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Theorem ltmul2 9607
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.)
Assertion
Ref Expression
ltmul2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( C  x.  A )  <  ( C  x.  B ) ) )

Proof of Theorem ltmul2
StepHypRef Expression
1 ltmul1 9606 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
2 recn 8827 . . . 4  |-  ( C  e.  RR  ->  C  e.  CC )
3 recn 8827 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 mulcom 8823 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
53, 4sylan 457 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  CC )  ->  ( A  x.  C
)  =  ( C  x.  A ) )
653adant2 974 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  CC )  ->  ( A  x.  C )  =  ( C  x.  A ) )
7 recn 8827 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
8 mulcom 8823 . . . . . . 7  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
97, 8sylan 457 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  CC )  ->  ( B  x.  C
)  =  ( C  x.  B ) )
1093adant1 973 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  CC )  ->  ( B  x.  C )  =  ( C  x.  B ) )
116, 10breq12d 4036 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  CC )  ->  (
( A  x.  C
)  <  ( B  x.  C )  <->  ( C  x.  A )  <  ( C  x.  B )
) )
122, 11syl3an3 1217 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  x.  C
)  <  ( B  x.  C )  <->  ( C  x.  A )  <  ( C  x.  B )
) )
13123adant3r 1179 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <  ( B  x.  C )  <->  ( C  x.  A )  <  ( C  x.  B ) ) )
141, 13bitrd 244 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( C  x.  A )  <  ( C  x.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742    < clt 8867
This theorem is referenced by:  ltmul12a  9612  mulgt1  9615  ltmulgt11  9616  lt2msq1  9639  ltdiv2  9641  ltmul2i  9678  ltmul2d  10428  ef01bndlem  12464  cos01gt0  12471  sin4lt0  12475  iserodd  12888  pockthg  12953  prmreclem1  12963  prmreclem5  12967  blcvx  18304  dvcvx  19367  itgulm  19784  tangtx  19873  chtub  20451  bposlem1  20523  bposlem2  20524  bposlem7  20529  lgsdilem  20561  lgsquadlem1  20593  lgsquadlem2  20594  chebbnd1lem3  20620  chto1ub  20625  pntlemb  20746  irrapxlem1  26907  irrapxlem2  26908  irrapxlem5  26911  pellexlem2  26915  rmspecsqrnq  26991  stoweidlem11  27760  stoweidlem26  27775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040
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