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Theorem ltmul1 8539
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 8538 . . 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 8527 . . . 4  |-  ( ( C  e.  RR  /\  0  <  C )  ->  E. x  e.  RR  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) )
433ad2ant3 1020 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( C  e.  RR  /\  0  <  C ) )  ->  E. x  e.  RR  ( 0  <  x  /\  ( C  x.  x
)  =  1 ) )
5 simpl1 1000 . . . . . . . . . 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 1052 . . . . . . . . . 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 7978 . . . . . . . . 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 1001 . . . . . . . . . 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 7978 . . . . . . . . 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 1177 . . . . . . . 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 8538 . . . . . . . 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 7976 . . . . . . . . 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 7976 . . . . . . . . 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 7976 . . . . . . . . 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 7971 . . . . . . 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 7976 . . . . . . . . 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 7971 . . . . . . 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 4031 . . . . . 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 5885 . . . . . 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 5885 . . . . . 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 4031 . . . . 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 7965 . . . . 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 7965 . . . . 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 4031 . . . 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 2599 . 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 978    = wceq 1353    e. wcel 2148   E.wrex 2456   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803    x. cmul 7807    < clt 7982
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltadd 7918  ax-pre-mulgt0 7919
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-ltxr 7987  df-sub 8120  df-neg 8121
This theorem is referenced by:  lemul1  8540  reapmul1lem  8541  ltmul2  8802  ltdiv1  8814  ltdiv23  8838  recp1lt1  8845  ltmul1i  8866  ltmul1d  9725  mertenslemi1  11527  flodddiv4t2lthalf  11925  qnumgt0  12181  tangtx  13926
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