This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem sqrecd

Description: Square of reciprocal is reciprocal of square. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses expcld.1 ⊢ φ → A ∈ ℂ
sqrecd.1 ⊢ φ → A ≠ 0
Assertion sqrecd ⊢ φ → 1 A 2 = 1 A 2

Proof

Step Hyp Ref Expression
1 expcld.1 ⊢ φ → A ∈ ℂ
2 sqrecd.1 ⊢ φ → A ≠ 0
3 2z ⊢ 2 ∈ ℤ
4 3 a1i ⊢ φ → 2 ∈ ℤ
5 exprec ⊢ A ∈ ℂ ∧ A ≠ 0 ∧ 2 ∈ ℤ → 1 A 2 = 1 A 2
6 1 2 4 5 syl3anc ⊢ φ → 1 A 2 = 1 A 2