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Theorem lediv2 10455
Description: Division of a positive number by both sides of 'less than or equal to'. (Contributed by NM, 10-Jan-2006.)
Assertion
Ref Expression
lediv2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( C  /  B )  <_  ( C  /  A ) ) )

Proof of Theorem lediv2
StepHypRef Expression
1 gt0ne0 10038 . . . . 5  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
2 rereccl 10283 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  RR )
31, 2syldan 470 . . . 4  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( 1  /  B
)  e.  RR )
433ad2ant2 1018 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  B
)  e.  RR )
5 gt0ne0 10038 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
6 rereccl 10283 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )
75, 6syldan 470 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
873ad2ant1 1017 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  A
)  e.  RR )
9 simp3l 1024 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  RR )
10 simp3r 1025 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  C )
11 lemul2 10416 . . 3  |-  ( ( ( 1  /  B
)  e.  RR  /\  ( 1  /  A
)  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  ( (
1  /  B )  <_  ( 1  /  A )  <->  ( C  x.  ( 1  /  B
) )  <_  ( C  x.  ( 1  /  A ) ) ) )
124, 8, 9, 10, 11syl112anc 1232 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( 1  /  B )  <_  (
1  /  A )  <-> 
( C  x.  (
1  /  B ) )  <_  ( C  x.  ( 1  /  A
) ) ) )
13 lerec 10447 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )
14133adant3 1016 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )
15 recn 9599 . . . . . . 7  |-  ( C  e.  RR  ->  C  e.  CC )
16 recn 9599 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  CC )
1716adantr 465 . . . . . . . 8  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  CC )
1817, 1jca 532 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
19 divrec 10244 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( C  /  B )  =  ( C  x.  (
1  /  B ) ) )
20193expb 1197 . . . . . . 7  |-  ( ( C  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( C  /  B )  =  ( C  x.  ( 1  /  B ) ) )
2115, 18, 20syl2an 477 . . . . . 6  |-  ( ( C  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( C  /  B )  =  ( C  x.  ( 1  /  B ) ) )
22213adant2 1015 . . . . 5  |-  ( ( C  e.  RR  /\  ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( C  /  B
)  =  ( C  x.  ( 1  /  B ) ) )
23 recn 9599 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
2423adantr 465 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
2524, 5jca 532 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
26 divrec 10244 . . . . . . . 8  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  ( C  /  A )  =  ( C  x.  (
1  /  A ) ) )
27263expb 1197 . . . . . . 7  |-  ( ( C  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  ->  ( C  /  A )  =  ( C  x.  ( 1  /  A ) ) )
2815, 25, 27syl2an 477 . . . . . 6  |-  ( ( C  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( C  /  A )  =  ( C  x.  ( 1  /  A ) ) )
29283adant3 1016 . . . . 5  |-  ( ( C  e.  RR  /\  ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( C  /  A
)  =  ( C  x.  ( 1  /  A ) ) )
3022, 29breq12d 4469 . . . 4  |-  ( ( C  e.  RR  /\  ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( C  /  B )  <_  ( C  /  A )  <->  ( C  x.  ( 1  /  B
) )  <_  ( C  x.  ( 1  /  A ) ) ) )
31303coml 1203 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  C  e.  RR )  ->  (
( C  /  B
)  <_  ( C  /  A )  <->  ( C  x.  ( 1  /  B
) )  <_  ( C  x.  ( 1  /  A ) ) ) )
32313adant3r 1225 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( C  /  B )  <_  ( C  /  A )  <->  ( C  x.  ( 1  /  B
) )  <_  ( C  x.  ( 1  /  A ) ) ) )
3312, 14, 323bitr4d 285 1  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( C  /  B )  <_  ( C  /  A ) ) )
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   1c1 9510    x. cmul 9514    < clt 9645    <_ cle 9646    / 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:  lediv2d  11305  isprm6  14262  divdenle  14294  gexexlem  16985  znidomb  18727  aaliou2b  22863  log2tlbnd  23402  fsumharmonic  23467  bcmono  23678  dchrisum0lem1  23827  selberg3lem1  23868  pntrsumo1  23876  pntibndlem3  23903  nndivlub  30128  stoweidlem42  32027  stoweidlem51  32036  stoweidlem59  32044
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