ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  divalgmod Unicode version

Theorem divalgmod 11009
Description: The result of the  mod operator satisfies the requirements for the remainder  R in the division algorithm for a positive divisor (compare divalg2 11008 and divalgb 11007). This demonstration theorem justifies the use of  mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.)
Assertion
Ref Expression
divalgmod  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R ) ) ) ) )

Proof of Theorem divalgmod
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 zq 9080 . . . . . . . 8  |-  ( N  e.  ZZ  ->  N  e.  QQ )
21adantr 270 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  QQ )
3 nnq 9087 . . . . . . . 8  |-  ( D  e.  NN  ->  D  e.  QQ )
43adantl 271 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  QQ )
5 simpr 108 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  NN )
65nngt0d 8437 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  0  <  D )
72, 4, 6modqcld 9700 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  QQ )
8 snidg 3468 . . . . . 6  |-  ( ( N  mod  D )  e.  QQ  ->  ( N  mod  D )  e. 
{ ( N  mod  D ) } )
97, 8syl 14 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  { ( N  mod  D ) } )
10 eleq1 2150 . . . . 5  |-  ( R  =  ( N  mod  D )  ->  ( R  e.  { ( N  mod  D ) }  <->  ( N  mod  D )  e.  {
( N  mod  D
) } ) )
119, 10syl5ibrcom 155 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  ->  R  e.  {
( N  mod  D
) } ) )
12 elsni 3459 . . . 4  |-  ( R  e.  { ( N  mod  D ) }  ->  R  =  ( N  mod  D ) )
1311, 12impbid1 140 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
R  e.  { ( N  mod  D ) } ) )
14 modqlt 9705 . . . . . . . . 9  |-  ( ( N  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( N  mod  D )  < 
D )
152, 4, 6, 14syl3anc 1174 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  <  D )
16 znq 9078 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  /  D
)  e.  QQ )
1716flqcld 9649 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( |_ `  ( N  /  D ) )  e.  ZZ )
18 nnz 8739 . . . . . . . . . 10  |-  ( D  e.  NN  ->  D  e.  ZZ )
1918adantl 271 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  ZZ )
20 zmodcl 9716 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  NN0 )
2120nn0zd 8836 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  ZZ )
22 zsubcl 8761 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( N  mod  D )  e.  ZZ )  -> 
( N  -  ( N  mod  D ) )  e.  ZZ )
2321, 22syldan 276 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  -  ( N  mod  D ) )  e.  ZZ )
24 nncn 8402 . . . . . . . . . . . 12  |-  ( D  e.  NN  ->  D  e.  CC )
2524adantl 271 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  CC )
2617zcnd 8839 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( |_ `  ( N  /  D ) )  e.  CC )
2725, 26mulcomd 7488 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  =  ( ( |_
`  ( N  /  D ) )  x.  D ) )
28 modqval 9696 . . . . . . . . . . . 12  |-  ( ( N  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( N  mod  D )  =  ( N  -  ( D  x.  ( |_ `  ( N  /  D
) ) ) ) )
292, 4, 6, 28syl3anc 1174 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  =  ( N  -  ( D  x.  ( |_ `  ( N  /  D ) ) ) ) )
3020nn0cnd 8698 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  CC )
31 zmulcl 8773 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ZZ  /\  ( |_ `  ( N  /  D ) )  e.  ZZ )  -> 
( D  x.  ( |_ `  ( N  /  D ) ) )  e.  ZZ )
3218, 17, 31syl2an2 561 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  e.  ZZ )
3332zcnd 8839 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  e.  CC )
34 zcn 8725 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
3534adantr 270 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  CC )
3630, 33, 35subexsub 7829 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( N  mod  D )  =  ( N  -  ( D  x.  ( |_ `  ( N  /  D ) ) ) )  <->  ( D  x.  ( |_ `  ( N  /  D ) ) )  =  ( N  -  ( N  mod  D ) ) ) )
3729, 36mpbid 145 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  =  ( N  -  ( N  mod  D ) ) )
3827, 37eqtr3d 2122 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( |_ `  ( N  /  D
) )  x.  D
)  =  ( N  -  ( N  mod  D ) ) )
