Theorem sqrtth

Description: Square root theorem over the complex numbers. Theorem I.35 of Apostol p. 29. (Contributed by Mario Carneiro, 10-Jul-2013)

	Ref	Expression
Assertion	sqrtth	⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		sqrtthlem	⊢ ( 𝐴 ∈ ℂ → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ∧ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) )
2	1	simp1d	⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )