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Theorem divalgmod 11472
Description: The result of the  mod operator satisfies the requirements for the remainder  R in the division algorithm for a positive divisor (compare divalg2 11471 and divalgb 11470). 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 9320 . . . . . . . 8  |-  ( N  e.  ZZ  ->  N  e.  QQ )
21adantr 272 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  QQ )
3 nnq 9327 . . . . . . . 8  |-  ( D  e.  NN  ->  D  e.  QQ )
43adantl 273 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  QQ )
5 simpr 109 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  NN )
65nngt0d 8674 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  0  <  D )
72, 4, 6modqcld 9994 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  QQ )
8 snidg 3520 . . . . . 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 2177 . . . . 5  |-  ( R  =  ( N  mod  D )  ->  ( R  e.  { ( N  mod  D ) }  <->  ( N  mod  D )  e.  {
( N  mod  D
) } ) )
119, 10syl5ibrcom 156 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  ->  R  e.  {
( N  mod  D
) } ) )
12 elsni 3511 . . . 4  |-  ( R  e.  { ( N  mod  D ) }  ->  R  =  ( N  mod  D ) )
1311, 12impbid1 141 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
R  e.  { ( N  mod  D ) } ) )
14 modqlt 9999 . . . . . . . . 9  |-  ( ( N  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( N  mod  D )  < 
D )
152, 4, 6, 14syl3anc 1199 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  <  D )
16 znq 9318 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  /  D
)  e.  QQ )
1716flqcld 9943 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( |_ `  ( N  /  D ) )  e.  ZZ )
18 nnz 8977 . . . . . . . . . 10  |-  ( D  e.  NN  ->  D  e.  ZZ )
1918adantl 273 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  ZZ )
20 zmodcl 10010 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  NN0 )
2120nn0zd 9075 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  ZZ )
22 zsubcl 8999 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( N  mod  D )  e.  ZZ )  -> 
( N  -  ( N  mod  D ) )  e.  ZZ )
2321, 22syldan 278 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  -  ( N  mod  D ) )  e.  ZZ )
24 nncn 8638 . . . . . . . . . . . 12  |-  ( D  e.  NN  ->  D  e.  CC )
2524adantl 273 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  CC )
2617zcnd 9078 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( |_ `  ( N  /  D ) )  e.  CC )
2725, 26mulcomd 7711 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  =  ( ( |_
`  ( N  /  D ) )  x.  D ) )
28 modqval 9990 . . . . . . . . . . . 12  |-  ( ( N  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( N  mod  D )  =  ( N  -  ( D  x.  ( |_ `  ( N  /  D
) ) ) ) )
292, 4, 6, 28syl3anc 1199 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  =  ( N  -  ( D  x.  ( |_ `  ( N  /  D ) ) ) ) )
3020nn0cnd 8936 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  CC )
31 zmulcl 9011 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ZZ  /\  ( |_ `  ( N  /  D ) )  e.  ZZ )  -> 
( D  x.  ( |_ `  ( N  /  D ) ) )  e.  ZZ )
3218, 17, 31syl2an2 566 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  e.  ZZ )
3332zcnd 9078 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  e.  CC )
34 zcn 8963 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
3534adantr 272 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  CC )
3630, 33, 35subexsub 8053 . . . . . . . . . . 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 146 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  =  ( N  -  ( N  mod  D ) ) )
3827, 37eqtr3d 2149 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( |_ `  ( N  /  D
) )  x.  D
)  =  ( N  -  ( N  mod  D ) ) )
39 dvds0lem 11351 . . . . . . . . 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 1202 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  ||  ( N  -  ( N  mod  D ) ) )
41 divalg2 11471 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) )
42 breq1 3898 . . . . . . . . . . 11  |-  ( z  =  ( N  mod  D )  ->  ( z  <  D  <->  ( N  mod  D )  <  D ) )
43 oveq2 5736 . . . . . . . . . . . 12  |-  ( z  =  ( N  mod  D )  ->  ( N  -  z )  =  ( N  -  ( N  mod  D ) ) )
4443breq2d 3907 . . . . . . . . . . 11  |-  ( z  =  ( N  mod  D )  ->  ( D  ||  ( N  -  z
)  <->  D  ||  ( N  -  ( N  mod  D ) ) ) )
4542, 44anbi12d 462 . . . . . . . . . 10  |-  ( z  =  ( N  mod  D )  ->  ( (
z  <  D  /\  D  ||  ( N  -  z ) )  <->  ( ( N  mod  D )  < 
D  /\  D  ||  ( N  -  ( N  mod  D ) ) ) ) )
4645riota2 5706 . . . . . . . . 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 406 . . . . . . . 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 910 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) )  =  ( N  mod  D ) )
4948eqcomd 2120 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  =  ( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z
) ) ) )
5049sneqd 3506 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  { ( N  mod  D ) }  =  {
( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) ) } )
51 snriota 5713 . . . . . 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 2150 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  { ( N  mod  D ) }  =  {
z  e.  NN0  | 
( z  <  D  /\  D  ||  ( N  -  z ) ) } )
5453eleq2d 2184 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  e.  {
( N  mod  D
) }  <->  R  e.  { z  e.  NN0  | 
( z  <  D  /\  D  ||  ( N  -  z ) ) } ) )
5513, 54bitrd 187 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
R  e.  { z  e.  NN0  |  (
z  <  D  /\  D  ||  ( N  -  z ) ) } ) )
56 breq1 3898 . . . 4  |-  ( z  =  R  ->  (
z  <  D  <->  R  <  D ) )
57 oveq2 5736 . . . . 5  |-  ( z  =  R  ->  ( N  -  z )  =  ( N  -  R ) )
5857breq2d 3907 . . . 4  |-  ( z  =  R  ->  ( D  ||  ( N  -  z )  <->  D  ||  ( N  -  R )
) )
5956, 58anbi12d 462 . . 3  |-  ( z  =  R  ->  (
( z  <  D  /\  D  ||  ( N  -  z ) )  <-> 
( R  <  D  /\  D  ||  ( N  -  R ) ) ) )
6059elrab 2809 . 2  |-  ( R  e.  { z  e. 
NN0  |  ( z  <  D  /\  D  ||  ( N  -  z
) ) }  <->  ( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) )
6155, 60syl6bb 195 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   E!wreu 2392   {crab 2394   {csn 3493   class class class wbr 3895   ` cfv 5081   iota_crio 5683  (class class class)co 5728   CCcc 7545   0cc0 7547    x. cmul 7552    < clt 7724    - cmin 7856    / cdiv 8345   NNcn 8630   NN0cn0 8881   ZZcz 8958   QQcq 9313   |_cfl 9934    mod cmo 9988    || cdvds 11341
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-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  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-nul 3330  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-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  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-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  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-uz 9229  df-q 9314  df-rp 9344  df-fl 9936  df-mod 9989  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663  df-dvds 11342
This theorem is referenced by:  divalgmodcl  11473
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