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

Theorem ltrec 9922
Description: The reciprocal of both sides of 'less than'. (Contributed by NM, 26-Sep-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
ltrec  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <  B  <->  ( 1  /  B )  <  ( 1  /  A ) ) )

Proof of Theorem ltrec
StepHypRef Expression
1 1re 9121 . . . . 5  |-  1  e.  RR
21a1i 11 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
1  e.  RR )
3 simprl 734 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  B  e.  RR )
4 simpll 732 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  A  e.  RR )
5 simplr 733 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  A )
6 ltmuldiv 9911 . . . 4  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( 1  x.  A )  <  B  <->  1  <  ( B  /  A ) ) )
72, 3, 4, 5, 6syl112anc 1189 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( 1  x.  A )  <  B  <->  1  <  ( B  /  A ) ) )
84recnd 9145 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  A  e.  CC )
98mulid2d 9137 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( 1  x.  A
)  =  A )
109breq1d 4247 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( 1  x.  A )  <  B  <->  A  <  B ) )
113recnd 9145 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  B  e.  CC )
125gt0ne0d 9622 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  A  =/=  0 )
1311, 8, 12divrecd 9824 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( B  /  A
)  =  ( B  x.  ( 1  /  A ) ) )
1413breq2d 4249 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( 1  <  ( B  /  A )  <->  1  <  ( B  x.  ( 1  /  A ) ) ) )
157, 10, 143bitr3d 276 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <  B  <->  1  <  ( B  x.  ( 1  /  A
) ) ) )
164, 12rereccld 9872 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( 1  /  A
)  e.  RR )
17 simprr 735 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  B )
18 ltdivmul 9913 . . 3  |-  ( ( 1  e.  RR  /\  ( 1  /  A
)  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( (
1  /  B )  <  ( 1  /  A )  <->  1  <  ( B  x.  ( 1  /  A ) ) ) )
192, 16, 3, 17, 18syl112anc 1189 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( 1  /  B )  <  (
1  /  A )  <->  1  <  ( B  x.  ( 1  /  A ) ) ) )
2015, 19bitr4d 249 1  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <  B  <->  ( 1  /  B )  <  ( 1  /  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1727   class class class wbr 4237  (class class class)co 6110   RRcr 9020   0cc0 9021   1c1 9022    x. cmul 9026    < clt 9151    / cdiv 9708
This theorem is referenced by:  lerec  9923  ltdiv2  9926  ltrec1  9928  reclt1  9936  recgt1  9937  ltreci  9952  nnrecl  10250  ltrecd  10697  chebbnd1  21197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-riota 6578  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709
  Copyright terms: Public domain W3C validator