Theorem sqrtval

Description: Value of square root function. (Contributed by Mario Carneiro, 8-Jul-2013)

	Ref	Expression
Assertion	sqrtval	⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 𝐴 ) = ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq2	⊢ ( 𝑦 = 𝐴 → ( ( 𝑥 ↑ 2 ) = 𝑦 ↔ ( 𝑥 ↑ 2 ) = 𝐴 ) )
2	1	3anbi1d	⊢ ( 𝑦 = 𝐴 → ( ( ( 𝑥 ↑ 2 ) = 𝑦 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ↔ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) )
3	2	riotabidv	⊢ ( 𝑦 = 𝐴 → ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝑦 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) )
4		df-sqrt	⊢ √ = ( 𝑦 ∈ ℂ ↦ ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝑦 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) )
5		riotaex	⊢ ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) ∈ V
6	3 4 5	fvmpt	⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 𝐴 ) = ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) )