Metamath Proof Explorer
Description: Swap denominators in a division. (Contributed by NM, 15-Sep-1999)
|
|
Ref |
Expression |
|
Hypotheses |
divclz.1 |
⊢ 𝐴 ∈ ℂ |
|
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
|
|
divmulz.3 |
⊢ 𝐶 ∈ ℂ |
|
|
divmul.4 |
⊢ 𝐵 ≠ 0 |
|
|
divdiv23.5 |
⊢ 𝐶 ≠ 0 |
|
Assertion |
divdiv32i |
⊢ ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) / 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
divclz.1 |
⊢ 𝐴 ∈ ℂ |
2 |
|
divclz.2 |
⊢ 𝐵 ∈ ℂ |
3 |
|
divmulz.3 |
⊢ 𝐶 ∈ ℂ |
4 |
|
divmul.4 |
⊢ 𝐵 ≠ 0 |
5 |
|
divdiv23.5 |
⊢ 𝐶 ≠ 0 |
6 |
1 2 3
|
divdiv23zi |
⊢ ( ( 𝐵 ≠ 0 ∧ 𝐶 ≠ 0 ) → ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) / 𝐵 ) ) |
7 |
4 5 6
|
mp2an |
⊢ ( ( 𝐴 / 𝐵 ) / 𝐶 ) = ( ( 𝐴 / 𝐶 ) / 𝐵 ) |