Step |
Hyp |
Ref |
Expression |
1 |
|
rereb |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℝ ↔ ( ℜ ‘ 𝐴 ) = 𝐴 ) ) |
2 |
1
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 ∈ ℝ ↔ ( ℜ ‘ 𝐴 ) = 𝐴 ) ) |
3 |
|
recl |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ ) |
4 |
3
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℂ ) |
5 |
4
|
3ad2ant1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( ℜ ‘ 𝐴 ) ∈ ℂ ) |
6 |
|
simp1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → 𝐴 ∈ ℂ ) |
7 |
|
recn |
⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ ℂ ) |
8 |
7
|
anim1i |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) |
9 |
8
|
3adant1 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) |
10 |
|
mulcan |
⊢ ( ( ( ℜ ‘ 𝐴 ) ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) = ( 𝐵 · 𝐴 ) ↔ ( ℜ ‘ 𝐴 ) = 𝐴 ) ) |
11 |
5 6 9 10
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) = ( 𝐵 · 𝐴 ) ↔ ( ℜ ‘ 𝐴 ) = 𝐴 ) ) |
12 |
7
|
adantr |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → 𝐵 ∈ ℂ ) |
13 |
4
|
adantl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( ℜ ‘ 𝐴 ) ∈ ℂ ) |
14 |
|
ax-icn |
⊢ i ∈ ℂ |
15 |
|
imcl |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℝ ) |
16 |
15
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) ∈ ℂ ) |
17 |
|
mulcl |
⊢ ( ( i ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) |
18 |
14 16 17
|
sylancr |
⊢ ( 𝐴 ∈ ℂ → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) |
19 |
18
|
adantl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( i · ( ℑ ‘ 𝐴 ) ) ∈ ℂ ) |
20 |
12 13 19
|
adddid |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 · ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) = ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) + ( 𝐵 · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) |
21 |
|
replim |
⊢ ( 𝐴 ∈ ℂ → 𝐴 = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) |
22 |
21
|
adantl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → 𝐴 = ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) |
23 |
22
|
oveq2d |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 · 𝐴 ) = ( 𝐵 · ( ( ℜ ‘ 𝐴 ) + ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) |
24 |
|
mul12 |
⊢ ( ( i ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( ℑ ‘ 𝐴 ) ∈ ℂ ) → ( i · ( 𝐵 · ( ℑ ‘ 𝐴 ) ) ) = ( 𝐵 · ( i · ( ℑ ‘ 𝐴 ) ) ) ) |
25 |
14 7 16 24
|
mp3an3an |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( i · ( 𝐵 · ( ℑ ‘ 𝐴 ) ) ) = ( 𝐵 · ( i · ( ℑ ‘ 𝐴 ) ) ) ) |
26 |
25
|
oveq2d |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) + ( i · ( 𝐵 · ( ℑ ‘ 𝐴 ) ) ) ) = ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) + ( 𝐵 · ( i · ( ℑ ‘ 𝐴 ) ) ) ) ) |
27 |
20 23 26
|
3eqtr4d |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 · 𝐴 ) = ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) + ( i · ( 𝐵 · ( ℑ ‘ 𝐴 ) ) ) ) ) |
28 |
27
|
fveq2d |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( ℜ ‘ ( 𝐵 · 𝐴 ) ) = ( ℜ ‘ ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) + ( i · ( 𝐵 · ( ℑ ‘ 𝐴 ) ) ) ) ) ) |
29 |
|
remulcl |
⊢ ( ( 𝐵 ∈ ℝ ∧ ( ℜ ‘ 𝐴 ) ∈ ℝ ) → ( 𝐵 · ( ℜ ‘ 𝐴 ) ) ∈ ℝ ) |
30 |
3 29
|
sylan2 |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 · ( ℜ ‘ 𝐴 ) ) ∈ ℝ ) |
31 |
|
remulcl |
⊢ ( ( 𝐵 ∈ ℝ ∧ ( ℑ ‘ 𝐴 ) ∈ ℝ ) → ( 𝐵 · ( ℑ ‘ 𝐴 ) ) ∈ ℝ ) |
32 |
15 31
|
sylan2 |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 · ( ℑ ‘ 𝐴 ) ) ∈ ℝ ) |
33 |
|
crre |
⊢ ( ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) ∈ ℝ ∧ ( 𝐵 · ( ℑ ‘ 𝐴 ) ) ∈ ℝ ) → ( ℜ ‘ ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) + ( i · ( 𝐵 · ( ℑ ‘ 𝐴 ) ) ) ) ) = ( 𝐵 · ( ℜ ‘ 𝐴 ) ) ) |
34 |
30 32 33
|
syl2anc |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( ℜ ‘ ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) + ( i · ( 𝐵 · ( ℑ ‘ 𝐴 ) ) ) ) ) = ( 𝐵 · ( ℜ ‘ 𝐴 ) ) ) |
35 |
28 34
|
eqtr2d |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 · ( ℜ ‘ 𝐴 ) ) = ( ℜ ‘ ( 𝐵 · 𝐴 ) ) ) |
36 |
35
|
eqeq1d |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) = ( 𝐵 · 𝐴 ) ↔ ( ℜ ‘ ( 𝐵 · 𝐴 ) ) = ( 𝐵 · 𝐴 ) ) ) |
37 |
|
mulcl |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 · 𝐴 ) ∈ ℂ ) |
38 |
7 37
|
sylan |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( 𝐵 · 𝐴 ) ∈ ℂ ) |
39 |
|
rereb |
⊢ ( ( 𝐵 · 𝐴 ) ∈ ℂ → ( ( 𝐵 · 𝐴 ) ∈ ℝ ↔ ( ℜ ‘ ( 𝐵 · 𝐴 ) ) = ( 𝐵 · 𝐴 ) ) ) |
40 |
38 39
|
syl |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐵 · 𝐴 ) ∈ ℝ ↔ ( ℜ ‘ ( 𝐵 · 𝐴 ) ) = ( 𝐵 · 𝐴 ) ) ) |
41 |
36 40
|
bitr4d |
⊢ ( ( 𝐵 ∈ ℝ ∧ 𝐴 ∈ ℂ ) → ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) = ( 𝐵 · 𝐴 ) ↔ ( 𝐵 · 𝐴 ) ∈ ℝ ) ) |
42 |
41
|
ancoms |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ) → ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) = ( 𝐵 · 𝐴 ) ↔ ( 𝐵 · 𝐴 ) ∈ ℝ ) ) |
43 |
42
|
3adant3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( ( 𝐵 · ( ℜ ‘ 𝐴 ) ) = ( 𝐵 · 𝐴 ) ↔ ( 𝐵 · 𝐴 ) ∈ ℝ ) ) |
44 |
2 11 43
|
3bitr2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0 ) → ( 𝐴 ∈ ℝ ↔ ( 𝐵 · 𝐴 ) ∈ ℝ ) ) |