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Theorem ltmul2 9609
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 9608 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
2 recn 8829 . . . 4  |-  ( C  e.  RR  ->  C  e.  CC )
3 recn 8829 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 mulcom 8825 . . . . . . 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 8829 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
8 mulcom 8825 . . . . . . 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 4038 . . . 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 1625    e. wcel 1686   class class class wbr 4025  (class class class)co 5860   CCcc 8737   RRcr 8738   0cc0 8739    x. cmul 8744    < clt 8869
This theorem is referenced by:  ltmul12a  9614  mulgt1  9617  ltmulgt11  9618  lt2msq1  9641  ltdiv2  9643  ltmul2i  9680  ltmul2d  10430  ef01bndlem  12466  cos01gt0  12473  sin4lt0  12477  iserodd  12890  pockthg  12955  prmreclem1  12965  prmreclem5  12969  blcvx  18306  dvcvx  19369  itgulm  19786  tangtx  19875  chtub  20453  bposlem1  20525  bposlem2  20526  bposlem7  20531  lgsdilem  20563  lgsquadlem1  20595  lgsquadlem2  20596  chebbnd1lem3  20622  chto1ub  20627  pntlemb  20748  irrapxlem1  26918  irrapxlem2  26919  irrapxlem5  26922  pellexlem2  26926  rmspecsqrnq  27002  stoweidlem11  27771  stoweidlem26  27786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-ltxr 8874  df-sub 9041  df-neg 9042
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