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Metamath Proof Explorer

Theorem sqrtdivd

Description: Square root distributes over division. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypotheses resqrcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
resqrcld.2 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
sqrdivd.3 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
Assertion sqrtdivd ( 𝜑 → ( √ ‘ ( 𝐴 / 𝐵 ) ) = ( ( √ ‘ 𝐴 ) / ( √ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 resqrcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 resqrcld.2 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
3 sqrdivd.3 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
4 sqrtdiv ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ 𝐵 ∈ ℝ+ ) → ( √ ‘ ( 𝐴 / 𝐵 ) ) = ( ( √ ‘ 𝐴 ) / ( √ ‘ 𝐵 ) ) )
5 1 2 3 4 syl21anc ⊢ ( 𝜑 → ( √ ‘ ( 𝐴 / 𝐵 ) ) = ( ( √ ‘ 𝐴 ) / ( √ ‘ 𝐵 ) ) )