Step |
Hyp |
Ref |
Expression |
1 |
|
sqval.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
sqreci.1 |
⊢ 𝐴 ≠ 0 |
3 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
4 |
3 1 3 1 2 2
|
divmuldivi |
⊢ ( ( 1 / 𝐴 ) · ( 1 / 𝐴 ) ) = ( ( 1 · 1 ) / ( 𝐴 · 𝐴 ) ) |
5 |
|
1t1e1 |
⊢ ( 1 · 1 ) = 1 |
6 |
5
|
oveq1i |
⊢ ( ( 1 · 1 ) / ( 𝐴 · 𝐴 ) ) = ( 1 / ( 𝐴 · 𝐴 ) ) |
7 |
4 6
|
eqtri |
⊢ ( ( 1 / 𝐴 ) · ( 1 / 𝐴 ) ) = ( 1 / ( 𝐴 · 𝐴 ) ) |
8 |
1 2
|
reccli |
⊢ ( 1 / 𝐴 ) ∈ ℂ |
9 |
8
|
sqvali |
⊢ ( ( 1 / 𝐴 ) ↑ 2 ) = ( ( 1 / 𝐴 ) · ( 1 / 𝐴 ) ) |
10 |
1
|
sqvali |
⊢ ( 𝐴 ↑ 2 ) = ( 𝐴 · 𝐴 ) |
11 |
10
|
oveq2i |
⊢ ( 1 / ( 𝐴 ↑ 2 ) ) = ( 1 / ( 𝐴 · 𝐴 ) ) |
12 |
7 9 11
|
3eqtr4i |
⊢ ( ( 1 / 𝐴 ) ↑ 2 ) = ( 1 / ( 𝐴 ↑ 2 ) ) |