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Theorem ltmul1 9574
Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
ltmul1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )

Proof of Theorem ltmul1
StepHypRef Expression
1 ltmul1a 9573 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  A  < 
B )  ->  ( A  x.  C )  <  ( B  x.  C
) )
21ex 425 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  ->  ( A  x.  C
)  <  ( B  x.  C ) ) )
3 oveq1 5799 . . . . . 6  |-  ( A  =  B  ->  ( A  x.  C )  =  ( B  x.  C ) )
43a1i 12 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  =  B  ->  ( A  x.  C )  =  ( B  x.  C ) ) )
5 ltmul1a 9573 . . . . . . 7  |-  ( ( ( B  e.  RR  /\  A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  B  < 
A )  ->  ( B  x.  C )  <  ( A  x.  C
) )
65ex 425 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  ->  ( B  x.  C
)  <  ( A  x.  C ) ) )
763com12 1160 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  <  A  ->  ( B  x.  C
)  <  ( A  x.  C ) ) )
84, 7orim12d 814 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  =  B  \/  B  < 
A )  ->  (
( A  x.  C
)  =  ( B  x.  C )  \/  ( B  x.  C
)  <  ( A  x.  C ) ) ) )
98con3d 127 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( -.  ( ( A  x.  C )  =  ( B  x.  C )  \/  ( B  x.  C )  <  ( A  x.  C
) )  ->  -.  ( A  =  B  \/  B  <  A ) ) )
10 simp1 960 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  A  e.  RR )
11 simp3l 988 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  C  e.  RR )
1210, 11remulcld 8831 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  x.  C
)  e.  RR )
13 simp2 961 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  B  e.  RR )
1413, 11remulcld 8831 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( B  x.  C
)  e.  RR )
1512, 14lttrid 8925 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <  ( B  x.  C )  <->  -.  ( ( A  x.  C )  =  ( B  x.  C )  \/  ( B  x.  C )  <  ( A  x.  C )
) ) )
1610, 13lttrid 8925 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  -.  ( A  =  B  \/  B  <  A
) ) )
179, 15, 163imtr4d 261 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <  ( B  x.  C )  ->  A  <  B ) )
182, 17impbid 185 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   class class class wbr 3997  (class class class)co 5792   RRcr 8704   0cc0 8705    x. cmul 8710    < clt 8835
This theorem is referenced by:  ltmul2  9575  lemul1  9576  ltdiv1  9588  ltdiv23  9615  recp1lt1  9622  ltmul1i  9643  ltdivp1i  9651  ltmul1d  10394  expmulnbnd  11199  discr1  11203  mertenslem1  12302  qnumgt0  12783  4sqlem12  12965  pgpfaclem2  15279  mbfi1fseqlem4  19035  itg2monolem1  19067  dgrcolem2  19617  tangtx  19835  ftalem1  20272  basellem4  20283  lgsquadlem1  20555  lgsquadlem2  20556  pntpbnd1  20697  ostth2lem1  20729  nn0prpwlem  25605  pellexlem2  26282  stoweidlem11  27095  stoweidlem13  27097  stoweidlem26  27110  stoweidlem34  27118  stoweidlem59  27143
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-iota 6225  df-riota 6272  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-ltxr 8840  df-sub 9007  df-neg 9008
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