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Metamath Proof Explorer

Theorem mvllmuli

Description: Move the left term in a product on the LHS to the RHS, inference form. Uses divcan4i . (Contributed by David A. Wheeler, 11-Oct-2018)

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

Proof

Step	Hyp	Ref	Expression
1		mvllmuli.1	⊢ 𝐴 ∈ ℂ
2		mvllmuli.2	⊢ 𝐵 ∈ ℂ
3		mvllmuli.3	⊢ 𝐴 ≠ 0
4		mvllmuli.4	⊢ ( 𝐴 · 𝐵 ) = 𝐶
5	2 1 3	divcan4i	⊢ ( ( 𝐵 · 𝐴 ) / 𝐴 ) = 𝐵
6	1 2 4	mulcomli	⊢ ( 𝐵 · 𝐴 ) = 𝐶
7	6	oveq1i	⊢ ( ( 𝐵 · 𝐴 ) / 𝐴 ) = ( 𝐶 / 𝐴 )
8	5 7	eqtr3i	⊢ 𝐵 = ( 𝐶 / 𝐴 )