Theorem rdiv

Description: Right-division. (Contributed by BJ, 6-Jun-2019)

	Ref	Expression
Hypotheses	ldiv.a	$$⊢\phi \to A\in ℂ$$
	ldiv.b	$$⊢\phi \to B\in ℂ$$
	ldiv.c	$$⊢\phi \to C\in ℂ$$
	rdiv.an0	$$⊢\phi \to A\ne 0$$
Assertion	rdiv	$$⊢\phi \to \left(AB=C\leftrightarrow B=\frac{C}{A}\right)$$

Proof

Step	Hyp	Ref	Expression
1		ldiv.a	$$⊢\phi \to A\in ℂ$$
2		ldiv.b	$$⊢\phi \to B\in ℂ$$
3		ldiv.c	$$⊢\phi \to C\in ℂ$$
4		rdiv.an0	$$⊢\phi \to A\ne 0$$
5	1 2	mulcomd	$$⊢\phi \to AB=BA$$
6	5	eqeq1d	$$⊢\phi \to \left(AB=C\leftrightarrow BA=C\right)$$
7	2 1 3 4	ldiv	$$⊢\phi \to \left(BA=C\leftrightarrow B=\frac{C}{A}\right)$$
8	6 7	bitrd	$$⊢\phi \to \left(AB=C\leftrightarrow B=\frac{C}{A}\right)$$