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Theorem ltdiv1 10402
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 996 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  RR )
2 simp2 997 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  RR )
3 simp3l 1024 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  RR )
4 simp3r 1025 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  C )
54gt0ne0d 10113 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  =/=  0 )
63, 5rereccld 10367 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( 1  /  C
)  e.  RR )
7 recgt0 10382 . . . 4  |-  ( ( C  e.  RR  /\  0  <  C )  -> 
0  <  ( 1  /  C ) )
873ad2ant3 1019 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
0  <  ( 1  /  C ) )
9 ltmul1 10388 . . 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 1232 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  ( 1  /  C ) )  <  ( B  x.  ( 1  /  C
) ) ) )
111recnd 9618 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  CC )
123recnd 9618 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  CC )
1311, 12, 5divrecd 10319 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  /  C
)  =  ( A  x.  ( 1  /  C ) ) )
142recnd 9618 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  CC )
1514, 12, 5divrecd 10319 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  /  C
)  =  ( B  x.  ( 1  /  C ) ) )
1613, 15breq12d 4460 . 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 256 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  /  C )  <  ( B  /  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767   class class class wbr 4447  (class class class)co 6282   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493    < clt 9624    / cdiv 10202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203
This theorem is referenced by:  lediv1  10403  gt0div  10404  ltmuldiv  10411  ltdivmul  10413  ltdiv23  10432  ltdiv1i  10461  ltdiv1d  11293  flltdivnn0lt  11929  quoremz  11946  quoremnn0ALT  11948  fldiv  11951  hashdvds  14160  dvcvx  22156  sinq12gt0  22633  tanord1  22657  atanlogsublem  22974  basellem4  23085  chtub  23215  bposlem7  23293  lgsquadlem1  23357  lgsquadlem2  23358  chebbnd1lem3  23384  cvmliftlem6  28375  cvmliftlem7  28376  cvmliftlem8  28377  cvmliftlem9  28378  cvmliftlem10  28379  nndivsub  29499  tan2h  29624  dvtanlem  29641  nn0prpwlem  29717  reglogltb  30431  hashgcdlem  30762  stoweidlem14  31314  stoweidlem26  31326
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