Theorem sqrtmsq2i

Description: Relationship between square root and squares. (Contributed by NM, 31-Jul-1999)

	Ref	Expression
Hypotheses	sqrtthi.1	⊢ 𝐴 ∈ ℝ
	sqr11.1	⊢ 𝐵 ∈ ℝ
Assertion	sqrtmsq2i	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐴 ) = 𝐵 ↔ 𝐴 = ( 𝐵 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	⊢ 𝐴 ∈ ℝ
2		sqr11.1	⊢ 𝐵 ∈ ℝ
3		sqrtsq2	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( ( √ ‘ 𝐴 ) = 𝐵 ↔ 𝐴 = ( 𝐵 ↑ 2 ) ) )
4	2 3	mpanr1	⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐴 ) = 𝐵 ↔ 𝐴 = ( 𝐵 ↑ 2 ) ) )
5	1 4	mpanl1	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐴 ) = 𝐵 ↔ 𝐴 = ( 𝐵 ↑ 2 ) ) )
6	2	recni	⊢ 𝐵 ∈ ℂ
7	6	sqvali	⊢ ( 𝐵 ↑ 2 ) = ( 𝐵 · 𝐵 )
8	7	eqeq2i	⊢ ( 𝐴 = ( 𝐵 ↑ 2 ) ↔ 𝐴 = ( 𝐵 · 𝐵 ) )
9	5 8	bitrdi	⊢ ( ( 0 ≤ 𝐴 ∧ 0 ≤ 𝐵 ) → ( ( √ ‘ 𝐴 ) = 𝐵 ↔ 𝐴 = ( 𝐵 · 𝐵 ) ) )