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Theorem lerec 9728
Description: The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 3-Oct-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
lerec  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )

Proof of Theorem lerec
StepHypRef Expression
1 ltrec 9727 . . . 4  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( A  e.  RR  /\  0  < 
A ) )  -> 
( B  <  A  <->  ( 1  /  A )  <  ( 1  /  B ) ) )
21ancoms 439 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( B  <  A  <->  ( 1  /  A )  <  ( 1  /  B ) ) )
32notbid 285 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( -.  B  < 
A  <->  -.  ( 1  /  A )  < 
( 1  /  B
) ) )
4 simpll 730 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  A  e.  RR )
5 simprl 732 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  B  e.  RR )
64, 5lenltd 9055 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <_  B  <->  -.  B  <  A ) )
7 simprr 733 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  B )
87gt0ne0d 9427 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  B  =/=  0 )
95, 8rereccld 9677 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( 1  /  B
)  e.  RR )
10 simplr 731 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  A )
1110gt0ne0d 9427 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  A  =/=  0 )
124, 11rereccld 9677 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( 1  /  A
)  e.  RR )
139, 12lenltd 9055 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( 1  /  B )  <_  (
1  /  A )  <->  -.  ( 1  /  A
)  <  ( 1  /  B ) ) )
143, 6, 133bitr4d 276 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:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    e. wcel 1710   class class class wbr 4104  (class class class)co 5945   RRcr 8826   0cc0 8827   1c1 8828    < clt 8957    <_ cle 8958    / cdiv 9513
This theorem is referenced by:  lerec2  9734  ledivdiv  9735  lediv2  9736  lediv12a  9739  lereci  9758  lerecd  10501  resqrex  11832  harmonic  12414  gzrngunit  16543  evth  18561  dyadss  19053  vitali  19072  itg2gt0  19219  logtayllem  20117  leibpi  20349  harmonicbnd4  20416  basellem6  20435  dchrmusumlema  20754  minvecolem3  21569  subfacval3  24124  lediv2aALT  24417  irrapxlem4  26233  irrapxlem5  26234  pellexlem2  26238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514
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