Theorem resqrtth

Description: Square root theorem over the reals. Theorem I.35 of Apostol p. 29. (Contributed by Mario Carneiro, 9-Jul-2013)

	Ref	Expression
Assertion	resqrtth	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		resqrtthlem	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ∧ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) )
2	1	simp1d	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 )