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Metamath Proof Explorer

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+ ) ) )