Metamath Proof Explorer


Theorem mvlrmuld

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 ( 𝜑𝐴 ∈ ℂ )
mvlrmuld.2 ( 𝜑𝐵 ∈ ℂ )
mvlrmuld.3 ( 𝜑𝐵 ≠ 0 )
mvlrmuld.4 ( 𝜑 → ( 𝐴 · 𝐵 ) = 𝐶 )
Assertion mvlrmuld ( 𝜑𝐴 = ( 𝐶 / 𝐵 ) )

Proof

Step Hyp Ref Expression
1 mvlrmuld.1 ( 𝜑𝐴 ∈ ℂ )
2 mvlrmuld.2 ( 𝜑𝐵 ∈ ℂ )
3 mvlrmuld.3 ( 𝜑𝐵 ≠ 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 ( 𝜑𝐴 = ( 𝐶 / 𝐵 ) )