Theorem mdiv

Description: A division law. (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	$$⊢\phi \to C\in ℂ$$
	mdiv.an0	$$⊢\phi \to A\ne 0$$
	mdiv.bn0	$$⊢\phi \to B\ne 0$$
Assertion	mdiv	$$⊢\phi \to \left(A=\frac{C}{B}\leftrightarrow B=\frac{C}{A}\right)$$

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		mdiv.an0	$$⊢\phi \to A\ne 0$$
5		mdiv.bn0	$$⊢\phi \to B\ne 0$$
6	1 2 3 5	ldiv	$$⊢\phi \to \left(AB=C\leftrightarrow A=\frac{C}{B}\right)$$
7	1 2 3 4	rdiv	$$⊢\phi \to \left(AB=C\leftrightarrow B=\frac{C}{A}\right)$$
8	6 7	bitr3d	$$⊢\phi \to \left(A=\frac{C}{B}\leftrightarrow B=\frac{C}{A}\right)$$