Step |
Hyp |
Ref |
Expression |
1 |
|
sqdivzi.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
sqdivzi.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
oveq2 |
⊢ ( 𝐵 = if ( 𝐵 ≠ 0 , 𝐵 , 1 ) → ( 𝐴 / 𝐵 ) = ( 𝐴 / if ( 𝐵 ≠ 0 , 𝐵 , 1 ) ) ) |
4 |
3
|
oveq1d |
⊢ ( 𝐵 = if ( 𝐵 ≠ 0 , 𝐵 , 1 ) → ( ( 𝐴 / 𝐵 ) ↑ 2 ) = ( ( 𝐴 / if ( 𝐵 ≠ 0 , 𝐵 , 1 ) ) ↑ 2 ) ) |
5 |
|
oveq1 |
⊢ ( 𝐵 = if ( 𝐵 ≠ 0 , 𝐵 , 1 ) → ( 𝐵 ↑ 2 ) = ( if ( 𝐵 ≠ 0 , 𝐵 , 1 ) ↑ 2 ) ) |
6 |
5
|
oveq2d |
⊢ ( 𝐵 = if ( 𝐵 ≠ 0 , 𝐵 , 1 ) → ( ( 𝐴 ↑ 2 ) / ( 𝐵 ↑ 2 ) ) = ( ( 𝐴 ↑ 2 ) / ( if ( 𝐵 ≠ 0 , 𝐵 , 1 ) ↑ 2 ) ) ) |
7 |
4 6
|
eqeq12d |
⊢ ( 𝐵 = if ( 𝐵 ≠ 0 , 𝐵 , 1 ) → ( ( ( 𝐴 / 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) / ( 𝐵 ↑ 2 ) ) ↔ ( ( 𝐴 / if ( 𝐵 ≠ 0 , 𝐵 , 1 ) ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) / ( if ( 𝐵 ≠ 0 , 𝐵 , 1 ) ↑ 2 ) ) ) ) |
8 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
9 |
2 8
|
ifcli |
⊢ if ( 𝐵 ≠ 0 , 𝐵 , 1 ) ∈ ℂ |
10 |
|
elimne0 |
⊢ if ( 𝐵 ≠ 0 , 𝐵 , 1 ) ≠ 0 |
11 |
1 9 10
|
sqdivi |
⊢ ( ( 𝐴 / if ( 𝐵 ≠ 0 , 𝐵 , 1 ) ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) / ( if ( 𝐵 ≠ 0 , 𝐵 , 1 ) ↑ 2 ) ) |
12 |
7 11
|
dedth |
⊢ ( 𝐵 ≠ 0 → ( ( 𝐴 / 𝐵 ) ↑ 2 ) = ( ( 𝐴 ↑ 2 ) / ( 𝐵 ↑ 2 ) ) ) |