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Theorem ltrec 9682
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 8882 . . . . 5  |-  1  e.  RR
21a1i 10 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
1  e.  RR )
3 simprl 732 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  B  e.  RR )
4 simpll 730 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  A  e.  RR )
5 simplr 731 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  A )
6 ltmuldiv 9671 . . . 4  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( ( 1  x.  A )  <  B  <->  1  <  ( B  /  A ) ) )
72, 3, 4, 5, 6syl112anc 1186 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( 1  x.  A )  <  B  <->  1  <  ( B  /  A ) ) )
84recnd 8906 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  A  e.  CC )
98mulid2d 8898 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( 1  x.  A
)  =  A )
109breq1d 4070 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( 1  x.  A )  <  B  <->  A  <  B ) )
113recnd 8906 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  B  e.  CC )
125gt0ne0d 9382 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  ->  A  =/=  0 )
1311, 8, 12divrecd 9584 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( B  /  A
)  =  ( B  x.  ( 1  /  A ) ) )
1413breq2d 4072 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( 1  <  ( B  /  A )  <->  1  <  ( B  x.  ( 1  /  A ) ) ) )
157, 10, 143bitr3d 274 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <  B  <->  1  <  ( B  x.  ( 1  /  A
) ) ) )
164, 12rereccld 9632 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( 1  /  A
)  e.  RR )
17 simprr 733 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
0  <  B )
18 ltdivmul 9673 . . 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 1186 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( 1  /  B )  <  (
1  /  A )  <->  1  <  ( B  x.  ( 1  /  A ) ) ) )
2015, 19bitr4d 247 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 176    /\ wa 358    e. wcel 1701   class class class wbr 4060  (class class class)co 5900   RRcr 8781   0cc0 8782   1c1 8783    x. cmul 8787    < clt 8912    / cdiv 9468
This theorem is referenced by:  lerec  9683  ltdiv2  9686  ltrec1  9688  reclt1  9696  recgt1  9697  ltreci  9712  nnrecl  10010  ltrecd  10455  chebbnd1  20674
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
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