Theorem mvllmuld

Description: Move LHS left multiplication to RHS. (Contributed by David A. Wheeler, 15-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 ) )