Theorem redivrec2d

Description: Relationship between division and reciprocal. (Contributed by SN, 9-Apr-2026)

	Ref	Expression
Hypotheses	redivrec2d.a	$$⊢\phi \to A\in ℝ$$
	redivrec2d.b	$$⊢\phi \to B\in ℝ$$
	redivrec2d.z	$$⊢\phi \to B\ne 0$$
Assertion	redivrec2d	$$⊢\phi \to A{/}_{ℝ}B=\left(1{/}_{ℝ}B\right)A$$

Proof

Step	Hyp	Ref	Expression
1		redivrec2d.a	$$⊢\phi \to A\in ℝ$$
2		redivrec2d.b	$$⊢\phi \to B\in ℝ$$
3		redivrec2d.z	$$⊢\phi \to B\ne 0$$
4	2 3	rerecidd	$$⊢\phi \to B\left(1{/}_{ℝ}B\right)=1$$
5	4	oveq1d	$$⊢\phi \to B\left(1{/}_{ℝ}B\right)A=1A$$
6	2	recnd	$$⊢\phi \to B\in ℂ$$
7	2 3	sn-rereccld	$$⊢\phi \to 1{/}_{ℝ}B\in ℝ$$
8	7	recnd	$$⊢\phi \to 1{/}_{ℝ}B\in ℂ$$
9	1	recnd	$$⊢\phi \to A\in ℂ$$
10	6 8 9	mulassd	$$⊢\phi \to B\left(1{/}_{ℝ}B\right)A=B\left(1{/}_{ℝ}B\right)A$$
11		remullid	$$⊢A\in ℝ\to 1A=A$$
12	1 11	syl	$$⊢\phi \to 1A=A$$
13	5 10 12	3eqtr3d	$$⊢\phi \to B\left(1{/}_{ℝ}B\right)A=A$$
14	7 1	remulcld	$$⊢\phi \to \left(1{/}_{ℝ}B\right)A\in ℝ$$
15	1 14 2 3	redivmuld	$$⊢\phi \to \left(A{/}_{ℝ}B=\left(1{/}_{ℝ}B\right)A\leftrightarrow B\left(1{/}_{ℝ}B\right)A=A\right)$$
16	13 15	mpbird	$$⊢\phi \to A{/}_{ℝ}B=\left(1{/}_{ℝ}B\right)A$$