Theorem divsqrtid

Description: A real number divided by its square root. (Contributed by Thierry Arnoux, 1-Jan-2022)

	Ref	Expression
Assertion	divsqrtid	⊢ ( 𝐴 ∈ ℝ+ → ( 𝐴 / ( √ ‘ 𝐴 ) ) = ( √ ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		rpre	⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ )
2		rpge0	⊢ ( 𝐴 ∈ ℝ+ → 0 ≤ 𝐴 )
3		remsqsqrt	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐴 ) ) = 𝐴 )
4	1 2 3	syl2anc	⊢ ( 𝐴 ∈ ℝ+ → ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐴 ) ) = 𝐴 )
5	4	oveq1d	⊢ ( 𝐴 ∈ ℝ+ → ( ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐴 ) ) / ( √ ‘ 𝐴 ) ) = ( 𝐴 / ( √ ‘ 𝐴 ) ) )
6	1	recnd	⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ )
7	6	sqrtcld	⊢ ( 𝐴 ∈ ℝ+ → ( √ ‘ 𝐴 ) ∈ ℂ )
8		rpsqrtcl	⊢ ( 𝐴 ∈ ℝ+ → ( √ ‘ 𝐴 ) ∈ ℝ+ )
9	8	rpne0d	⊢ ( 𝐴 ∈ ℝ+ → ( √ ‘ 𝐴 ) ≠ 0 )
10	7 7 9	divcan4d	⊢ ( 𝐴 ∈ ℝ+ → ( ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐴 ) ) / ( √ ‘ 𝐴 ) ) = ( √ ‘ 𝐴 ) )
11	5 10	eqtr3d	⊢ ( 𝐴 ∈ ℝ+ → ( 𝐴 / ( √ ‘ 𝐴 ) ) = ( √ ‘ 𝐴 ) )