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Theorem divelunit 24966
Description: A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.)
Assertion
Ref Expression
divelunit  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  e.  ( 0 [,] 1
)  <->  A  <_  B ) )

Proof of Theorem divelunit
StepHypRef Expression
1 0re 9026 . . . 4  |-  0  e.  RR
2 1re 9025 . . . 4  |-  1  e.  RR
31, 2elicc2i 10910 . . 3  |-  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
) )
4 df-3an 938 . . 3  |-  ( ( ( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B )  /\  ( A  /  B )  <_ 
1 )  <->  ( (
( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B ) )  /\  ( A  /  B
)  <_  1 ) )
53, 4bitri 241 . 2  |-  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  ( (
( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B ) )  /\  ( A  /  B
)  <_  1 ) )
6 ledivmul 9817 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <_  1  <->  A  <_  ( B  x.  1 ) ) )
72, 6mp3an2 1267 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  <_ 
1  <->  A  <_  ( B  x.  1 ) ) )
87adantlr 696 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  <_ 
1  <->  A  <_  ( B  x.  1 ) ) )
9 simpll 731 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  A  e.  RR )
10 simprl 733 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  e.  RR )
11 gt0ne0 9427 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
1211adantl 453 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  =/=  0 )
139, 10, 12redivcld 9776 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  /  B )  e.  RR )
14 divge0 9813 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
1513, 14jca 519 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
) ) )
1615biantrurd 495 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  <_ 
1  <->  ( ( ( A  /  B )  e.  RR  /\  0  <_  ( A  /  B
) )  /\  ( A  /  B )  <_ 
1 ) ) )
17 recn 9015 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
1817ad2antrl 709 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  e.  CC )
1918mulid1d 9040 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( B  x.  1 )  =  B )
2019breq2d 4167 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  <_  ( B  x.  1 )  <->  A  <_  B ) )
218, 16, 203bitr3d 275 . 2  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( (
( ( A  /  B )  e.  RR  /\  0  <_  ( A  /  B ) )  /\  ( A  /  B
)  <_  1 )  <-> 
A  <_  B )
)
225, 21syl5bb 249 1  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  e.  ( 0 [,] 1
)  <->  A  <_  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717    =/= wne 2552   class class class wbr 4155  (class class class)co 6022   CCcc 8923   RRcr 8924   0cc0 8925   1c1 8926    x. cmul 8930    < clt 9055    <_ cle 9056    / cdiv 9611   [,]cicc 10853
This theorem is referenced by:  brbtwn2  25560  axsegconlem7  25578  axcontlem2  25620  axcontlem4  25622  axcontlem7  25625  axcontlem8  25626
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-po 4446  df-so 4447  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-riota 6487  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-div 9612  df-icc 10857
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