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

Theorem modlt 11145
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 8974 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
2 rpcnne0 10522 . . . . 5  |-  ( B  e.  RR+  ->  ( B  e.  CC  /\  B  =/=  0 ) )
3 divcan2 9579 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  /  B ) )  =  A )
433expb 1153 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( B  x.  ( A  /  B
) )  =  A )
51, 2, 4syl2an 463 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( A  /  B ) )  =  A )
65oveq1d 5996 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( B  x.  ( A  /  B
) )  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
7 rpcn 10513 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  CC )
87adantl 452 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  CC )
9 rerpdivcl 10532 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
109recnd 9008 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  CC )
11 reflcl 11092 . . . . . 6  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  RR )
129, 11syl 15 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
1312recnd 9008 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
148, 10, 13subdid 9382 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  =  ( ( B  x.  ( A  /  B ) )  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
15 modval 11139 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
166, 14, 153eqtr4rd 2409 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) ) )
17 fraclt1 11098 . . . . 5  |-  ( ( A  /  B )  e.  RR  ->  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) )  <  1 )
189, 17syl 15 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  1 )
19 divid 9598 . . . . . 6  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  /  B
)  =  1 )
202, 19syl 15 . . . . 5  |-  ( B  e.  RR+  ->  ( B  /  B )  =  1 )
2120adantl 452 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  /  B
)  =  1 )
2218, 21breqtrrd 4151 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) )
239, 12resubcld 9358 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  e.  RR )
24 rpre 10511 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  RR )
2524adantl 452 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
26 rpregt0 10518 . . . . 5  |-  ( B  e.  RR+  ->  ( B  e.  RR  /\  0  <  B ) )
2726adantl 452 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  e.  RR  /\  0  <  B ) )
28 ltmuldiv2 9774 . . . 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 1183 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) )  <  B  <->  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) ) )
3022, 29mpbird 223 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  <  B )
3116, 30eqbrtrd 4145 1  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  <  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   class class class wbr 4125   ` cfv 5358  (class class class)co 5981   CCcc 8882   RRcr 8883   0cc0 8884   1c1 8885    x. cmul 8889    < clt 9014    - cmin 9184    / cdiv 9570   RR+crp 10505   |_cfl 11088    mod cmo 11137
This theorem is referenced by:  zmodfz  11155  modid2  11158  modabs  11161  modsubdir  11172  digit1  11400  divalgmod  12813  bitsmod  12835  bitsinv1lem  12840  bezoutlem3  12927  eucalglt  12963  odzdvds  13068  fldivp1  13153  4sqlem6  13198  4sqlem12  13211  mndodcong  15067  oddvds  15072  gexdvds  15105  zlpirlem3  16660  sineq0  20107  efif1olem2  20123  lgseisenlem1  20811  modelico  26497  irrapxlem1  26498  pellfund14  26574  jm2.19  26677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-pre-sup 8962
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-recs 6530  df-rdg 6565  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-sup 7341  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-div 9571  df-nn 9894  df-n0 10115  df-z 10176  df-uz 10382  df-rp 10506  df-fl 11089  df-mod 11138
  Copyright terms: Public domain W3C validator