Step |
Hyp |
Ref |
Expression |
1 |
|
cnelprrecn |
⊢ ℂ ∈ { ℝ , ℂ } |
2 |
1
|
a1i |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ℂ ∈ { ℝ , ℂ } ) |
3 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → 𝐴 ∈ ℂ ) |
4 |
|
0cnd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → 0 ∈ ℂ ) |
5 |
1
|
a1i |
⊢ ( 𝐴 ∈ ℂ → ℂ ∈ { ℝ , ℂ } ) |
6 |
|
id |
⊢ ( 𝐴 ∈ ℂ → 𝐴 ∈ ℂ ) |
7 |
5 6
|
dvmptc |
⊢ ( 𝐴 ∈ ℂ → ( ℂ D ( 𝑦 ∈ ℂ ↦ 𝐴 ) ) = ( 𝑦 ∈ ℂ ↦ 0 ) ) |
8 |
7
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℂ D ( 𝑦 ∈ ℂ ↦ 𝐴 ) ) = ( 𝑦 ∈ ℂ ↦ 0 ) ) |
9 |
|
mulcl |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝐵 · 𝑦 ) ∈ ℂ ) |
10 |
9
|
sincld |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( sin ‘ ( 𝐵 · 𝑦 ) ) ∈ ℂ ) |
11 |
10
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( sin ‘ ( 𝐵 · 𝑦 ) ) ∈ ℂ ) |
12 |
|
simpl |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → 𝐵 ∈ ℂ ) |
13 |
9
|
coscld |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( cos ‘ ( 𝐵 · 𝑦 ) ) ∈ ℂ ) |
14 |
12 13
|
mulcld |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝐵 · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ∈ ℂ ) |
15 |
14
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( 𝐵 · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ∈ ℂ ) |
16 |
|
dvsinax |
⊢ ( 𝐵 ∈ ℂ → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( sin ‘ ( 𝐵 · 𝑦 ) ) ) ) = ( 𝑦 ∈ ℂ ↦ ( 𝐵 · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( sin ‘ ( 𝐵 · 𝑦 ) ) ) ) = ( 𝑦 ∈ ℂ ↦ ( 𝐵 · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) ) |
18 |
2 3 4 8 11 15 17
|
dvmptmul |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( 𝐴 · ( sin ‘ ( 𝐵 · 𝑦 ) ) ) ) ) = ( 𝑦 ∈ ℂ ↦ ( ( 0 · ( sin ‘ ( 𝐵 · 𝑦 ) ) ) + ( ( 𝐵 · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) · 𝐴 ) ) ) ) |
19 |
11
|
mul02d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( 0 · ( sin ‘ ( 𝐵 · 𝑦 ) ) ) = 0 ) |
20 |
12
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → 𝐵 ∈ ℂ ) |
21 |
13
|
adantll |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( cos ‘ ( 𝐵 · 𝑦 ) ) ∈ ℂ ) |
22 |
20 21 3
|
mul32d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( ( 𝐵 · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) · 𝐴 ) = ( ( 𝐵 · 𝐴 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) |
23 |
|
simpr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐵 ∈ ℂ ) |
24 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → 𝐴 ∈ ℂ ) |
25 |
23 24
|
mulcomd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐵 · 𝐴 ) = ( 𝐴 · 𝐵 ) ) |
26 |
25
|
adantr |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( 𝐵 · 𝐴 ) = ( 𝐴 · 𝐵 ) ) |
27 |
26
|
oveq1d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( ( 𝐵 · 𝐴 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) = ( ( 𝐴 · 𝐵 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) |
28 |
22 27
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( ( 𝐵 · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) · 𝐴 ) = ( ( 𝐴 · 𝐵 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) |
29 |
19 28
|
oveq12d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( ( 0 · ( sin ‘ ( 𝐵 · 𝑦 ) ) ) + ( ( 𝐵 · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) · 𝐴 ) ) = ( 0 + ( ( 𝐴 · 𝐵 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) ) |
30 |
3 20
|
mulcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ ) |
31 |
30 21
|
mulcld |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( ( 𝐴 · 𝐵 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ∈ ℂ ) |
32 |
31
|
addid2d |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( 0 + ( ( 𝐴 · 𝐵 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) = ( ( 𝐴 · 𝐵 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) |
33 |
29 32
|
eqtrd |
⊢ ( ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) ∧ 𝑦 ∈ ℂ ) → ( ( 0 · ( sin ‘ ( 𝐵 · 𝑦 ) ) ) + ( ( 𝐵 · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) · 𝐴 ) ) = ( ( 𝐴 · 𝐵 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) |
34 |
33
|
mpteq2dva |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝑦 ∈ ℂ ↦ ( ( 0 · ( sin ‘ ( 𝐵 · 𝑦 ) ) ) + ( ( 𝐵 · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) · 𝐴 ) ) ) = ( 𝑦 ∈ ℂ ↦ ( ( 𝐴 · 𝐵 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) ) |
35 |
18 34
|
eqtrd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( ℂ D ( 𝑦 ∈ ℂ ↦ ( 𝐴 · ( sin ‘ ( 𝐵 · 𝑦 ) ) ) ) ) = ( 𝑦 ∈ ℂ ↦ ( ( 𝐴 · 𝐵 ) · ( cos ‘ ( 𝐵 · 𝑦 ) ) ) ) ) |