rediveq0d

Description: A ratio is zero iff the numerator is zero. (Contributed by SN, 25-Nov-2025)

	Ref	Expression
Hypotheses	redivcan2d.a	$⊢ φ \to A \in ℝ$
	redivcan2d.b	$⊢ φ \to B \in ℝ$
	redivcan2d.z	$⊢ φ \to B \neq 0$
Assertion	rediveq0d	Could not format assertion : No typesetting found for \|- ( ph -> ( ( A /R B ) = 0 <-> A = 0 ) ) with typecode \|-

Step	Hyp	Ref	Expression
1		redivcan2d.a	$⊢ φ \to A \in ℝ$
2		redivcan2d.b	$⊢ φ \to B \in ℝ$
3		redivcan2d.z	$⊢ φ \to B \neq 0$
4		0red	$⊢ φ \to 0 \in ℝ$
5	1 4 2 3	redivmul2d	Could not format ( ph -> ( ( A /R B ) = 0 <-> A = ( B x. 0 ) ) ) : No typesetting found for \|- ( ph -> ( ( A /R B ) = 0 <-> A = ( B x. 0 ) ) ) with typecode \|-
6		remul01	$⊢ B \in ℝ \to B \cdot 0 = 0$
7	2 6	syl	$⊢ φ \to B \cdot 0 = 0$
8	7	eqeq2d	$⊢ φ \to (A = B \cdot 0 \leftrightarrow A = 0)$
9	5 8	bitrd	Could not format ( ph -> ( ( A /R B ) = 0 <-> A = 0 ) ) : No typesetting found for \|- ( ph -> ( ( A /R B ) = 0 <-> A = 0 ) ) with typecode \|-

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