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Theorem modifeq2int 9758
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 943 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  A  e.  NN0 )
2 nn0z 8740 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  ZZ )
3 zq 9080 . . . . . . 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 270 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  e.  QQ )
7 nnq 9087 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  QQ )
873ad2ant2 965 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  QQ )
98adantr 270 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  B  e.  QQ )
101nn0ge0d 8699 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  0  <_  A )
1110adantr 270 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
0  <_  A )
12 simpr 108 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  <  B )
13 modqid 9721 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( 0  <_  A  /\  A  <  B
) )  ->  ( A  mod  B )  =  A )
146, 9, 11, 12, 13syl22anc 1175 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
( A  mod  B
)  =  A )
15 iftrue 3394 . . . . 5  |-  ( A  <  B  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  A )
1615eqcomd 2093 . . . 4  |-  ( A  <  B  ->  A  =  if ( A  < 
B ,  A , 
( A  -  B
) ) )
1716adantl 271 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
1814, 17eqtrd 2120 . 2  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
( A  mod  B
)  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
195adantr 270 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  A  e.  QQ )
208adantr 270 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  e.  QQ )
21 simp2 944 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  NN )
2221adantr 270 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  e.  NN )
2322nngt0d 8437 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  0  <  B
)
2421nnred 8407 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  RR )
251nn0red 8697 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  A  e.  RR )
2624, 25lenltd 7580 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( B  <_  A  <->  -.  A  <  B ) )
2726biimpar 291 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  <_  A
)
28 simpl3 948 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  A  <  (
2  x.  B ) )
29 q2submod 9757 . . . 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 1182 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  mod  B )  =  ( A  -  B ) )
31 iffalse 3397 . . . . 5  |-  ( -.  A  <  B  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  ( A  -  B ) )
3231adantl 271 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  ( A  -  B
) )
3332eqcomd 2093 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  -  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
3430, 33eqtrd 2120 . 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 8837 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  ZZ )
37 zdclt 8794 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  <  B )
38 exmiddc 782 . . . 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 403 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( A  <  B  \/  -.  A  <  B ) )
4118, 34, 40mpjaodan 747 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 102    \/ wo 664  DECID wdc 780    /\ w3a 924    = wceq 1289    e. wcel 1438   ifcif 3389   class class class wbr 3837  (class class class)co 5634   0cc0 7329    x. cmul 7334    < clt 7501    <_ cle 7502    - cmin 7632   NNcn 8394   2c2 8444   NN0cn0 8643   ZZcz 8720   QQcq 9073    mod cmo 9694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-n0 8644  df-z 8721  df-q 9074  df-rp 9104  df-fl 9642  df-mod 9695
This theorem is referenced by: (None)
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