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	⊢ ( 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 ) )