Theorem sqrtthi

Description: Square root theorem. Theorem I.35 of Apostol p. 29. (Contributed by NM, 26-May-1999) (Revised by Mario Carneiro, 6-Sep-2013)

	Ref	Expression
Hypothesis	sqrtthi.1	⊢ 𝐴 ∈ ℝ
Assertion	sqrtthi	⊢ ( 0 ≤ 𝐴 → ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐴 ) ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	⊢ 𝐴 ∈ ℝ
2		remsqsqrt	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐴 ) ) = 𝐴 )
3	1 2	mpan	⊢ ( 0 ≤ 𝐴 → ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐴 ) ) = 𝐴 )