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Metamath Proof Explorer

Theorem mdiv

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

		Ref	Expression
	Hypotheses	ldiv.a	$⊢ φ \to A \in ℂ$
		ldiv.b	$⊢ φ \to B \in ℂ$
		ldiv.c	$⊢ φ \to C \in ℂ$
		mdiv.an0	$⊢ φ \to A \neq 0$
		mdiv.bn0	$⊢ φ \to B \neq 0$
	Assertion	mdiv	$⊢ φ \to (A = \frac{C}{B} \leftrightarrow B = \frac{C}{A})$

Proof

Step	Hyp	Ref	Expression
1		ldiv.a	$⊢ φ \to A \in ℂ$
2		ldiv.b	$⊢ φ \to B \in ℂ$
3		ldiv.c	$⊢ φ \to C \in ℂ$
4		mdiv.an0	$⊢ φ \to A \neq 0$
5		mdiv.bn0	$⊢ φ \to B \neq 0$
6	1 2 3 5	ldiv	$⊢ φ \to (A B = C \leftrightarrow A = \frac{C}{B})$
7	1 2 3 4	rdiv	$⊢ φ \to (A B = C \leftrightarrow B = \frac{C}{A})$
8	6 7	bitr3d	$⊢ φ \to (A = \frac{C}{B} \leftrightarrow B = \frac{C}{A})$