Metamath Proof Explorer
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 ) ) ) |