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Theorem modifeq2int 10404
Description: If a nonnegative integer is less than twice a positive integer, the nonnegative integer modulo the positive integer equals the nonnegative integer or the nonnegative integer minus the positive integer. (Contributed by Alexander van der Vekens, 21-May-2018.)
Assertion
Ref Expression
modifeq2int  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( A  mod  B )  =  if ( A  < 
B ,  A , 
( A  -  B
) ) )

Proof of Theorem modifeq2int
StepHypRef Expression
1 simp1 999 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  A  e.  NN0 )
2 nn0z 9291 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  ZZ )
3 zq 9644 . . . . . . 7  |-  ( A  e.  ZZ  ->  A  e.  QQ )
42, 3syl 14 . . . . . 6  |-  ( A  e.  NN0  ->  A  e.  QQ )
51, 4syl 14 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  A  e.  QQ )
65adantr 276 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  e.  QQ )
7 nnq 9651 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  QQ )
873ad2ant2 1021 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  QQ )
98adantr 276 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  B  e.  QQ )
101nn0ge0d 9250 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  0  <_  A )
1110adantr 276 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
0  <_  A )
12 simpr 110 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  <  B )
13 modqid 10367 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( 0  <_  A  /\  A  <  B
) )  ->  ( A  mod  B )  =  A )
146, 9, 11, 12, 13syl22anc 1250 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
( A  mod  B
)  =  A )
15 iftrue 3554 . . . . 5  |-  ( A  <  B  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  A )
1615eqcomd 2195 . . . 4  |-  ( A  <  B  ->  A  =  if ( A  < 
B ,  A , 
( A  -  B
) ) )
1716adantl 277 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
1814, 17eqtrd 2222 . 2  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
( A  mod  B
)  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
195adantr 276 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  A  e.  QQ )
208adantr 276 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  e.  QQ )
21 simp2 1000 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  NN )
2221adantr 276 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  e.  NN )
2322nngt0d 8981 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  0  <  B
)
2421nnred 8950 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  RR )
251nn0red 9248 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  A  e.  RR )
2624, 25lenltd 8093 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( B  <_  A  <->  -.  A  <  B ) )
2726biimpar 297 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  <_  A
)
28 simpl3 1004 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  A  <  (
2  x.  B ) )
29 q2submod 10403 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ  /\  0  <  B )  /\  ( B  <_  A  /\  A  <  ( 2  x.  B ) ) )  ->  ( A  mod  B )  =  ( A  -  B ) )
3019, 20, 23, 27, 28, 29syl32anc 1257 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  mod  B )  =  ( A  -  B ) )
31 iffalse 3557 . . . . 5  |-  ( -.  A  <  B  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  ( A  -  B ) )
3231adantl 277 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  ( A  -  B
) )
3332eqcomd 2195 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  -  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
3430, 33eqtrd 2222 . 2  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  mod  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
351, 2syl 14 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  A  e.  ZZ )
3621nnzd 9392 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  ZZ )
37 zdclt 9348 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  <  B )
38 exmiddc 837 . . . 4  |-  (DECID  A  < 
B  ->  ( A  <  B  \/  -.  A  <  B ) )
3937, 38syl 14 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  <  B  \/  -.  A  <  B
) )
4035, 36, 39syl2anc 411 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( A  <  B  \/  -.  A  <  B ) )
4118, 34, 40mpjaodan 799 1  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( A  mod  B )  =  if ( A  < 
B ,  A , 
( A  -  B
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    \/ wo 709  DECID wdc 835    /\ w3a 980    = wceq 1364    e. wcel 2160   ifcif 3549   class class class wbr 4018  (class class class)co 5891   0cc0 7829    x. cmul 7834    < clt 8010    <_ cle 8011    - cmin 8146   NNcn 8937   2c2 8988   NN0cn0 9194   ZZcz 9271   QQcq 9637    mod cmo 10340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1cn 7922  ax-1re 7923  ax-icn 7924  ax-addcl 7925  ax-addrcl 7926  ax-mulcl 7927  ax-mulrcl 7928  ax-addcom 7929  ax-mulcom 7930  ax-addass 7931  ax-mulass 7932  ax-distr 7933  ax-i2m1 7934  ax-0lt1 7935  ax-1rid 7936  ax-0id 7937  ax-rnegex 7938  ax-precex 7939  ax-cnre 7940  ax-pre-ltirr 7941  ax-pre-ltwlin 7942  ax-pre-lttrn 7943  ax-pre-apti 7944  ax-pre-ltadd 7945  ax-pre-mulgt0 7946  ax-pre-mulext 7947  ax-arch 7948
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-pnf 8012  df-mnf 8013  df-xr 8014  df-ltxr 8015  df-le 8016  df-sub 8148  df-neg 8149  df-reap 8550  df-ap 8557  df-div 8648  df-inn 8938  df-2 8996  df-n0 9195  df-z 9272  df-q 9638  df-rp 9672  df-fl 10288  df-mod 10341
This theorem is referenced by: (None)
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