Metamath Proof Explorer
Description: Relationship between division and multiplication. (Contributed by NM, 8-May-1999) (Revised by Mario Carneiro, 17-Feb-2014)
|
|
Ref |
Expression |
|
Hypotheses |
divclz.1 |
⊢ 𝐴 ∈ ℂ |
|
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
|
|
divmulz.3 |
⊢ 𝐶 ∈ ℂ |
|
Assertion |
divmulzi |
⊢ ( 𝐵 ≠ 0 → ( ( 𝐴 / 𝐵 ) = 𝐶 ↔ ( 𝐵 · 𝐶 ) = 𝐴 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
divclz.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
divmulz.3 |
⊢ 𝐶 ∈ ℂ |
4 |
|
divmul |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) → ( ( 𝐴 / 𝐵 ) = 𝐶 ↔ ( 𝐵 · 𝐶 ) = 𝐴 ) ) |
5 |
1 3 4
|
mp3an12 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) = 𝐶 ↔ ( 𝐵 · 𝐶 ) = 𝐴 ) ) |
6 |
2 5
|
mpan |
⊢ ( 𝐵 ≠ 0 → ( ( 𝐴 / 𝐵 ) = 𝐶 ↔ ( 𝐵 · 𝐶 ) = 𝐴 ) ) |