# Metamath Proof Explorer

## Theorem ledivmul2

Description: 'Less than or equal to' relationship between division and multiplication. (Contributed by NM, 9-Dec-2005)

Ref Expression
Assertion ledivmul2 ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left(\frac{{A}}{{C}}\le {B}↔{A}\le {B}{C}\right)$

### Proof

Step Hyp Ref Expression
1 ledivmul ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left(\frac{{A}}{{C}}\le {B}↔{A}\le {C}{B}\right)$
2 recn ${⊢}{B}\in ℝ\to {B}\in ℂ$
3 recn ${⊢}{C}\in ℝ\to {C}\in ℂ$
4 mulcom ${⊢}\left({B}\in ℂ\wedge {C}\in ℂ\right)\to {B}{C}={C}{B}$
5 2 3 4 syl2an ${⊢}\left({B}\in ℝ\wedge {C}\in ℝ\right)\to {B}{C}={C}{B}$
6 5 adantrr ${⊢}\left({B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to {B}{C}={C}{B}$
7 6 3adant1 ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to {B}{C}={C}{B}$
8 7 breq2d ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left({A}\le {B}{C}↔{A}\le {C}{B}\right)$
9 1 8 bitr4d ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left(\frac{{A}}{{C}}\le {B}↔{A}\le {B}{C}\right)$