Metamath Proof Explorer
Description: A cancellation law for division. (Eliminates a hypothesis of divcan3i with the weak deduction theorem.) (Contributed by NM, 3-Feb-2004)
|
|
Ref |
Expression |
|
Hypotheses |
divclz.1 |
⊢ 𝐴 ∈ ℂ |
|
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
|
Assertion |
divcan3zi |
⊢ ( 𝐵 ≠ 0 → ( ( 𝐵 · 𝐴 ) / 𝐵 ) = 𝐴 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
divclz.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
divcan3 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( ( 𝐵 · 𝐴 ) / 𝐵 ) = 𝐴 ) |
4 |
1 2 3
|
mp3an12 |
⊢ ( 𝐵 ≠ 0 → ( ( 𝐵 · 𝐴 ) / 𝐵 ) = 𝐴 ) |