Step |
Hyp |
Ref |
Expression |
1 |
|
negicn |
⊢ - i ∈ ℂ |
2 |
1
|
a1i |
⊢ ( 𝐴 ∈ ℂ → - i ∈ ℂ ) |
3 |
|
id |
⊢ ( 𝐴 ∈ ℂ → 𝐴 ∈ ℂ ) |
4 |
2 3
|
mulcld |
⊢ ( 𝐴 ∈ ℂ → ( - i · 𝐴 ) ∈ ℂ ) |
5 |
|
absrele |
⊢ ( ( - i · 𝐴 ) ∈ ℂ → ( abs ‘ ( ℜ ‘ ( - i · 𝐴 ) ) ) ≤ ( abs ‘ ( - i · 𝐴 ) ) ) |
6 |
4 5
|
syl |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ℜ ‘ ( - i · 𝐴 ) ) ) ≤ ( abs ‘ ( - i · 𝐴 ) ) ) |
7 |
|
imre |
⊢ ( 𝐴 ∈ ℂ → ( ℑ ‘ 𝐴 ) = ( ℜ ‘ ( - i · 𝐴 ) ) ) |
8 |
7
|
fveq2d |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ℑ ‘ 𝐴 ) ) = ( abs ‘ ( ℜ ‘ ( - i · 𝐴 ) ) ) ) |
9 |
|
absmul |
⊢ ( ( - i ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( abs ‘ ( - i · 𝐴 ) ) = ( ( abs ‘ - i ) · ( abs ‘ 𝐴 ) ) ) |
10 |
1 9
|
mpan |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( - i · 𝐴 ) ) = ( ( abs ‘ - i ) · ( abs ‘ 𝐴 ) ) ) |
11 |
|
ax-icn |
⊢ i ∈ ℂ |
12 |
|
absneg |
⊢ ( i ∈ ℂ → ( abs ‘ - i ) = ( abs ‘ i ) ) |
13 |
11 12
|
ax-mp |
⊢ ( abs ‘ - i ) = ( abs ‘ i ) |
14 |
|
absi |
⊢ ( abs ‘ i ) = 1 |
15 |
13 14
|
eqtri |
⊢ ( abs ‘ - i ) = 1 |
16 |
15
|
oveq1i |
⊢ ( ( abs ‘ - i ) · ( abs ‘ 𝐴 ) ) = ( 1 · ( abs ‘ 𝐴 ) ) |
17 |
|
abscl |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℝ ) |
18 |
17
|
recnd |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℂ ) |
19 |
18
|
mulid2d |
⊢ ( 𝐴 ∈ ℂ → ( 1 · ( abs ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) ) |
20 |
16 19
|
eqtrid |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ - i ) · ( abs ‘ 𝐴 ) ) = ( abs ‘ 𝐴 ) ) |
21 |
10 20
|
eqtr2d |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) = ( abs ‘ ( - i · 𝐴 ) ) ) |
22 |
6 8 21
|
3brtr4d |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ ( ℑ ‘ 𝐴 ) ) ≤ ( abs ‘ 𝐴 ) ) |