39 dvds0lem 10888 . . . . . . . . 9  |-  ( ( ( ( |_ `  ( N  /  D
) )  e.  ZZ  /\  D  e.  ZZ  /\  ( N  -  ( N  mod  D ) )  e.  ZZ )  /\  ( ( |_ `  ( N  /  D
) )  x.  D
)  =  ( N  -  ( N  mod  D ) ) )  ->  D  ||  ( N  -  ( N  mod  D ) ) )
4017, 19, 23, 38, 39syl31anc 1177 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  ||  ( N  -  ( N  mod  D ) ) )
41 divalg2 11008 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) )
42 breq1 3840 . . . . . . . . . . 11  |-  ( z  =  ( N  mod  D )  ->  ( z  <  D  <->  ( N  mod  D )  <  D ) )
43 oveq2 5642 . . . . . . . . . . . 12  |-  ( z  =  ( N  mod  D )  ->  ( N  -  z )  =  ( N  -  ( N  mod  D ) ) )
4443breq2d 3849 . . . . . . . . . . 11  |-  ( z  =  ( N  mod  D )  ->  ( D  ||  ( N  -  z
)  <->  D  ||  ( N  -  ( N  mod  D ) ) ) )
4542, 44anbi12d 457 . . . . . . . . . 10  |-  ( z  =  ( N  mod  D )  ->  ( (
z  <  D  /\  D  ||  ( N  -  z ) )  <->  ( ( N  mod  D )  < 
D  /\  D  ||  ( N  -  ( N  mod  D ) ) ) ) )
4645riota2 5612 . . . . . . . . 9  |-  ( ( ( N  mod  D
)  e.  NN0  /\  E! z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) )  ->  ( (
( N  mod  D
)  <  D  /\  D  ||  ( N  -  ( N  mod  D ) ) )  <->  ( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z
) ) )  =  ( N  mod  D
) ) )
4720, 41, 46syl2anc 403 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( ( N  mod  D )  < 
D  /\  D  ||  ( N  -  ( N  mod  D ) ) )  <-> 
( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) )  =  ( N  mod  D ) ) )
4815, 40, 47mpbi2and 889 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) )  =  ( N  mod  D ) )
4948eqcomd 2093 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  =  ( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z
) ) ) )
5049sneqd 3454 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  { ( N  mod  D ) }  =  {
( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) ) } )
51 snriota 5619 . . . . . 6  |-  ( E! z  e.  NN0  (
z  <  D  /\  D  ||  ( N  -  z ) )  ->  { z  e.  NN0  |  ( z  <  D  /\  D  ||  ( N  -  z ) ) }  =  { (
iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) ) } )
5241, 51syl 14 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  { z  e.  NN0  |  ( z  <  D  /\  D  ||  ( N  -  z ) ) }  =  { (
iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) ) } )
5350, 52eqtr4d 2123 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  { ( N  mod  D ) }  =  {
z  e.  NN0  | 
( z  <  D  /\  D  ||  ( N  -  z ) ) } )
5453eleq2d 2157 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  e.  {
( N  mod  D
) }  <->  R  e.  { z  e.  NN0  | 
( z  <  D  /\  D  ||  ( N  -  z ) ) } ) )
5513, 54bitrd 186 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
R  e.  { z  e.  NN0  |  (
z  <  D  /\  D  ||  ( N  -  z ) ) } ) )
56 breq1 3840 . . . 4  |-  ( z  =  R  ->  (
z  <  D  <->  R  <  D ) )
57 oveq2 5642 . . . . 5  |-  ( z  =  R  ->  ( N  -  z )  =  ( N  -  R ) )
5857breq2d 3849 . . . 4  |-  ( z  =  R  ->  ( D  ||  ( N  -  z )  <->  D  ||  ( N  -  R )
) )
5956, 58anbi12d 457 . . 3  |-  ( z  =  R  ->  (
( z  <  D  /\  D  ||  ( N  -  z ) )  <-> 
( R  <  D  /\  D  ||  ( N  -  R ) ) ) )
6059elrab 2769 . 2  |-  ( R  e.  { z  e. 
NN0  |  ( z  <  D  /\  D  ||  ( N  -  z
) ) }  <->  ( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) )
6155, 60syl6bb 194 1  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   E!wreu 2361   {crab 2363   {csn 3441   class class class wbr 3837   ` cfv 5002   iota_crio 5589  (class class class)co 5634   CCcc 7327   0cc0 7329    x. cmul 7334    < clt 7501    - cmin 7632    / cdiv 8113   NNcn 8394   NN0cn0 8643   ZZcz 8720   QQcq 9073   |_cfl 9640    mod cmo 9694    || cdvds 10878
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-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  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-nul 3285  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-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  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-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  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-uz 8989  df-q 9074  df-rp 9104  df-fl 9642  df-mod 9695  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-dvds 10879
This theorem is referenced by:  divalgmodcl  11010
  Copyright terms: Public domain W3C validator