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Theorem lediv2 8406
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 simp2l 970 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  RR )
2 simp2r 971 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  B )
31, 2gt0ap0d 8159 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B #  0 )
41, 3rerecclapd 8354 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  B
)  e.  RR )
5 simp1l 968 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  RR )
6 simp1r 969 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  A )
75, 6gt0ap0d 8159 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A #  0 )
85, 7rerecclapd 8354 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  A
)  e.  RR )
9 simp3l 972 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  RR )
10 simp3r 973 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  C )
11 lemul2 8372 . . 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 1179 . 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 8399 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )
14133adant3 964 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )
159recnd 7570 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  CC )
161recnd 7570 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  CC )
1715, 16, 3divrecapd 8314 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  /  B
)  =  ( C  x.  ( 1  /  B ) ) )
185recnd 7570 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  CC )
1915, 18, 7divrecapd 8314 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( C  /  A
)  =  ( C  x.  ( 1  /  A ) ) )
2017, 19breq12d 3864 . 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 ) ) ) )
2112, 14, 203bitr4d 219 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    /\ wa 103    <-> wb 104    /\ w3a 925    e. wcel 1439   class class class wbr 3851  (class class class)co 5666   RRcr 7403   0cc0 7404   1c1 7405    x. cmul 7409    < clt 7576    <_ cle 7577    / cdiv 8193
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 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7490  ax-resscn 7491  ax-1cn 7492  ax-1re 7493  ax-icn 7494  ax-addcl 7495  ax-addrcl 7496  ax-mulcl 7497  ax-mulrcl 7498  ax-addcom 7499  ax-mulcom 7500  ax-addass 7501  ax-mulass 7502  ax-distr 7503  ax-i2m1 7504  ax-0lt1 7505  ax-1rid 7506  ax-0id 7507  ax-rnegex 7508  ax-precex 7509  ax-cnre 7510  ax-pre-ltirr 7511  ax-pre-ltwlin 7512  ax-pre-lttrn 7513  ax-pre-apti 7514  ax-pre-ltadd 7515  ax-pre-mulgt0 7516  ax-pre-mulext 7517
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-id 4129  df-po 4132  df-iso 4133  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-iota 4993  df-fun 5030  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7578  df-mnf 7579  df-xr 7580  df-ltxr 7581  df-le 7582  df-sub 7709  df-neg 7710  df-reap 8106  df-ap 8113  df-div 8194
This theorem is referenced by:  lediv2d  9252  nnledivrp  9291  isprm6  11458  divdenle  11507
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