# Metamath Proof Explorer

## Theorem lediv1

Description: Division of both sides of a less than or equal to relation by a positive number. (Contributed by NM, 18-Nov-2004)

Ref Expression
Assertion lediv1 ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left({A}\le {B}↔\frac{{A}}{{C}}\le \frac{{B}}{{C}}\right)$

### Proof

Step Hyp Ref Expression
1 ltdiv1 ${⊢}\left({B}\in ℝ\wedge {A}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left({B}<{A}↔\frac{{B}}{{C}}<\frac{{A}}{{C}}\right)$
2 1 3com12 ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left({B}<{A}↔\frac{{B}}{{C}}<\frac{{A}}{{C}}\right)$
3 2 notbid ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left(¬{B}<{A}↔¬\frac{{B}}{{C}}<\frac{{A}}{{C}}\right)$
4 lenlt ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\right)\to \left({A}\le {B}↔¬{B}<{A}\right)$
5 4 3adant3 ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left({A}\le {B}↔¬{B}<{A}\right)$
6 gt0ne0 ${⊢}\left({C}\in ℝ\wedge 0<{C}\right)\to {C}\ne 0$
7 6 3adant1 ${⊢}\left({A}\in ℝ\wedge {C}\in ℝ\wedge 0<{C}\right)\to {C}\ne 0$
8 redivcl ${⊢}\left({A}\in ℝ\wedge {C}\in ℝ\wedge {C}\ne 0\right)\to \frac{{A}}{{C}}\in ℝ$
9 7 8 syld3an3 ${⊢}\left({A}\in ℝ\wedge {C}\in ℝ\wedge 0<{C}\right)\to \frac{{A}}{{C}}\in ℝ$
10 9 3expb ${⊢}\left({A}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \frac{{A}}{{C}}\in ℝ$
11 10 3adant2 ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \frac{{A}}{{C}}\in ℝ$
12 6 3adant1 ${⊢}\left({B}\in ℝ\wedge {C}\in ℝ\wedge 0<{C}\right)\to {C}\ne 0$
13 redivcl ${⊢}\left({B}\in ℝ\wedge {C}\in ℝ\wedge {C}\ne 0\right)\to \frac{{B}}{{C}}\in ℝ$
14 12 13 syld3an3 ${⊢}\left({B}\in ℝ\wedge {C}\in ℝ\wedge 0<{C}\right)\to \frac{{B}}{{C}}\in ℝ$
15 14 3expb ${⊢}\left({B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \frac{{B}}{{C}}\in ℝ$
16 15 3adant1 ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \frac{{B}}{{C}}\in ℝ$
17 11 16 lenltd ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left(\frac{{A}}{{C}}\le \frac{{B}}{{C}}↔¬\frac{{B}}{{C}}<\frac{{A}}{{C}}\right)$
18 3 5 17 3bitr4d ${⊢}\left({A}\in ℝ\wedge {B}\in ℝ\wedge \left({C}\in ℝ\wedge 0<{C}\right)\right)\to \left({A}\le {B}↔\frac{{A}}{{C}}\le \frac{{B}}{{C}}\right)$