Theorem msqsqrtd

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	msqsqrtd	⊢ ( 𝜑 → ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐴 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2	1	sqrtcld	⊢ ( 𝜑 → ( √ ‘ 𝐴 ) ∈ ℂ )
3	2	sqvald	⊢ ( 𝜑 → ( ( √ ‘ 𝐴 ) ↑ 2 ) = ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐴 ) ) )
4	1	sqsqrtd	⊢ ( 𝜑 → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )
5	3 4	eqtr3d	⊢ ( 𝜑 → ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐴 ) ) = 𝐴 )