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Theorem ltmul1 7657
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 7656 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  /\  A  < 
B )  ->  ( A  x.  C )  <  ( B  x.  C
) )
21ex 112 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( A  <  B  ->  ( A  x.  C
)  <  ( B  x.  C ) ) )
3 recexgt0 7645 . . . 4  |-  ( ( C  e.  RR  /\  0  <  C )  ->  E. x  e.  RR  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) )
433ad2ant3 938 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  E. x  e.  RR  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) )
5 simpl1 918 . . . . . . . . . 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 970 . . . . . . . . . 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 7115 . . . . . . . . 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 919 . . . . . . . . . 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 7115 . . . . . . . . 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 491 . . . . . . . . . 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 499 . . . . . . . . . 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 294 . . . . . . . . 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 1095 . . . . . . . 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 7656 . . . . . . . 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 271 . . . . . . 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 7113 . . . . . . . . 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 265 . . . . . . . 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 7113 . . . . . . . . 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 265 . . . . . . . 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 7113 . . . . . . . . 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 265 . . . . . . . 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 7108 . . . . . . 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 7113 . . . . . . . . 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 265 . . . . . . . 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 7108 . . . . . . 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 3821 . . . . . 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 500 . . . . . . . 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 265 . . . . . . 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 5556 . . . . . 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 5556 . . . . . 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 3821 . . . . 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 7102 . . . . 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 7102 . . . . 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 3821 . . . 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 112 . . 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 2454 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  -> 
( ( A  x.  C )  <  ( B  x.  C )  ->  A  <  B ) )
372, 36impbid 124 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 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   E.wrex 2324   class class class wbr 3792  (class class class)co 5540   CCcc 6945   RRcr 6946   0cc0 6947   1c1 6948    x. cmul 6952    < clt 7119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltadd 7058  ax-pre-mulgt0 7059
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-ltxr 7124  df-sub 7247  df-neg 7248
This theorem is referenced by:  lemul1  7658  reapmul1lem  7659  ltmul2  7897  ltdiv1  7909  ltdiv23  7933  recp1lt1  7940  ltmul1i  7961  ltmul1d  8762  flodddiv4t2lthalf  10249
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