Metamath Proof Explorer
Description: A cancellation law for division. (Contributed by NM, 18-May-1999)
|
|
Ref |
Expression |
|
Hypotheses |
divclz.1 |
⊢ 𝐴 ∈ ℂ |
|
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
|
|
divcl.3 |
⊢ 𝐵 ≠ 0 |
|
Assertion |
divcan1i |
⊢ ( ( 𝐴 / 𝐵 ) · 𝐵 ) = 𝐴 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
divclz.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
divcl.3 |
⊢ 𝐵 ≠ 0 |
4 |
1 2 3
|
divcli |
⊢ ( 𝐴 / 𝐵 ) ∈ ℂ |
5 |
1 2 3
|
divcan2i |
⊢ ( 𝐵 · ( 𝐴 / 𝐵 ) ) = 𝐴 |
6 |
2 4 5
|
mulcomli |
⊢ ( ( 𝐴 / 𝐵 ) · 𝐵 ) = 𝐴 |