Theorem sqrecd

Description: Square of reciprocal. (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 ) ) )