MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  modlt Structured version   Unicode version

Theorem modlt 11830
Description: The modulo operation is less than its second argument. (Contributed by NM, 10-Nov-2008.)
Assertion
Ref Expression
modlt  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  <  B )

Proof of Theorem modlt
StepHypRef Expression
1 recn 9478 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
2 rpcnne0 11114 . . . . 5  |-  ( B  e.  RR+  ->  ( B  e.  CC  /\  B  =/=  0 ) )
3 divcan2 10108 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  /  B ) )  =  A )
433expb 1189 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( B  x.  ( A  /  B
) )  =  A )
51, 2, 4syl2an 477 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( A  /  B ) )  =  A )
65oveq1d 6210 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( B  x.  ( A  /  B
) )  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
7 rpcn 11105 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  CC )
87adantl 466 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  CC )
9 rerpdivcl 11124 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
109recnd 9518 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  CC )
11 reflcl 11758 . . . . . 6  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  RR )
129, 11syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
1312recnd 9518 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
148, 10, 13subdid 9906 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  =  ( ( B  x.  ( A  /  B ) )  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
15 modval 11822 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
166, 14, 153eqtr4rd 2504 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) ) )
17 fraclt1 11764 . . . . 5  |-  ( ( A  /  B )  e.  RR  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  1 )
189, 17syl 16 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  1 )
19 divid 10127 . . . . . 6  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  /  B
)  =  1 )
202, 19syl 16 . . . . 5  |-  ( B  e.  RR+  ->  ( B  /  B )  =  1 )
2120adantl 466 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  /  B
)  =  1 )
2218, 21breqtrrd 4421 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) )
239, 12resubcld 9882 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  e.  RR )
24 rpre 11103 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  RR )
2524adantl 466 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
26 rpregt0 11110 . . . . 5  |-  ( B  e.  RR+  ->  ( B  e.  RR  /\  0  <  B ) )
2726adantl 466 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  e.  RR  /\  0  <  B ) )
28 ltmuldiv2 10309 . . . 4  |-  ( ( ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  e.  RR  /\  B  e.  RR  /\  ( B  e.  RR  /\  0  <  B ) )  -> 
( ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) )  <  B  <->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) ) )
2923, 25, 27, 28syl3anc 1219 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) )  <  B  <->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) ) )
3022, 29mpbird 232 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  <  B )
3116, 30eqbrtrd 4415 1  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  <  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   CCcc 9386   RRcr 9387   0cc0 9388   1c1 9389    x. cmul 9393    < clt 9524    - cmin 9701    / cdiv 10099   RR+crp 11097   |_cfl 11752    mod cmo 11820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fl 11754  df-mod 11821
This theorem is referenced by:  zmodfz  11841  modid2  11847  modabs  11853  modaddmodup  11874  modsubdir  11879  digit1  12110  cshwidxmod  12553  repswcshw  12559  divalgmod  13723  bitsmod  13745  bitsinv1lem  13750  bezoutlem3  13837  eucalglt  13873  odzdvds  13980  fldivp1  14072  4sqlem6  14117  4sqlem12  14130  mndodcong  16161  oddvds  16166  gexdvds  16199  zringlpirlem3  18025  zlpirlem3  18030  sineq0  22111  efif1olem2  22127  lgseisenlem1  22816  modelico  29305  irrapxlem1  29306  pellfund14  29382  jm2.19  29485  sineq0ALT  31986
  Copyright terms: Public domain W3C validator