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Theorem divelunit 9089
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 7170 . . . 4  |-  0  e.  RR
2 1re 7169 . . . 4  |-  1  e.  RR
31, 2elicc2i 9027 . . 3  |-  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
) )
4 df-3an 922 . . 3  |-  ( ( ( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B )  /\  ( A  /  B )  <_ 
1 )  <->  ( (
( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B ) )  /\  ( A  /  B
)  <_  1 ) )
53, 4bitri 182 . 2  |-  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  ( (
( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B ) )  /\  ( A  /  B
)  <_  1 ) )
6 ledivmul 8011 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <_  1  <->  A  <_  ( B  x.  1 ) ) )
72, 6mp3an2 1257 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  <_ 
1  <->  A  <_  ( B  x.  1 ) ) )
87adantlr 461 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  <_ 
1  <->  A  <_  ( B  x.  1 ) ) )
9 simpll 496 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  A  e.  RR )
10 simprl 498 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  e.  RR )
11 gt0ap0 7781 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B #  0 )
1211adantl 271 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B #  0
)
139, 10, 12redivclapd 7976 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  /  B )  e.  RR )
14 divge0 8007 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
1513, 14jca 300 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
) ) )
1615biantrurd 299 . . 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 7157 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
1817ad2antrl 474 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  e.  CC )
1918mulid1d 7187 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( B  x.  1 )  =  B )
2019breq2d 3799 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  <_  ( B  x.  1 )  <->  A  <_  B ) )
218, 16, 203bitr3d 216 . 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 190 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    /\ wa 102    <-> wb 103    /\ w3a 920    e. wcel 1434   class class class wbr 3787  (class class class)co 5537   CCcc 7030   RRcr 7031   0cc0 7032   1c1 7033    x. cmul 7037    < clt 7204    <_ cle 7205   # cap 7737    / cdiv 7816   [,]cicc 8979
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144  ax-pre-mulext 7145
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-po 4053  df-iso 4054  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-reap 7731  df-ap 7738  df-div 7817  df-icc 8983
This theorem is referenced by: (None)
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