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Theorem modifeq2int 10127
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 966 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  A  e.  NN0 )
2 nn0z 9042 . . . . . . 7  |-  ( A  e.  NN0  ->  A  e.  ZZ )
3 zq 9386 . . . . . . 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 274 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  e.  QQ )
7 nnq 9393 . . . . . 6  |-  ( B  e.  NN  ->  B  e.  QQ )
873ad2ant2 988 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  QQ )
98adantr 274 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  B  e.  QQ )
101nn0ge0d 9001 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  0  <_  A )
1110adantr 274 . . . 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 10090 . . . 4  |-  ( ( ( A  e.  QQ  /\  B  e.  QQ )  /\  ( 0  <_  A  /\  A  <  B
) )  ->  ( A  mod  B )  =  A )
146, 9, 11, 12, 13syl22anc 1202 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
( A  mod  B
)  =  A )
15 iftrue 3449 . . . . 5  |-  ( A  <  B  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  A )
1615eqcomd 2123 . . . 4  |-  ( A  <  B  ->  A  =  if ( A  < 
B ,  A , 
( A  -  B
) ) )
1716adantl 275 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  ->  A  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
1814, 17eqtrd 2150 . 2  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  A  <  B )  -> 
( A  mod  B
)  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
195adantr 274 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  A  e.  QQ )
208adantr 274 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  e.  QQ )
21 simp2 967 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  NN )
2221adantr 274 . . . . 5  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  e.  NN )
2322nngt0d 8732 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  0  <  B
)
2421nnred 8701 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  RR )
251nn0red 8999 . . . . . 6  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  A  e.  RR )
2624, 25lenltd 7848 . . . . 5  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( B  <_  A  <->  -.  A  <  B ) )
2726biimpar 295 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  B  <_  A
)
28 simpl3 971 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  A  <  (
2  x.  B ) )
29 q2submod 10126 . . . 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 1209 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  mod  B )  =  ( A  -  B ) )
31 iffalse 3452 . . . . 5  |-  ( -.  A  <  B  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  ( A  -  B ) )
3231adantl 275 . . . 4  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  if ( A  <  B ,  A ,  ( A  -  B ) )  =  ( A  -  B
) )
3332eqcomd 2123 . . 3  |-  ( ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B ) )  /\  -.  A  <  B )  ->  ( A  -  B )  =  if ( A  <  B ,  A ,  ( A  -  B ) ) )
3430, 33eqtrd 2150 . 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 9140 . . 3  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  B  e.  ZZ )
37 zdclt 9096 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  -> DECID  A  <  B )
38 exmiddc 806 . . . 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 408 . 2  |-  ( ( A  e.  NN0  /\  B  e.  NN  /\  A  <  ( 2  x.  B
) )  ->  ( A  <  B  \/  -.  A  <  B ) )
4118, 34, 40mpjaodan 772 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 682  DECID wdc 804    /\ w3a 947    = wceq 1316    e. wcel 1465   ifcif 3444   class class class wbr 3899  (class class class)co 5742   0cc0 7588    x. cmul 7593    < clt 7768    <_ cle 7769    - cmin 7901   NNcn 8688   2c2 8739   NN0cn0 8945   ZZcz 9022   QQcq 9379    mod cmo 10063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689  df-2 8747  df-n0 8946  df-z 9023  df-q 9380  df-rp 9410  df-fl 10011  df-mod 10064
This theorem is referenced by: (None)
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