Theorem sqr00d

Description: A square root is zero iff its argument is 0. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	sqrt00d.2	⊢ ( 𝜑 → ( √ ‘ 𝐴 ) = 0 )
Assertion	sqr00d	⊢ ( 𝜑 → 𝐴 = 0 )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		sqrt00d.2	⊢ ( 𝜑 → ( √ ‘ 𝐴 ) = 0 )
3	1	sqsqrtd	⊢ ( 𝜑 → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )
4	2	sq0id	⊢ ( 𝜑 → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 0 )
5	3 4	eqtr3d	⊢ ( 𝜑 → 𝐴 = 0 )