Metamath Proof Explorer
Description: Move the right term in a product on the LHS to the RHS, deduction form.
(Contributed by David A. Wheeler, 11-Oct-2018)
|
|
Ref |
Expression |
|
Hypotheses |
mvlrmuld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
|
|
mvlrmuld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
|
|
mvlrmuld.3 |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
|
|
mvlrmuld.4 |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = 𝐶 ) |
|
Assertion |
mvlrmuld |
⊢ ( 𝜑 → 𝐴 = ( 𝐶 / 𝐵 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
mvlrmuld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
2 |
|
mvlrmuld.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
3 |
|
mvlrmuld.3 |
⊢ ( 𝜑 → 𝐵 ≠ 0 ) |
4 |
|
mvlrmuld.4 |
⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = 𝐶 ) |
5 |
1 2 3
|
divcan4d |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) / 𝐵 ) = 𝐴 ) |
6 |
4
|
oveq1d |
⊢ ( 𝜑 → ( ( 𝐴 · 𝐵 ) / 𝐵 ) = ( 𝐶 / 𝐵 ) ) |
7 |
5 6
|
eqtr3d |
⊢ ( 𝜑 → 𝐴 = ( 𝐶 / 𝐵 ) ) |