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Theorem divelunit 9672
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 7684 . . . 4  |-  0  e.  RR
2 1re 7683 . . . 4  |-  1  e.  RR
31, 2elicc2i 9609 . . 3  |-  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
)  /\  ( A  /  B )  <_  1
) )
4 df-3an 945 . . 3  |-  ( ( ( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B )  /\  ( A  /  B )  <_ 
1 )  <->  ( (
( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B ) )  /\  ( A  /  B
)  <_  1 ) )
53, 4bitri 183 . 2  |-  ( ( A  /  B )  e.  ( 0 [,] 1 )  <->  ( (
( A  /  B
)  e.  RR  /\  0  <_  ( A  /  B ) )  /\  ( A  /  B
)  <_  1 ) )
6 ledivmul 8539 . . . . 5  |-  ( ( A  e.  RR  /\  1  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <_  1  <->  A  <_  ( B  x.  1 ) ) )
72, 6mp3an2 1284 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  <_ 
1  <->  A  <_  ( B  x.  1 ) ) )
87adantlr 466 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  <_ 
1  <->  A  <_  ( B  x.  1 ) ) )
9 simpll 501 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  A  e.  RR )
10 simprl 503 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  e.  RR )
11 gt0ap0 8300 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B #  0 )
1211adantl 273 . . . . . 6  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B #  0
)
139, 10, 12redivclapd 8501 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  /  B )  e.  RR )
14 divge0 8535 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  0  <_  ( A  /  B ) )
1513, 14jca 302 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  e.  RR  /\  0  <_ 
( A  /  B
) ) )
1615biantrurd 301 . . 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 7671 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
1817ad2antrl 479 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  e.  CC )
1918mulid1d 7701 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( B  x.  1 )  =  B )
2019breq2d 3905 . . 3  |-  ( ( ( A  e.  RR  /\  0  <_  A )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  <_  ( B  x.  1 )  <->  A  <_  B ) )
218, 16, 203bitr3d 217 . 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 191 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 103    <-> wb 104    /\ w3a 943    e. wcel 1461   class class class wbr 3893  (class class class)co 5726   CCcc 7539   RRcr 7540   0cc0 7541   1c1 7542    x. cmul 7546    < clt 7718    <_ cle 7719   # cap 8255    / cdiv 8339   [,]cicc 9561
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657
This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-id 4173  df-po 4176  df-iso 4177  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340  df-icc 9565
This theorem is referenced by: (None)
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