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Metamath Proof Explorer

Theorem sqrtmuld

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

Ref Expression
Hypotheses resqrcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
resqrcld.2 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
sqr11d.3 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
sqr11d.4 ⊢ ( 𝜑 → 0 ≤ 𝐵 )
Assertion sqrtmuld ( 𝜑 → ( √ ‘ ( 𝐴 · 𝐵 ) ) = ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 resqrcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 resqrcld.2 ⊢ ( 𝜑 → 0 ≤ 𝐴 )
3 sqr11d.3 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4 sqr11d.4 ⊢ ( 𝜑 → 0 ≤ 𝐵 )
5 sqrtmul ⊢ ( ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) ∧ ( 𝐵 ∈ ℝ ∧ 0 ≤ 𝐵 ) ) → ( √ ‘ ( 𝐴 · 𝐵 ) ) = ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐵 ) ) )
6 1 2 3 4 5 syl22anc ⊢ ( 𝜑 → ( √ ‘ ( 𝐴 · 𝐵 ) ) = ( ( √ ‘ 𝐴 ) · ( √ ‘ 𝐵 ) ) )