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Theorem modlt 11970
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 9578 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
2 rpcnne0 11233 . . . . 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 1197 . . . . 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 6297 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( B  x.  ( A  /  B
) )  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
7 rpcn 11224 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  CC )
87adantl 466 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  CC )
9 rerpdivcl 11243 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
109recnd 9618 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  CC )
11 reflcl 11897 . . . . . 6  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  RR )
129, 11syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
1312recnd 9618 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
148, 10, 13subdid 10008 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  =  ( ( B  x.  ( A  /  B ) )  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
15 modval 11962 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
166, 14, 153eqtr4rd 2519 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) ) )
17 fraclt1 11903 . . . . 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 10230 . . . . . 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 4473 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) )
239, 12resubcld 9983 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  e.  RR )
24 rpre 11222 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  RR )
2524adantl 466 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
26 rpregt0 11229 . . . . 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 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 ) ) )
2923, 25, 27, 28syl3anc 1228 . . 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 4467 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 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    x. cmul 9493    < clt 9624    - cmin 9801    / cdiv 10202   RR+crp 11216   |_cfl 11891    mod cmo 11960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-rp 11217  df-fl 11893  df-mod 11961
This theorem is referenced by:  zmodfz  11981  modid2  11987  modabs  11993  modaddmodup  12014  modsubdir  12019  digit1  12264  cshwidxmod  12733  repswcshw  12739  divalgmod  13919  bitsmod  13941  bitsinv1lem  13946  bezoutlem3  14033  eucalglt  14069  odzdvds  14177  fldivp1  14271  4sqlem6  14316  4sqlem12  14329  mndodcong  16362  oddvds  16367  gexdvds  16400  zringlpirlem3  18278  zlpirlem3  18283  sineq0  22647  efif1olem2  22663  lgseisenlem1  23352  modelico  30361  irrapxlem1  30362  pellfund14  30438  jm2.19  30539  fourierswlem  31531  fouriersw  31532  sineq0ALT  32817
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