Metamath Proof Explorer
Description: Relationship between division and multiplication. (Contributed by Mario Carneiro, 27-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
div1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
divcld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
|
divmuld.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
|
|
divassd.4 |
⊢ ( 𝜑 → 𝐶 ≠ 0 ) |
|
Assertion |
divmul2d |
⊢ ( 𝜑 → ( ( 𝐴 / 𝐶 ) = 𝐵 ↔ 𝐴 = ( 𝐶 · 𝐵 ) ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
div1d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
divcld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
divmuld.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
4 |
|
divassd.4 |
⊢ ( 𝜑 → 𝐶 ≠ 0 ) |
5 |
|
divmul2 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = 𝐵 ↔ 𝐴 = ( 𝐶 · 𝐵 ) ) ) |
6 |
1 2 3 4 5
|
syl112anc |
⊢ ( 𝜑 → ( ( 𝐴 / 𝐶 ) = 𝐵 ↔ 𝐴 = ( 𝐶 · 𝐵 ) ) ) |