
95 97 102
|
3eqtr4rd |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) = ( ( ( sin ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 104 |
94 103
|
oveq12d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) = ( ( ( ( cos ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) + ( ( ( sin ‘ 𝑥 ) ↑ 2 ) / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) ) |
| 105 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
| 106 |
|
expneg |
⊢ ( ( ( cos ‘ 𝑥 ) ∈ ℂ ∧ 2 ∈ ℕ0 ) → ( ( cos ‘ 𝑥 ) ↑ - 2 ) = ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 107 |
8 105 106
|
sylancl |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( cos ‘ 𝑥 ) ↑ - 2 ) = ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) ) |
| 108 |
91 104 107
|
3eqtr4d |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) → ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) = ( ( cos ‘ 𝑥 ) ↑ - 2 ) ) |
| 109 |
108
|
rgen |
⊢ ∀ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) = ( ( cos ‘ 𝑥 ) ↑ - 2 ) |
| 110 |
|
mpteq12 |
⊢ ( ( ( ◡ cos “ ( ℂ ∖ { 0 } ) ) = dom tan ∧ ∀ 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) = ( ( cos ‘ 𝑥 ) ↑ - 2 ) ) → ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ dom tan ↦ ( ( cos ‘ 𝑥 ) ↑ - 2 ) ) ) |
| 111 |
79 109 110
|
mp2an |
⊢ ( 𝑥 ∈ ( ◡ cos “ ( ℂ ∖ { 0 } ) ) ↦ ( ( ( cos ‘ 𝑥 ) · ( 1 / ( cos ‘ 𝑥 ) ) ) + ( ( - ( 1 / ( ( cos ‘ 𝑥 ) ↑ 2 ) ) · - ( sin ‘ 𝑥 ) ) · ( sin ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ dom tan ↦ ( ( cos ‘ 𝑥 ) ↑ - 2 ) ) |
| 112 |
18 76 111
|
3eqtri |
⊢ ( ℂ D tan ) = ( 𝑥 ∈ dom tan ↦ ( ( cos ‘ 𝑥 ) ↑ - 2 ) ) |