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Metamath Proof Explorer

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 )