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Theorem lediv2 10326
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 9908 . . . . 5  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
2 rereccl 10153 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  RR )
31, 2syldan 470 . . . 4  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( 1  /  B
)  e.  RR )
433ad2ant2 1010 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  B
)  e.  RR )
5 gt0ne0 9908 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
6 rereccl 10153 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )
75, 6syldan 470 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
873ad2ant1 1009 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  A
)  e.  RR )
9 simp3l 1016 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  RR )
10 simp3r 1017 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  C )
11 lemul2 10286 . . 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 1223 . 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 10318 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )
14133adant3 1008 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )
15 recn 9476 . . . . . . 7  |-  ( C  e.  RR  ->  C  e.  CC )
16 recn 9476 . . . . . . . . 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 10114 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( C  /  B )  =  ( C  x.  (
1  /  B ) ) )
20193expb 1189 . . . . . . 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 1007 . . . . 5  |-  ( ( C  e.  RR  /\  ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( C  /  B
)  =  ( C  x.  ( 1  /  B ) ) )
23 recn 9476 . . . . . . . . 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 10114 . . . . . . . 8  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  ( C  /  A )  =  ( C  x.  (
1  /  A ) ) )
27263expb 1189 . . . . . . 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 1008 . . . . 5  |-  ( ( C  e.  RR  /\  ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( C  /  A
)  =  ( C  x.  ( 1  /  A ) ) )
3022, 29breq12d 4406 . . . 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 1195 . . 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 1216 . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   1c1 9387    x. cmul 9391    < clt 9522    <_ cle 9523    / cdiv 10097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098
This theorem is referenced by:  lediv2d  11155  isprm6  13906  divdenle  13938  gexexlem  16447  znidomb  18112  aaliou2b  21933  log2tlbnd  22466  fsumharmonic  22531  bcmono  22742  dchrisum0lem1  22891  selberg3lem1  22932  pntrsumo1  22940  pntibndlem3  22967  nndivlub  28441  stoweidlem42  29978  stoweidlem51  29987  stoweidlem59  29995
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