Metamath Proof Explorer


Theorem sqrecii

Description: Square of reciprocal. (Contributed by NM, 17-Sep-1999)

Ref Expression
Hypotheses sqval.1 𝐴 ∈ ℂ
sqreci.1 𝐴 ≠ 0
Assertion sqrecii ( ( 1 / 𝐴 ) ↑ 2 ) = ( 1 / ( 𝐴 ↑ 2 ) )

Proof

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 ) )