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Theorem divalg 12602
Description: The division algorithm (theorem). Dividing an integer  N by a nonzero integer  D produces a (unique) quotient  q and a unique remainder  0  <_  r  <  ( abs `  D
). The proof does not use  /,  |_ or  mod. (Contributed by Paul Chapman, 21-Mar-2011.)
Assertion
Ref Expression
divalg  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  ->  E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) ) )
Distinct variable groups:    D, q,
r    N, q, r

Proof of Theorem divalg
StepHypRef Expression
1 eqeq1 2289 . . . . . 6  |-  ( N  =  if ( N  e.  ZZ ,  N ,  1 )  -> 
( N  =  ( ( q  x.  D
)  +  r )  <-> 
if ( N  e.  ZZ ,  N , 
1 )  =  ( ( q  x.  D
)  +  r ) ) )
213anbi3d 1258 . . . . 5  |-  ( N  =  if ( N  e.  ZZ ,  N ,  1 )  -> 
( ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) )  <->  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  D )  +  r ) ) ) )
32rexbidv 2564 . . . 4  |-  ( N  =  if ( N  e.  ZZ ,  N ,  1 )  -> 
( E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) )  <->  E. q  e.  ZZ  ( 0  <_  r  /\  r  <  ( abs `  D )  /\  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  D )  +  r ) ) ) )
43reubidv 2724 . . 3  |-  ( N  =  if ( N  e.  ZZ ,  N ,  1 )  -> 
( E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) )  <->  E! r  e.  ZZ  E. q  e.  ZZ  (
0  <_  r  /\  r  <  ( abs `  D
)  /\  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  D )  +  r ) ) ) )
5 fveq2 5525 . . . . . . 7  |-  ( D  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( abs `  D
)  =  ( abs `  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) ) )
65breq2d 4035 . . . . . 6  |-  ( D  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( r  < 
( abs `  D
)  <->  r  <  ( abs `  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) ) ) )
7 oveq2 5866 . . . . . . . 8  |-  ( D  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( q  x.  D )  =  ( q  x.  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) ) )
87oveq1d 5873 . . . . . . 7  |-  ( D  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( ( q  x.  D )  +  r )  =  ( ( q  x.  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) )  +  r ) )
98eqeq2d 2294 . . . . . 6  |-  ( D  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  D )  +  r )  <->  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) )  +  r ) ) )
106, 93anbi23d 1255 . . . . 5  |-  ( D  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( ( 0  <_  r  /\  r  <  ( abs `  D
)  /\  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  D )  +  r ) )  <->  ( 0  <_  r  /\  r  <  ( abs `  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) )  /\  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) )  +  r ) ) ) )
1110rexbidv 2564 . . . 4  |-  ( D  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  D )  +  r ) )  <->  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) )  /\  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) )  +  r ) ) ) )
1211reubidv 2724 . . 3  |-  ( D  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  D )  +  r ) )  <->  E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) )  /\  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) )  +  r ) ) ) )
13 1z 10053 . . . . 5  |-  1  e.  ZZ
1413elimel 3617 . . . 4  |-  if ( N  e.  ZZ ,  N ,  1 )  e.  ZZ
15 simpl 443 . . . . 5  |-  ( ( D  e.  ZZ  /\  D  =/=  0 )  ->  D  e.  ZZ )
16 eleq1 2343 . . . . 5  |-  ( D  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( D  e.  ZZ  <->  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  e.  ZZ ) )
17 eleq1 2343 . . . . 5  |-  ( 1  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( 1  e.  ZZ  <->  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  e.  ZZ ) )
1815, 16, 17, 13elimdhyp 3618 . . . 4  |-  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  e.  ZZ
19 simpr 447 . . . . 5  |-  ( ( D  e.  ZZ  /\  D  =/=  0 )  ->  D  =/=  0 )
20 neeq1 2454 . . . . 5  |-  ( D  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( D  =/=  0  <->  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  =/=  0 ) )
21 neeq1 2454 . . . . 5  |-  ( 1  =  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ->  ( 1  =/=  0  <->  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  =/=  0 ) )
22 ax-1ne0 8806 . . . . 5  |-  1  =/=  0
2319, 20, 21, 22elimdhyp 3618 . . . 4  |-  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  =/=  0
24 eqid 2283 . . . 4  |-  { r  e.  NN0  |  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ||  ( if ( N  e.  ZZ ,  N , 
1 )  -  r
) }  =  {
r  e.  NN0  |  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 )  ||  ( if ( N  e.  ZZ ,  N , 
1 )  -  r
) }
2514, 18, 23, 24divalglem10 12601 . . 3  |-  E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) )  /\  if ( N  e.  ZZ ,  N ,  1 )  =  ( ( q  x.  if ( ( D  e.  ZZ  /\  D  =/=  0 ) ,  D ,  1 ) )  +  r ) )
264, 12, 25dedth2h 3607 . 2  |-  ( ( N  e.  ZZ  /\  ( D  e.  ZZ  /\  D  =/=  0 ) )  ->  E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) ) )
27263impb 1147 1  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  ->  E! r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544   E!wreu 2545   {crab 2547   ifcif 3565   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    - cmin 9037   NN0cn0 9965   ZZcz 10024   abscabs 11719    || cdivides 12531
This theorem is referenced by:  divalg2  12604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-dvds 12532
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