Theorem resqrtthlem

Description: Lemma for resqrtth . (Contributed by Mario Carneiro, 9-Jul-2013)

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

Proof

Step	Hyp	Ref	Expression
1		recn	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℂ )
2		sqrtval	⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 𝐴 ) = ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) )
3	2	eqcomd	⊢ ( 𝐴 ∈ ℂ → ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = ( √ ‘ 𝐴 ) )
4	1 3	syl	⊢ ( 𝐴 ∈ ℝ → ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = ( √ ‘ 𝐴 ) )
5	4	adantr	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = ( √ ‘ 𝐴 ) )
6		resqrtcl	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℝ )
7	6	recnd	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( √ ‘ 𝐴 ) ∈ ℂ )
8		resqreu	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ∃! 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) )
9		oveq1	⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( 𝑥 ↑ 2 ) = ( ( √ ‘ 𝐴 ) ↑ 2 ) )
10	9	eqeq1d	⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( ( 𝑥 ↑ 2 ) = 𝐴 ↔ ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ) )
11		fveq2	⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( ℜ ‘ 𝑥 ) = ( ℜ ‘ ( √ ‘ 𝐴 ) ) )
12	11	breq2d	⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( 0 ≤ ( ℜ ‘ 𝑥 ) ↔ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ) )
13		oveq2	⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( i · 𝑥 ) = ( i · ( √ ‘ 𝐴 ) ) )
14		neleq1	⊢ ( ( i · 𝑥 ) = ( i · ( √ ‘ 𝐴 ) ) → ( ( i · 𝑥 ) ∉ ℝ+ ↔ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) )
15	13 14	syl	⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( ( i · 𝑥 ) ∉ ℝ+ ↔ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) )
16	10 12 15	3anbi123d	⊢ ( 𝑥 = ( √ ‘ 𝐴 ) → ( ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ↔ ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ∧ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) ) )
17	16	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 ) ) )
18	7 8 17	syl2anc	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ∧ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) ↔ ( ℩ 𝑥 ∈ ℂ ( ( 𝑥 ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ 𝑥 ) ∧ ( i · 𝑥 ) ∉ ℝ+ ) ) = ( √ ‘ 𝐴 ) ) )
19	5 18	mpbird	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( ( √ ‘ 𝐴 ) ↑ 2 ) = 𝐴 ∧ 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) ∧ ( i · ( √ ‘ 𝐴 ) ) ∉ ℝ+ ) )