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Theorem ltdiv1 9665
Description: Division of both sides of 'less than' by a positive number. (Contributed by NM, 10-Oct-2004.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
ltdiv1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  /  C )  <  ( B  /  C ) ) )

Proof of Theorem ltdiv1
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  RR )
2 simp2 956 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  RR )
3 simp3l 983 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  RR )
4 simp3r 984 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  C )
54gt0ne0d 9382 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  =/=  0 )
63, 5rereccld 9632 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  C
)  e.  RR )
7 recgt0 9645 . . . 4  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
0  <  ( 1  /  C ) )
873ad2ant3 978 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  ( 1  /  C ) )
9 ltmul1 9651 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  (
( 1  /  C
)  e.  RR  /\  0  <  ( 1  /  C ) ) )  ->  ( A  < 
B  <->  ( A  x.  ( 1  /  C
) )  <  ( B  x.  ( 1  /  C ) ) ) )
101, 2, 6, 8, 9syl112anc 1186 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  ( 1  /  C ) )  <  ( B  x.  ( 1  /  C
) ) ) )
111recnd 8906 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  CC )
123recnd 8906 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  CC )
1311, 12, 5divrecd 9584 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  /  C
)  =  ( A  x.  ( 1  /  C ) ) )
142recnd 8906 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  CC )
1514, 12, 5divrecd 9584 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  /  C
)  =  ( B  x.  ( 1  /  C ) ) )
1613, 15breq12d 4073 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  /  C )  <  ( B  /  C )  <->  ( A  x.  ( 1  /  C
) )  <  ( B  x.  ( 1  /  C ) ) ) )
1710, 16bitr4d 247 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  /  C )  <  ( B  /  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1701   class class class wbr 4060  (class class class)co 5900   RRcr 8781   0cc0 8782   1c1 8783    x. cmul 8787    < clt 8912    / cdiv 9468
This theorem is referenced by:  lediv1  9666  gt0div  9667  ltmuldiv  9671  ltdivmul  9673  ltdiv23  9692  ltdiv1i  9721  ltdiv1d  10478  quoremz  11006  quoremnn0ALT  11008  fldiv  11011  hashdvds  12890  dvcvx  19420  sinq12gt0  19928  tanord1  19952  atanlogsublem  20264  basellem4  20374  chtub  20504  bposlem7  20582  lgsquadlem1  20646  lgsquadlem2  20647  chebbnd1lem3  20673  cvmliftlem6  24105  cvmliftlem7  24106  cvmliftlem8  24107  cvmliftlem9  24108  cvmliftlem10  24109  nndivsub  25282  nn0prpwlem  25387  reglogltb  26124  hashgcdlem  26664  stoweidlem14  26911  stoweidlem26  26923
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469
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