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Theorem ltdiv23 10456
Description: Swap denominator with other side of 'less than'. (Contributed by NM, 3-Oct-1999.)
Assertion
Ref Expression
ltdiv23  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <  C  <->  ( A  /  C )  <  B ) )

Proof of Theorem ltdiv23
StepHypRef Expression
1 simpl 457 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  RR )
2 gt0ne0 10038 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
31, 2jca 532 . . . . . 6  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  RR  /\  B  =/=  0 ) )
4 redivcl 10284 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A  /  B )  e.  RR )
543expb 1197 . . . . . 6  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( A  /  B )  e.  RR )
63, 5sylan2 474 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( A  /  B )  e.  RR )
763adant3 1016 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( A  /  B )  e.  RR )
8 simp3 998 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  C  e.  RR )
9 simp2 997 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( B  e.  RR  /\  0  < 
B ) )
10 ltmul1 10413 . . . 4  |-  ( ( ( A  /  B
)  e.  RR  /\  C  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( A  /  B )  <  C  <->  ( ( A  /  B
)  x.  B )  <  ( C  x.  B ) ) )
117, 8, 9, 10syl3anc 1228 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  C  e.  RR )  ->  ( ( A  /  B )  < 
C  <->  ( ( A  /  B )  x.  B )  <  ( C  x.  B )
) )
12113adant3r 1225 . 2  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <  C  <->  ( ( A  /  B
)  x.  B )  <  ( C  x.  B ) ) )
13 recn 9599 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
1413adantr 465 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  A  e.  CC )
15 recn 9599 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
1615ad2antrl 727 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  e.  CC )
172adantl 466 . . . . 5  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  B  =/=  0 )
1814, 16, 17divcan1d 10342 . . . 4  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( ( A  /  B )  x.  B )  =  A )
19183adant3 1016 . . 3  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  x.  B
)  =  A )
2019breq1d 4466 . 2  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( ( A  /  B )  x.  B )  <  ( C  x.  B )  <->  A  <  ( C  x.  B ) ) )
21 remulcl 9594 . . . . . . . 8  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  x.  B
)  e.  RR )
2221ancoms 453 . . . . . . 7  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  x.  B
)  e.  RR )
2322adantrr 716 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( C  x.  B )  e.  RR )
24233adant1 1014 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  x.  B
)  e.  RR )
25 ltdiv1 10427 . . . . 5  |-  ( ( A  e.  RR  /\  ( C  x.  B
)  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( A  <  ( C  x.  B
)  <->  ( A  /  C )  <  (
( C  x.  B
)  /  C ) ) )
2624, 25syld3an2 1275 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  ( ( C  x.  B )  /  C ) ) )
27 recn 9599 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
2827adantr 465 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  e.  CC )
29 gt0ne0 10038 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <  C )  ->  C  =/=  0 )
3028, 29jca 532 . . . . . . 7  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
( C  e.  CC  /\  C  =/=  0 ) )
31 divcan3 10252 . . . . . . . 8  |-  ( ( B  e.  CC  /\  C  e.  CC  /\  C  =/=  0 )  ->  (
( C  x.  B
)  /  C )  =  B )
32313expb 1197 . . . . . . 7  |-  ( ( B  e.  CC  /\  ( C  e.  CC  /\  C  =/=  0 ) )  ->  ( ( C  x.  B )  /  C )  =  B )
3315, 30, 32syl2an 477 . . . . . 6  |-  ( ( B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( ( C  x.  B )  /  C )  =  B )
34333adant1 1014 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  x.  B )  /  C
)  =  B )
3534breq2d 4468 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  (
( C  x.  B
)  /  C )  <-> 
( A  /  C
)  <  B )
)
3626, 35bitrd 253 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  B ) )
37363adant2r 1223 . 2  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( A  <  ( C  x.  B )  <->  ( A  /  C )  <  B ) )
3812, 20, 373bitrd 279 1  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  < 
C ) )  -> 
( ( A  /  B )  <  C  <->  ( A  /  C )  <  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   class class class wbr 4456  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509    x. cmul 9514    < clt 9645    / cdiv 10227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228
This theorem is referenced by:  ltdiv23i  10490  ltdiv23d  11337  divrcnv  13676  prmind2  14240  lebnumii  21592  bposlem2  23686  pntibndlem1  23900  stoweidlem7  31992
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