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Theorem modlt 11714
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 9368 . . . . 5  |-  ( A  e.  RR  ->  A  e.  CC )
2 rpcnne0 11004 . . . . 5  |-  ( B  e.  RR+  ->  ( B  e.  CC  /\  B  =/=  0 ) )
3 divcan2 9998 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  B  =/=  0 )  ->  ( B  x.  ( A  /  B ) )  =  A )
433expb 1183 . . . . 5  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  B  =/=  0 ) )  ->  ( B  x.  ( A  /  B
) )  =  A )
51, 2, 4syl2an 474 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  ( A  /  B ) )  =  A )
65oveq1d 6105 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( B  x.  ( A  /  B
) )  -  ( B  x.  ( |_ `  ( A  /  B
) ) ) )  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
7 rpcn 10995 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  CC )
87adantl 463 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  CC )
9 rerpdivcl 11014 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
109recnd 9408 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  CC )
11 reflcl 11642 . . . . . 6  |-  ( ( A  /  B )  e.  RR  ->  ( |_ `  ( A  /  B ) )  e.  RR )
129, 11syl 16 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  RR )
1312recnd 9408 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( |_ `  ( A  /  B ) )  e.  CC )
148, 10, 13subdid 9796 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  x.  (
( A  /  B
)  -  ( |_
`  ( A  /  B ) ) ) )  =  ( ( B  x.  ( A  /  B ) )  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
15 modval 11706 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( A  -  ( B  x.  ( |_ `  ( A  /  B ) ) ) ) )
166, 14, 153eqtr4rd 2484 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  mod  B
)  =  ( B  x.  ( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) ) ) )
17 fraclt1 11648 . . . . 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 10017 . . . . . 6  |-  ( ( B  e.  CC  /\  B  =/=  0 )  -> 
( B  /  B
)  =  1 )
202, 19syl 16 . . . . 5  |-  ( B  e.  RR+  ->  ( B  /  B )  =  1 )
2120adantl 463 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  /  B
)  =  1 )
2218, 21breqtrrd 4315 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  <  ( B  /  B ) )
239, 12resubcld 9772 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  -  ( |_ `  ( A  /  B ) ) )  e.  RR )
24 rpre 10993 . . . . 5  |-  ( B  e.  RR+  ->  B  e.  RR )
2524adantl 463 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  ->  B  e.  RR )
26 rpregt0 11000 . . . . 5  |-  ( B  e.  RR+  ->  ( B  e.  RR  /\  0  <  B ) )
2726adantl 463 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( B  e.  RR  /\  0  <  B ) )
28 ltmuldiv2 10199 . . . 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 1213 . . 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 4309 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 1364    e. wcel 1761    =/= wne 2604   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    x. cmul 9283    < clt 9414    - cmin 9591    / cdiv 9989   RR+crp 10987   |_cfl 11636    mod cmo 11704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-sup 7687  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-rp 10988  df-fl 11638  df-mod 11705
This theorem is referenced by:  zmodfz  11725  modid2  11731  modabs  11737  modaddmodup  11758  modsubdir  11763  digit1  11994  cshwidxmod  12436  repswcshw  12442  divalgmod  13606  bitsmod  13628  bitsinv1lem  13633  bezoutlem3  13720  eucalglt  13756  odzdvds  13863  fldivp1  13955  4sqlem6  14000  4sqlem12  14013  mndodcong  16038  oddvds  16043  gexdvds  16076  zringlpirlem3  17864  zlpirlem3  17869  sineq0  21942  efif1olem2  21958  lgseisenlem1  22647  modelico  29087  irrapxlem1  29088  pellfund14  29164  jm2.19  29267  sineq0ALT  31507
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