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Theorem modlt 11988
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 9571 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
2 rpcnne0 11238 . . . . 5  |-  ( B  e.  RR+  ->  ( B  e.  CC  /\  B  =/=  0 ) )
3 divcan2 10211 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  /  B ) )  =  A )
433expb 1195 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( B  x.  ( A  /  B
) )  =  A )
51, 2, 4syl2an 475 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( A  /  B ) )  =  A )
65oveq1d 6285 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( B  x.  ( A  /  B
) )  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
7 rpcn 11229 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  CC )
87adantl 464 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  CC )
9 rerpdivcl 11249 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
109recnd 9611 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  CC )
11 refldivcl 11939 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
1211recnd 9611 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
138, 10, 12subdid 10008 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  =  ( ( B  x.  ( A  /  B ) )  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
14 modval 11980 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
156, 13, 143eqtr4rd 2506 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) ) )
16 fraclt1 11920 . . . . 5  |-  ( ( A  /  B )  e.  RR  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  1 )
179, 16syl 16 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  1 )
18 divid 10230 . . . . . 6  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  /  B
)  =  1 )
192, 18syl 16 . . . . 5  |-  ( B  e.  RR+  ->  ( B  /  B )  =  1 )
2019adantl 464 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  /  B
)  =  1 )
2117, 20breqtrrd 4465 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) )
229, 11resubcld 9983 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  e.  RR )
23 rpre 11227 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  RR )
2423adantl 464 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
25 rpregt0 11234 . . . . 5  |-  ( B  e.  RR+  ->  ( B  e.  RR  /\  0  <  B ) )
2625adantl 464 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  e.  RR  /\  0  <  B ) )
27 ltmuldiv2 10412 . . . 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 ) ) )
2822, 24, 26, 27syl3anc 1226 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) )  <  B  <->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) ) )
2921, 28mpbird 232 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  <  B )
3015, 29eqbrtrd 4459 1  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  <  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    x. cmul 9486    < clt 9617    - cmin 9796    / cdiv 10202   RR+crp 11221   |_cfl 11908    mod cmo 11978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fl 11910  df-mod 11979
This theorem is referenced by:  zmodfz  11999  modid2  12005  modabs  12011  modaddmodup  12032  modsubdir  12037  digit1  12282  cshwidxmod  12765  repswcshw  12771  divalgmod  14148  bitsmod  14170  bitsinv1lem  14175  bezoutlem3  14262  eucalglt  14298  odzdvds  14406  fldivp1  14500  4sqlem6  14545  4sqlem12  14558  mndodcong  16765  oddvds  16770  gexdvds  16803  zringlpirlem3  18699  sineq0  23080  efif1olem2  23096  lgseisenlem1  23822  modelico  30996  irrapxlem1  30997  pellfund14  31073  jm2.19  31174  fourierswlem  32252  fouriersw  32253  sineq0ALT  34138
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