MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lediv2a Unicode version

Theorem lediv2a 9695
Description: Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
Assertion
Ref Expression
lediv2a  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  -> 
( C  /  B
)  <_  ( C  /  A ) )

Proof of Theorem lediv2a
StepHypRef Expression
1 pm3.2 434 . . . . . . 7  |-  ( C  e.  RR  ->  ( C  e.  RR  ->  ( C  e.  RR  /\  C  e.  RR )
) )
21pm2.43i 43 . . . . . 6  |-  ( C  e.  RR  ->  ( C  e.  RR  /\  C  e.  RR ) )
32adantr 451 . . . . 5  |-  ( ( C  e.  RR  /\  0  <_  C )  -> 
( C  e.  RR  /\  C  e.  RR ) )
4 leid 8961 . . . . . . 7  |-  ( C  e.  RR  ->  C  <_  C )
54anim2i 552 . . . . . 6  |-  ( ( 0  <_  C  /\  C  e.  RR )  ->  ( 0  <_  C  /\  C  <_  C ) )
65ancoms 439 . . . . 5  |-  ( ( C  e.  RR  /\  0  <_  C )  -> 
( 0  <_  C  /\  C  <_  C ) )
73, 6jca 518 . . . 4  |-  ( ( C  e.  RR  /\  0  <_  C )  -> 
( ( C  e.  RR  /\  C  e.  RR )  /\  (
0  <_  C  /\  C  <_  C ) ) )
87ad2antlr 707 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  -> 
( ( C  e.  RR  /\  C  e.  RR )  /\  (
0  <_  C  /\  C  <_  C ) ) )
983adantl2 1112 . 2  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  -> 
( ( C  e.  RR  /\  C  e.  RR )  /\  (
0  <_  C  /\  C  <_  C ) ) )
10 id 19 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  e.  RR  /\  B  e.  RR ) )
1110ad2ant2r 727 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  e.  RR  /\  B  e.  RR ) )
1211adantr 451 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  A  <_  B )  -> 
( A  e.  RR  /\  B  e.  RR ) )
13 simplr 731 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  A )
1413anim1i 551 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  A  <_  B )  -> 
( 0  <  A  /\  A  <_  B ) )
1512, 14jca 518 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  A  <_  B )  -> 
( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <  A  /\  A  <_  B ) ) )
16153adantl3 1113 . 2  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  -> 
( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <  A  /\  A  <_  B ) ) )
17 lediv12a 9694 . 2  |-  ( ( ( ( C  e.  RR  /\  C  e.  RR )  /\  (
0  <_  C  /\  C  <_  C ) )  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <  A  /\  A  <_  B ) ) )  ->  ( C  /  B )  <_  ( C  /  A ) )
189, 16, 17syl2anc 642 1  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  -> 
( C  /  B
)  <_  ( C  /  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    e. wcel 1701   class class class wbr 4060  (class class class)co 5900   RRcr 8781   0cc0 8782    < clt 8912    <_ cle 8913    / cdiv 9468
This theorem is referenced by:  lediv2ad  10459  dchrisum0lem1b  20717  pntrmax  20766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469
  Copyright terms: Public domain W3C validator