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Theorem ltmul2 9623
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 9622 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  <->  ( A  x.  C )  <  ( B  x.  C ) ) )
2 recn 8843 . . . 4  |-  ( C  e.  RR  ->  C  e.  CC )
3 recn 8843 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
4 mulcom 8839 . . . . . . 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 8843 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
8 mulcom 8839 . . . . . . 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 4052 . . . 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 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758    < clt 8883
This theorem is referenced by:  ltmul12a  9628  mulgt1  9631  ltmulgt11  9632  lt2msq1  9655  ltdiv2  9657  ltmul2i  9694  ltmul2d  10444  ef01bndlem  12480  cos01gt0  12487  sin4lt0  12491  iserodd  12904  pockthg  12969  prmreclem1  12979  prmreclem5  12983  blcvx  18320  dvcvx  19383  itgulm  19800  tangtx  19889  chtub  20467  bposlem1  20539  bposlem2  20540  bposlem7  20545  lgsdilem  20577  lgsquadlem1  20609  lgsquadlem2  20610  chebbnd1lem3  20636  chto1ub  20641  pntlemb  20762  irrapxlem1  27010  irrapxlem2  27011  irrapxlem5  27014  pellexlem2  27018  rmspecsqrnq  27094  stoweidlem11  27863  stoweidlem26  27878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056
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