Theorem divge1b

Description: The ratio of a real number to a positive real number is greater than or equal to 1 iff the divisor (the positive real number) is less than or equal to the dividend (the real number). (Contributed by AV, 26-May-2020)

	Ref	Expression
Assertion	divge1b	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\right)\to \left(A\le B\leftrightarrow 1\le \frac{B}{A}\right)$$

Proof

Step	Hyp	Ref	Expression
1		rpcn	$$⊢A\in {ℝ}^{+}\to A\in ℂ$$
2	1	mullidd	$$⊢A\in {ℝ}^{+}\to 1A=A$$
3	2	eqcomd	$$⊢A\in {ℝ}^{+}\to A=1A$$
4	3	adantr	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\right)\to A=1A$$
5	4	breq1d	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\right)\to \left(A\le B\leftrightarrow 1A\le B\right)$$
6		1red	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\right)\to 1\in ℝ$$
7		simpr	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\right)\to B\in ℝ$$
8		rpregt0	$$⊢A\in {ℝ}^{+}\to \left(A\in ℝ\wedge 0<A\right)$$
9	8	adantr	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\right)\to \left(A\in ℝ\wedge 0<A\right)$$
10		lemuldiv	$$⊢\left(1\in ℝ\wedge B\in ℝ\wedge \left(A\in ℝ\wedge 0<A\right)\right)\to \left(1A\le B\leftrightarrow 1\le \frac{B}{A}\right)$$
11	6 7 9 10	syl3anc	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\right)\to \left(1A\le B\leftrightarrow 1\le \frac{B}{A}\right)$$
12	5 11	bitrd	$$⊢\left(A\in {ℝ}^{+}\wedge B\in ℝ\right)\to \left(A\le B\leftrightarrow 1\le \frac{B}{A}\right)$$