redivmuld

Description: Relationship between division and multiplication. (Contributed by SN, 25-Nov-2025)

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

Step	Hyp	Ref	Expression
1		redivmuld.a	$⊢ φ \to A \in ℝ$
2		redivmuld.b	$⊢ φ \to B \in ℝ$
3		redivmuld.c	$⊢ φ \to C \in ℝ$
4		redivmuld.z	$⊢ φ \to C \neq 0$
5	1 3 4	redivvald	Could not format ( ph -> ( A /R C ) = ( iota_ x e. RR ( C x. x ) = A ) ) : No typesetting found for \|- ( ph -> ( A /R C ) = ( iota_ x e. RR ( C x. x ) = A ) ) with typecode \|-
6	5	eqeq1d	Could not format ( ph -> ( ( A /R C ) = B <-> ( iota_ x e. RR ( C x. x ) = A ) = B ) ) : No typesetting found for \|- ( ph -> ( ( A /R C ) = B <-> ( iota_ x e. RR ( C x. x ) = A ) = B ) ) with typecode \|-
7	1 3 4	rediveud	$⊢ φ \to \exists! x \in ℝ C x = A$
8		oveq2	$⊢ x = B \to C x = C B$
9	8	eqeq1d	$⊢ x = B \to (C x = A \leftrightarrow C B = A)$
10	9	riota2	$⊢ (B \in ℝ \land \exists! x \in ℝ C x = A) \to (C B = A \leftrightarrow (ι x \in ℝ \| C x = A) = B)$
11	2 7 10	syl2anc	$⊢ φ \to (C B = A \leftrightarrow (ι x \in ℝ \| C x = A) = B)$
12	6 11	bitr4d	Could not format ( ph -> ( ( A /R C ) = B <-> ( C x. B ) = A ) ) : No typesetting found for \|- ( ph -> ( ( A /R C ) = B <-> ( C x. B ) = A ) ) with typecode \|-

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