Theorem eqsqrt2d

Description: A deduction for showing that a number equals the square root of another. (Contributed by Mario Carneiro, 3-Apr-2015)

	Ref	Expression
Hypotheses	eqsqrtd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	eqsqrtd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
	eqsqrtd.3	⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) = 𝐵 )
	eqsqrt2d.4	⊢ ( 𝜑 → 0 < ( ℜ ‘ 𝐴 ) )
Assertion	eqsqrt2d	⊢ ( 𝜑 → 𝐴 = ( √ ‘ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqsqrtd.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		eqsqrtd.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		eqsqrtd.3	⊢ ( 𝜑 → ( 𝐴 ↑ 2 ) = 𝐵 )
4		eqsqrt2d.4	⊢ ( 𝜑 → 0 < ( ℜ ‘ 𝐴 ) )
5		0re	⊢ 0 ∈ ℝ
6	1	recld	⊢ ( 𝜑 → ( ℜ ‘ 𝐴 ) ∈ ℝ )
7		ltle	⊢ ( ( 0 ∈ ℝ ∧ ( ℜ ‘ 𝐴 ) ∈ ℝ ) → ( 0 < ( ℜ ‘ 𝐴 ) → 0 ≤ ( ℜ ‘ 𝐴 ) ) )
8	5 6 7	sylancr	⊢ ( 𝜑 → ( 0 < ( ℜ ‘ 𝐴 ) → 0 ≤ ( ℜ ‘ 𝐴 ) ) )
9	4 8	mpd	⊢ ( 𝜑 → 0 ≤ ( ℜ ‘ 𝐴 ) )
10		reim	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) = ( ℑ ‘ ( i · 𝐴 ) ) )
11	1 10	syl	⊢ ( 𝜑 → ( ℜ ‘ 𝐴 ) = ( ℑ ‘ ( i · 𝐴 ) ) )
12	4	gt0ne0d	⊢ ( 𝜑 → ( ℜ ‘ 𝐴 ) ≠ 0 )
13	11 12	eqnetrrd	⊢ ( 𝜑 → ( ℑ ‘ ( i · 𝐴 ) ) ≠ 0 )
14		rpre	⊢ ( ( i · 𝐴 ) ∈ ℝ+ → ( i · 𝐴 ) ∈ ℝ )
15	14	reim0d	⊢ ( ( i · 𝐴 ) ∈ ℝ+ → ( ℑ ‘ ( i · 𝐴 ) ) = 0 )
16	15	necon3ai	⊢ ( ( ℑ ‘ ( i · 𝐴 ) ) ≠ 0 → ¬ ( i · 𝐴 ) ∈ ℝ+ )
17	13 16	syl	⊢ ( 𝜑 → ¬ ( i · 𝐴 ) ∈ ℝ+ )
18	1 2 3 9 17	eqsqrtd	⊢ ( 𝜑 → 𝐴 = ( √ ‘ 𝐵 ) )