Theorem lediv23d

Description: Swap denominator with other side of 'less than or equal to'. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	ltdiv23d.1	$$⊢\phi \to A\in ℝ$$
	ltdiv23d.2	$$⊢\phi \to B\in {ℝ}^{+}$$
	ltdiv23d.3	$$⊢\phi \to C\in {ℝ}^{+}$$
	lediv23d.4	$$⊢\phi \to \frac{A}{B}\le C$$
Assertion	lediv23d	$$⊢\phi \to \frac{A}{C}\le B$$

Proof

Step	Hyp	Ref	Expression
1		ltdiv23d.1	$$⊢\phi \to A\in ℝ$$
2		ltdiv23d.2	$$⊢\phi \to B\in {ℝ}^{+}$$
3		ltdiv23d.3	$$⊢\phi \to C\in {ℝ}^{+}$$
4		lediv23d.4	$$⊢\phi \to \frac{A}{B}\le C$$
5	2	rpregt0d	$$⊢\phi \to \left(B\in ℝ\wedge 0<B\right)$$
6	3	rpregt0d	$$⊢\phi \to \left(C\in ℝ\wedge 0<C\right)$$
7		lediv23	$$⊢\left(A\in ℝ\wedge \left(B\in ℝ\wedge 0<B\right)\wedge \left(C\in ℝ\wedge 0<C\right)\right)\to \left(\frac{A}{B}\le C\leftrightarrow \frac{A}{C}\le B\right)$$
8	1 5 6 7	syl3anc	$$⊢\phi \to \left(\frac{A}{B}\le C\leftrightarrow \frac{A}{C}\le B\right)$$
9	4 8	mpbid	$$⊢\phi \to \frac{A}{C}\le B$$