This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem mvllmuld

Description: Move the left term in a product on the LHS to the RHS, deduction form. (Contributed by David A. Wheeler, 11-Oct-2018)

Ref Expression
Hypotheses mvllmuld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
mvllmuld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
mvllmuld.3 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
mvllmuld.4 ⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = 𝐶 )
Assertion mvllmuld ( 𝜑 → 𝐵 = ( 𝐶 / 𝐴 ) )

Proof

Step Hyp Ref Expression
1 mvllmuld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 mvllmuld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 mvllmuld.3 ⊢ ( 𝜑 → 𝐴 ≠ 0 )
4 mvllmuld.4 ⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = 𝐶 )
5 2 1 3 divcan4d ⊢ ( 𝜑 → ( ( 𝐵 · 𝐴 ) / 𝐴 ) = 𝐵 )
6 1 2 mulcomd ⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) = ( 𝐵 · 𝐴 ) )
7 6 4 eqtr3d ⊢ ( 𝜑 → ( 𝐵 · 𝐴 ) = 𝐶 )
8 7 oveq1d ⊢ ( 𝜑 → ( ( 𝐵 · 𝐴 ) / 𝐴 ) = ( 𝐶 / 𝐴 ) )
9 5 8 eqtr3d ⊢ ( 𝜑 → 𝐵 = ( 𝐶 / 𝐴 ) )