Metamath Proof Explorer
Description: One-to-one relationship for division. (Contributed by NM, 20-Aug-2001)
|
|
Ref |
Expression |
|
Hypotheses |
divclz.1 |
⊢ 𝐴 ∈ ℂ |
|
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
|
|
divmulz.3 |
⊢ 𝐶 ∈ ℂ |
|
|
divass.4 |
⊢ 𝐶 ≠ 0 |
|
Assertion |
div11i |
⊢ ( ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ↔ 𝐴 = 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
divclz.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
divmulz.3 |
⊢ 𝐶 ∈ ℂ |
4 |
|
divass.4 |
⊢ 𝐶 ≠ 0 |
5 |
3 4
|
pm3.2i |
⊢ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) |
6 |
|
div11 |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ ( 𝐶 ∈ ℂ ∧ 𝐶 ≠ 0 ) ) → ( ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ↔ 𝐴 = 𝐵 ) ) |
7 |
1 2 5 6
|
mp3an |
⊢ ( ( 𝐴 / 𝐶 ) = ( 𝐵 / 𝐶 ) ↔ 𝐴 = 𝐵 ) |