Step |
Hyp |
Ref |
Expression |
1 |
|
cjcj |
⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) = 𝐴 ) |
2 |
1
|
oveq2d |
⊢ ( 𝐴 ∈ ℂ → ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) ) = ( ( ∗ ‘ 𝐴 ) · 𝐴 ) ) |
3 |
|
cjcl |
⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) ∈ ℂ ) |
4 |
|
cjmul |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ∗ ‘ 𝐴 ) ∈ ℂ ) → ( ∗ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) ) ) |
5 |
3 4
|
mpdan |
⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( ( ∗ ‘ 𝐴 ) · ( ∗ ‘ ( ∗ ‘ 𝐴 ) ) ) ) |
6 |
|
mulcom |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ∗ ‘ 𝐴 ) ∈ ℂ ) → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ∗ ‘ 𝐴 ) · 𝐴 ) ) |
7 |
3 6
|
mpdan |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) = ( ( ∗ ‘ 𝐴 ) · 𝐴 ) ) |
8 |
2 5 7
|
3eqtr4d |
⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
9 |
|
mulcl |
⊢ ( ( 𝐴 ∈ ℂ ∧ ( ∗ ‘ 𝐴 ) ∈ ℂ ) → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℂ ) |
10 |
3 9
|
mpdan |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℂ ) |
11 |
|
cjreb |
⊢ ( ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℂ → ( ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ ↔ ( ∗ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) ) |
12 |
10 11
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ ↔ ( ∗ ‘ ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) ) |
13 |
8 12
|
mpbird |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ∈ ℝ ) |