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Metamath Proof Explorer

Theorem sqrtthlem

Description: Lemma for sqrtth . (Contributed by Mario Carneiro, 10-Jul-2013)

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

Proof

Step Hyp Ref Expression
1 sqrtval ⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 𝐴 ) = ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) )
2 1 eqcomd ⊢ ( 𝐴 ∈ ℂ → ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = ( √ ‘ 𝐴 ) )
3 sqrtcl ⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 𝐴 ) ∈ ℂ )
4 sqreu ⊢ ( 𝐴 ∈ ℂ → ∃! 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) )
5 oveq1 ⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( 𝑥 ↑ 2 ) = ( ( √ ‘ 𝐴 ) ↑ 2 ) )
6 5 eqeq1d ⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( ( 𝑥 ↑ 2 ) = 𝐴 ↔ ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ) )
7 fveq2 ⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( ℜ ‘ 𝑥 ) = ( ℜ ‘ ( √ ‘ 𝐴 ) ) )
8 7 breq2d ⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( 0 ≤ ( ℜ ‘ 𝑥 ) ↔ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ) )
9 oveq2 ⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( i · 𝑥 ) = ( i · ( √ ‘ 𝐴 ) ) )
10 neleq1 ⊢ ( ( i · 𝑥 ) = ( i · ( √ ‘ 𝐴 ) ) → ( ( i · 𝑥 ) ∉ ℝ+ ↔ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) )
11 9 10 syl ⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( ( i · 𝑥 ) ∉ ℝ+ ↔ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) )
12 6 8 11 3anbi123d ⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ↔ ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ∧ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) ) )
13 12 riota2 ⊢ ( ( ( √ ‘ 𝐴 ) ∈ ℂ ∧ ∃! 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) → ( ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ∧ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) ↔ ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = ( √ ‘ 𝐴 ) ) )
14 3 4 13 syl2anc ⊢ ( 𝐴 ∈ ℂ → ( ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ∧ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) ↔ ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = ( √ ‘ 𝐴 ) ) )
15 2 14 mpbird ⊢ ( 𝐴 ∈ ℂ → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ∧ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) )