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Theorem ltmul1 8272
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 8271 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  A  < 
B )  ->  ( A  x.  C )  <  ( B  x.  C
) )
21ex 114 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  ->  ( A  x.  C
)  <  ( B  x.  C ) ) )
3 recexgt0 8260 . . . 4  |-  ( ( C  e.  RR  /\  0  <  C )  ->  E. x  e.  RR  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) )
433ad2ant3 987 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  E. x  e.  RR  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) )
5 simpl1 967 . . . . . . . . . 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 1019 . . . . . . . . . 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 7720 . . . . . . . . 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 968 . . . . . . . . . 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 7720 . . . . . . . . 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 503 . . . . . . . . . 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 511 . . . . . . . . . 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 302 . . . . . . . . 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 1144 . . . . . . . 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 8271 . . . . . . . 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 279 . . . . . . 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 7718 . . . . . . . . 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 272 . . . . . . . 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 7718 . . . . . . . . 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 272 . . . . . . . 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 7718 . . . . . . . . 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 272 . . . . . . . 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 7713 . . . . . . 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 7718 . . . . . . . . 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 272 . . . . . . . 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 7713 . . . . . . 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 3924 . . . . . 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 512 . . . . . . . 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 272 . . . . . . 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 5744 . . . . . 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 5744 . . . . . 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 3924 . . . . 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 ) )
3217mulid1d 7707 . . . . 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 )
3324mulid1d 7707 . . . . 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 3924 . . . 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 114 . . 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 2528 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <  ( B  x.  C )  ->  A  <  B ) )
372, 36impbid 128 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 103    <-> wb 104    /\ w3a 945    = wceq 1314    e. wcel 1463   E.wrex 2391   class class class wbr 3895  (class class class)co 5728   CCcc 7545   RRcr 7546   0cc0 7547   1c1 7548    x. cmul 7552    < clt 7724
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltadd 7661  ax-pre-mulgt0 7662
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-iota 5046  df-fun 5083  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pnf 7726  df-mnf 7727  df-ltxr 7729  df-sub 7858  df-neg 7859
This theorem is referenced by:  lemul1  8273  reapmul1lem  8274  ltmul2  8524  ltdiv1  8536  ltdiv23  8560  recp1lt1  8567  ltmul1i  8588  ltmul1d  9424  mertenslemi1  11196  flodddiv4t2lthalf  11482  qnumgt0  11721
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