Theorem sqsqrtd

Description: Square root theorem. Theorem I.35 of Apostol p. 29. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	sqsqrtd	⊢ ( 𝜑 → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		sqrtth	⊢ ( 𝐴 ∈ ℂ → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )
3	1 2	syl	⊢ ( 𝜑 → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )