Step |
Hyp |
Ref |
Expression |
1 |
|
cjcl |
⊢ ( 𝐴 ∈ ℂ → ( ∗ ‘ 𝐴 ) ∈ ℂ ) |
2 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ∗ ‘ 𝐴 ) ∈ ℂ ) |
3 |
|
simpl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → 𝐴 ∈ ℂ ) |
4 |
2 3
|
mulcomd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( ∗ ‘ 𝐴 ) · 𝐴 ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
5 |
|
absvalsq |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
6 |
5
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( abs ‘ 𝐴 ) ↑ 2 ) = ( 𝐴 · ( ∗ ‘ 𝐴 ) ) ) |
7 |
4 6
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( ∗ ‘ 𝐴 ) · 𝐴 ) = ( ( abs ‘ 𝐴 ) ↑ 2 ) ) |
8 |
|
abscl |
⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℝ ) |
9 |
8
|
adantr |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( abs ‘ 𝐴 ) ∈ ℝ ) |
10 |
9
|
recnd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( abs ‘ 𝐴 ) ∈ ℂ ) |
11 |
10
|
sqcld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( abs ‘ 𝐴 ) ↑ 2 ) ∈ ℂ ) |
12 |
|
cjne0 |
⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ≠ 0 ↔ ( ∗ ‘ 𝐴 ) ≠ 0 ) ) |
13 |
12
|
biimpa |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ∗ ‘ 𝐴 ) ≠ 0 ) |
14 |
11 2 3 13
|
divmuld |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( ( ( abs ‘ 𝐴 ) ↑ 2 ) / ( ∗ ‘ 𝐴 ) ) = 𝐴 ↔ ( ( ∗ ‘ 𝐴 ) · 𝐴 ) = ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) |
15 |
7 14
|
mpbird |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) / ( ∗ ‘ 𝐴 ) ) = 𝐴 ) |
16 |
15
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / ( ( ( abs ‘ 𝐴 ) ↑ 2 ) / ( ∗ ‘ 𝐴 ) ) ) = ( 1 / 𝐴 ) ) |
17 |
|
abs00 |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) ) |
18 |
17
|
necon3bid |
⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) ≠ 0 ↔ 𝐴 ≠ 0 ) ) |
19 |
18
|
biimpar |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( abs ‘ 𝐴 ) ≠ 0 ) |
20 |
|
sqne0 |
⊢ ( ( abs ‘ 𝐴 ) ∈ ℂ → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) ≠ 0 ↔ ( abs ‘ 𝐴 ) ≠ 0 ) ) |
21 |
10 20
|
syl |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( ( abs ‘ 𝐴 ) ↑ 2 ) ≠ 0 ↔ ( abs ‘ 𝐴 ) ≠ 0 ) ) |
22 |
19 21
|
mpbird |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( ( abs ‘ 𝐴 ) ↑ 2 ) ≠ 0 ) |
23 |
11 2 22 13
|
recdivd |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / ( ( ( abs ‘ 𝐴 ) ↑ 2 ) / ( ∗ ‘ 𝐴 ) ) ) = ( ( ∗ ‘ 𝐴 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) |
24 |
16 23
|
eqtr3d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ) → ( 1 / 𝐴 ) = ( ( ∗ ‘ 𝐴 ) / ( ( abs ‘ 𝐴 ) ↑ 2 ) ) ) |