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Theorem modifeq2int 10052
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 964 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  A  e.  NN0 )
2 nn0z 8978 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  ZZ )
3 zq 9320 . . . . . . 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 272 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  e.  QQ )
7 nnq 9327 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  QQ )
873ad2ant2 986 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  QQ )
98adantr 272 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  B  e.  QQ )
101nn0ge0d 8937 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  0  <_  A )
1110adantr 272 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
0  <_  A )
12 simpr 109 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  <  B )
13 modqid 10015 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( 0  <_  A  /\  A  <  B
) )  ->  ( A  mod  B )  =  A )
146, 9, 11, 12, 13syl22anc 1200 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
( A  mod  B
)  =  A )
15 iftrue 3445 . . . . 5  |-  ( A  <  B  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  A )
1615eqcomd 2120 . . . 4  |-  ( A  <  B  ->  A  =  if ( A  < 
B ,  A , 
( A  -  B
) ) )
1716adantl 273 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
1814, 17eqtrd 2147 . 2  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
( A  mod  B
)  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
195adantr 272 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  A  e.  QQ )
208adantr 272 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  e.  QQ )
21 simp2 965 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  NN )
2221adantr 272 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  e.  NN )
2322nngt0d 8674 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  0  <  B
)
2421nnred 8643 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  RR )
251nn0red 8935 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  A  e.  RR )
2624, 25lenltd 7803 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( B  <_  A  <->  -.  A  <  B ) )
2726biimpar 293 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  <_  A
)
28 simpl3 969 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  A  <  (
2  x.  B ) )
29 q2submod 10051 . . . 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 1207 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  mod  B )  =  ( A  -  B ) )
31 iffalse 3448 . . . . 5  |-  ( -.  A  <  B  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  ( A  -  B ) )
3231adantl 273 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  ( A  -  B
) )
3332eqcomd 2120 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  -  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
3430, 33eqtrd 2147 . 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 9076 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  ZZ )
37 zdclt 9032 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  <  B )
38 exmiddc 804 . . . 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 406 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( A  <  B  \/  -.  A  <  B ) )
4118, 34, 40mpjaodan 770 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 103    \/ wo 680  DECID wdc 802    /\ w3a 945    = wceq 1314    e. wcel 1463   ifcif 3440   class class class wbr 3895  (class class class)co 5728   0cc0 7547    x. cmul 7552    < clt 7724    <_ cle 7725    - cmin 7856   NNcn 8630   2c2 8681   NN0cn0 8881   ZZcz 8958   QQcq 9313    mod cmo 9988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-n0 8882  df-z 8959  df-q 9314  df-rp 9344  df-fl 9936  df-mod 9989
This theorem is referenced by: (None)
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