Users' Mathboxes Mathbox for Paul Chapman < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lediv2aALT Unicode version

Theorem lediv2aALT 25078
Description: Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
Assertion
Ref Expression
lediv2aALT  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( A  <_  B  ->  ( C  /  B
)  <_  ( C  /  A ) ) )

Proof of Theorem lediv2aALT
StepHypRef Expression
1 gt0ne0 9457 . . . . . . . 8  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  =/=  0 )
2 rereccl 9696 . . . . . . . 8  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
( 1  /  B
)  e.  RR )
31, 2syldan 457 . . . . . . 7  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( 1  /  B
)  e.  RR )
4 gt0ne0 9457 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  =/=  0 )
5 rereccl 9696 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
( 1  /  A
)  e.  RR )
64, 5syldan 457 . . . . . . 7  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( 1  /  A
)  e.  RR )
73, 6anim12i 550 . . . . . 6  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( A  e.  RR  /\  0  < 
A ) )  -> 
( ( 1  /  B )  e.  RR  /\  ( 1  /  A
)  e.  RR ) )
87ancoms 440 . . . . 5  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( 1  /  B )  e.  RR  /\  ( 1  /  A
)  e.  RR ) )
983adant3 977 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( ( 1  /  B )  e.  RR  /\  ( 1  /  A
)  e.  RR ) )
10 simp3 959 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( C  e.  RR  /\  0  <_  C )
)
11 df-3an 938 . . . 4  |-  ( ( ( 1  /  B
)  e.  RR  /\  ( 1  /  A
)  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  <->  ( ( ( 1  /  B )  e.  RR  /\  (
1  /  A )  e.  RR )  /\  ( C  e.  RR  /\  0  <_  C )
) )
129, 10, 11sylanbrc 646 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( ( 1  /  B )  e.  RR  /\  ( 1  /  A
)  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
) )
13 lemul2a 9829 . . . 4  |-  ( ( ( ( 1  /  B )  e.  RR  /\  ( 1  /  A
)  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  /\  ( 1  /  B )  <_ 
( 1  /  A
) )  ->  ( C  x.  ( 1  /  B ) )  <_  ( C  x.  ( 1  /  A
) ) )
1413ex 424 . . 3  |-  ( ( ( 1  /  B
)  e.  RR  /\  ( 1  /  A
)  e.  RR  /\  ( C  e.  RR  /\  0  <_  C )
)  ->  ( (
1  /  B )  <_  ( 1  /  A )  ->  ( C  x.  ( 1  /  B ) )  <_  ( C  x.  ( 1  /  A
) ) ) )
1512, 14syl 16 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( ( 1  /  B )  <_  (
1  /  A )  ->  ( C  x.  ( 1  /  B
) )  <_  ( C  x.  ( 1  /  A ) ) ) )
16 lerec 9856 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )
17163adant3 977 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( A  <_  B  <->  ( 1  /  B )  <_  ( 1  /  A ) ) )
18 recn 9044 . . . . . . . . 9  |-  ( C  e.  RR  ->  C  e.  CC )
1918adantr 452 . . . . . . . 8  |-  ( ( C  e.  RR  /\  0  <_  C )  ->  C  e.  CC )
20 recn 9044 . . . . . . . . . 10  |-  ( B  e.  RR  ->  B  e.  CC )
2120adantr 452 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  0  <  B )  ->  B  e.  CC )
2221, 1jca 519 . . . . . . . 8  |-  ( ( B  e.  RR  /\  0  <  B )  -> 
( B  e.  CC  /\  B  =/=  0 ) )
2319, 22anim12i 550 . . . . . . 7  |-  ( ( ( C  e.  RR  /\  0  <_  C )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( C  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) ) )
24 3anass 940 . . . . . . 7  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  <->  ( C  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) ) )
2523, 24sylibr 204 . . . . . 6  |-  ( ( ( C  e.  RR  /\  0  <_  C )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( C  e.  CC  /\  B  e.  CC  /\  B  =/=  0 ) )
26 divrec 9658 . . . . . 6  |-  ( ( C  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( C  /  B )  =  ( C  x.  (
1  /  B ) ) )
2725, 26syl 16 . . . . 5  |-  ( ( ( C  e.  RR  /\  0  <_  C )  /\  ( B  e.  RR  /\  0  <  B ) )  ->  ( C  /  B )  =  ( C  x.  ( 1  /  B ) ) )
2827ancoms 440 . . . 4  |-  ( ( ( B  e.  RR  /\  0  <  B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( C  /  B
)  =  ( C  x.  ( 1  /  B ) ) )
29283adant1 975 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( C  /  B
)  =  ( C  x.  ( 1  /  B ) ) )
30 recn 9044 . . . . . . . . . 10  |-  ( A  e.  RR  ->  A  e.  CC )
3130adantr 452 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  0  <  A )  ->  A  e.  CC )
3231, 4jca 519 . . . . . . . 8  |-  ( ( A  e.  RR  /\  0  <  A )  -> 
( A  e.  CC  /\  A  =/=  0 ) )
3319, 32anim12i 550 . . . . . . 7  |-  ( ( ( C  e.  RR  /\  0  <_  C )  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( C  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) ) )
34 3anass 940 . . . . . . 7  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  <->  ( C  e.  CC  /\  ( A  e.  CC  /\  A  =/=  0 ) ) )
3533, 34sylibr 204 . . . . . 6  |-  ( ( ( C  e.  RR  /\  0  <_  C )  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( C  e.  CC  /\  A  e.  CC  /\  A  =/=  0 ) )
36 divrec 9658 . . . . . 6  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  A  =/=  0 )  ->  ( C  /  A )  =  ( C  x.  (
1  /  A ) ) )
3735, 36syl 16 . . . . 5  |-  ( ( ( C  e.  RR  /\  0  <_  C )  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( C  /  A )  =  ( C  x.  ( 1  /  A ) ) )
3837ancoms 440 . . . 4  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( C  /  A
)  =  ( C  x.  ( 1  /  A ) ) )
39383adant2 976 . . 3  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( C  /  A
)  =  ( C  x.  ( 1  /  A ) ) )
4029, 39breq12d 4193 . 2  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( ( C  /  B )  <_  ( C  /  A )  <->  ( C  x.  ( 1  /  B
) )  <_  ( C  x.  ( 1  /  A ) ) ) )
4115, 17, 403imtr4d 260 1  |-  ( ( ( A  e.  RR  /\  0  <  A )  /\  ( B  e.  RR  /\  0  < 
B )  /\  ( C  e.  RR  /\  0  <_  C ) )  -> 
( A  <_  B  ->  ( C  /  B
)  <_  ( C  /  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   class class class wbr 4180  (class class class)co 6048   CCcc 8952   RRcr 8953   0cc0 8954   1c1 8955    x. cmul 8959    < clt 9084    <_ cle 9085    / cdiv 9641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642
  Copyright terms: Public domain W3C validator