Theorem mvlrmuli

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

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

Proof

Step	Hyp	Ref	Expression
1		mvlrmuli.1	⊢ 𝐴 ∈ ℂ
2		mvlrmuli.2	⊢ 𝐵 ∈ ℂ
3		mvlrmuli.3	⊢ 𝐵 ≠ 0
4		mvlrmuli.4	⊢ ( 𝐴 · 𝐵 ) = 𝐶
5	1 2 3	divcan4i	⊢ ( ( 𝐴 · 𝐵 ) / 𝐵 ) = 𝐴
6	4	oveq1i	⊢ ( ( 𝐴 · 𝐵 ) / 𝐵 ) = ( 𝐶 / 𝐵 )
7	5 6	eqtr3i	⊢ 𝐴 = ( 𝐶 / 𝐵 )