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

Theorem lediv2a 10338
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 pm3.2 447 . . . . . . 7  |-  ( C  e.  RR  ->  ( C  e.  RR  ->  ( C  e.  RR  /\  C  e.  RR )
) )
21pm2.43i 47 . . . . . 6  |-  ( C  e.  RR  ->  ( C  e.  RR  /\  C  e.  RR ) )
32adantr 465 . . . . 5  |-  ( ( C  e.  RR  /\  0  <_  C )  -> 
( C  e.  RR  /\  C  e.  RR ) )
4 leid 9582 . . . . . . 7  |-  ( C  e.  RR  ->  C  <_  C )
54anim2i 569 . . . . . 6  |-  ( ( 0  <_  C  /\  C  e.  RR )  ->  ( 0  <_  C  /\  C  <_  C ) )
65ancoms 453 . . . . 5  |-  ( ( C  e.  RR  /\  0  <_  C )  -> 
( 0  <_  C  /\  C  <_  C ) )
73, 6jca 532 . . . 4  |-  ( ( C  e.  RR  /\  0  <_  C )  -> 
( ( C  e.  RR  /\  C  e.  RR )  /\  (
0  <_  C  /\  C  <_  C ) ) )
87ad2antlr 726 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( C  e.  RR  /\  0  <_  C ) )  /\  A  <_  B )  -> 
( ( C  e.  RR  /\  C  e.  RR )  /\  (
0  <_  C  /\  C  <_  C ) ) )
983adantl2 1145 . 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 22 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  e.  RR  /\  B  e.  RR ) )
1110ad2ant2r 746 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  e.  RR  /\  B  e.  RR ) )
1211adantr 465 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  A  <_  B )  -> 
( A  e.  RR  /\  B  e.  RR ) )
13 simplr 754 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  A )
1413anim1i 568 . . . 4  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  A  <_  B )  -> 
( 0  <  A  /\  A  <_  B ) )
1512, 14jca 532 . . 3  |-  ( ( ( ( A  e.  RR  /\  0  < 
A )  /\  ( B  e.  RR  /\  0  <  B ) )  /\  A  <_  B )  -> 
( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <  A  /\  A  <_  B ) ) )
16153adantl3 1146 . 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 10337 . 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 661 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758   class class class wbr 4401  (class class class)co 6201   RRcr 9393   0cc0 9394    < clt 9530    <_ cle 9531    / cdiv 10105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106
This theorem is referenced by:  lediv2ad  11161  dchrisum0lem1b  22898  pntrmax  22947
  Copyright terms: Public domain W3C validator