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Theorem ltmul1 8578
Description: Multiplication of both sides of 'less than' by a positive number. Theorem I.19 of [Apostol] p. 20. Part of Definition 11.2.7(vi) of [HoTT], p. (varies). (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
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ltmul1a 8577 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  A  < 
B )  ->  ( A  x.  C )  <  ( B  x.  C
) )
21ex 115 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  ->  ( A  x.  C
)  <  ( B  x.  C ) ) )
3 recexgt0 8566 . . . 4  |-  ( ( C  e.  RR  /\  0  <  C )  ->  E. x  e.  RR  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) )
433ad2ant3 1022 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  E. x  e.  RR  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) )
5 simpl1 1002 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  A  e.  RR )
6 simpl3l 1054 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  C  e.  RR )
75, 6remulcld 8017 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  ( A  x.  C )  e.  RR )
8 simpl2 1003 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  B  e.  RR )
98, 6remulcld 8017 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  ( B  x.  C )  e.  RR )
10 simprl 529 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  x  e.  RR )
11 simprrl 539 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  0  <  x )
1210, 11jca 306 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  (
x  e.  RR  /\  0  <  x ) )
137, 9, 123jca 1179 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  (
( A  x.  C
)  e.  RR  /\  ( B  x.  C
)  e.  RR  /\  ( x  e.  RR  /\  0  <  x ) ) )
14 ltmul1a 8577 . . . . . . . 8  |-  ( ( ( ( A  x.  C )  e.  RR  /\  ( B  x.  C
)  e.  RR  /\  ( x  e.  RR  /\  0  <  x ) )  /\  ( A  x.  C )  < 
( B  x.  C
) )  ->  (
( A  x.  C
)  x.  x )  <  ( ( B  x.  C )  x.  x ) )
1513, 14sylan 283 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  (
( A  x.  C
)  x.  x )  <  ( ( B  x.  C )  x.  x ) )
165recnd 8015 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  A  e.  CC )
1716adantr 276 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  A  e.  CC )
186recnd 8015 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  C  e.  CC )
1918adantr 276 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  C  e.  CC )
2010recnd 8015 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  x  e.  CC )
2120adantr 276 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  x  e.  CC )
2217, 19, 21mulassd 8010 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  (
( A  x.  C
)  x.  x )  =  ( A  x.  ( C  x.  x
) ) )
238recnd 8015 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  B  e.  CC )
2423adantr 276 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  B  e.  CC )
2524, 19, 21mulassd 8010 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  (
( B  x.  C
)  x.  x )  =  ( B  x.  ( C  x.  x
) ) )
2615, 22, 253brtr3d 4049 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  ( A  x.  ( C  x.  x ) )  < 
( B  x.  ( C  x.  x )
) )
27 simprrr 540 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  ( C  x.  x )  =  1 )
2827adantr 276 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  ( C  x.  x )  =  1 )
2928oveq2d 5911 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  ( A  x.  ( C  x.  x ) )  =  ( A  x.  1 ) )
3028oveq2d 5911 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  ( B  x.  ( C  x.  x ) )  =  ( B  x.  1 ) )
3126, 29, 303brtr3d 4049 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  ( A  x.  1 )  <  ( B  x.  1 ) )
3217mulridd 8003 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  ( A  x.  1 )  =  A )
3324mulridd 8003 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  ( B  x.  1 )  =  B )
3431, 32, 333brtr3d 4049 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  /\  ( A  x.  C )  <  ( B  x.  C
) )  ->  A  <  B )
3534ex 115 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  ( x  e.  RR  /\  (
0  <  x  /\  ( C  x.  x
)  =  1 ) ) )  ->  (
( A  x.  C
)  <  ( B  x.  C )  ->  A  <  B ) )
364, 35rexlimddv 2612 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <  ( B  x.  C )  ->  A  <  B ) )
372, 36impbid 129 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:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   E.wrex 2469   class class class wbr 4018  (class class class)co 5895   CCcc 7838   RRcr 7839   0cc0 7840   1c1 7841    x. cmul 7845    < clt 8021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltadd 7956  ax-pre-mulgt0 7957
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-sub 8159  df-neg 8160
This theorem is referenced by:  lemul1  8579  reapmul1lem  8580  ltmul2  8842  ltdiv1  8854  ltdiv23  8878  recp1lt1  8885  ltmul1i  8906  ltmul1d  9767  mertenslemi1  11574  flodddiv4t2lthalf  11973  qnumgt0  12229  4sqlem12  12433  tangtx  14711
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