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Theorem lediv2 9736
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 9329 . . . . 5  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
2 rereccl 9568 . . . . 5  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  RR )
31, 2syldan 456 . . . 4  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( 1  /  B
)  e.  RR )
433ad2ant2 977 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  B
)  e.  RR )
5 gt0ne0 9329 . . . . 5  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
6 rereccl 9568 . . . . 5  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )
75, 6syldan 456 . . . 4  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
873ad2ant1 976 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  A
)  e.  RR )
9 simp3l 983 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  RR )
10 simp3r 984 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  C )
11 lemul2 9699 . . 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 1186 . 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 9728 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )
14133adant3 975 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )
15 recn 8917 . . . . . . 7  |-  ( C  e.  RR  ->  C  e.  CC )
16 recn 8917 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  CC )
1716adantr 451 . . . . . . . 8  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  CC )
1817, 1jca 518 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
19 divrec 9530 . . . . . . . 8  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( C  /  B )  =  ( C  x.  (
1  /  B ) ) )
20193expb 1152 . . . . . . 7  |-  ( ( C  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( C  /  B )  =  ( C  x.  ( 1  /  B ) ) )
2115, 18, 20syl2an 463 . . . . . 6  |-  ( ( C  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( C  /  B )  =  ( C  x.  ( 1  /  B ) ) )
22213adant2 974 . . . . 5  |-  ( ( C  e.  RR  /\  ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( C  /  B
)  =  ( C  x.  ( 1  /  B ) ) )
23 recn 8917 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
2423adantr 451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
2524, 5jca 518 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
26 divrec 9530 . . . . . . . 8  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  ( C  /  A )  =  ( C  x.  (
1  /  A ) ) )
27263expb 1152 . . . . . . 7  |-  ( ( C  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) )  ->  ( C  /  A )  =  ( C  x.  ( 1  /  A ) ) )
2815, 25, 27syl2an 463 . . . . . 6  |-  ( ( C  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( C  /  A )  =  ( C  x.  ( 1  /  A ) ) )
29283adant3 975 . . . . 5  |-  ( ( C  e.  RR  /\  ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( C  /  A
)  =  ( C  x.  ( 1  /  A ) ) )
3022, 29breq12d 4117 . . . 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 1158 . . 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 1179 . 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 276 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   class class class wbr 4104  (class class class)co 5945   CCcc 8825   RRcr 8826   0cc0 8827   1c1 8828    x. cmul 8832    < clt 8957    <_ cle 8958    / cdiv 9513
This theorem is referenced by:  lediv2d  10506  isprm6  12885  divdenle  12917  gexexlem  15243  znidomb  16621  aaliou2b  19825  log2tlbnd  20352  fsumharmonic  20417  bcmono  20628  dchrisum0lem1  20777  selberg3lem1  20818  pntrsumo1  20826  pntibndlem3  20853  nndivlub  25456  stoweidlem42  27114  stoweidlem51  27123  stoweidlem59  27131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514
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