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Theorem divalgmod 11959
Description: The result of the  mod operator satisfies the requirements for the remainder  R in the division algorithm for a positive divisor (compare divalg2 11958 and divalgb 11957). 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 9651 . . . . . . . 8  |-  ( N  e.  ZZ  ->  N  e.  QQ )
21adantr 276 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  QQ )
3 nnq 9658 . . . . . . . 8  |-  ( D  e.  NN  ->  D  e.  QQ )
43adantl 277 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  QQ )
5 simpr 110 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  NN )
65nngt0d 8988 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  0  <  D )
72, 4, 6modqcld 10354 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  QQ )
8 snidg 3636 . . . . . 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 2252 . . . . 5  |-  ( R  =  ( N  mod  D )  ->  ( R  e.  { ( N  mod  D ) }  <->  ( N  mod  D )  e.  {
( N  mod  D
) } ) )
119, 10syl5ibrcom 157 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  ->  R  e.  {
( N  mod  D
) } ) )
12 elsni 3625 . . . 4  |-  ( R  e.  { ( N  mod  D ) }  ->  R  =  ( N  mod  D ) )
1311, 12impbid1 142 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
R  e.  { ( N  mod  D ) } ) )
14 modqlt 10359 . . . . . . . . 9  |-  ( ( N  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( N  mod  D )  < 
D )
152, 4, 6, 14syl3anc 1249 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  <  D )
16 znq 9649 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  /  D
)  e.  QQ )
1716flqcld 10303 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( |_ `  ( N  /  D ) )  e.  ZZ )
18 nnz 9297 . . . . . . . . . 10  |-  ( D  e.  NN  ->  D  e.  ZZ )
1918adantl 277 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  ZZ )
20 zmodcl 10370 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  NN0 )
2120nn0zd 9398 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  ZZ )
22 zsubcl 9319 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( N  mod  D )  e.  ZZ )  -> 
( N  -  ( N  mod  D ) )  e.  ZZ )
2321, 22syldan 282 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  -  ( N  mod  D ) )  e.  ZZ )
24 nncn 8952 . . . . . . . . . . . 12  |-  ( D  e.  NN  ->  D  e.  CC )
2524adantl 277 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  CC )
2617zcnd 9401 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( |_ `  ( N  /  D ) )  e.  CC )
2725, 26mulcomd 8004 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  =  ( ( |_
`  ( N  /  D ) )  x.  D ) )
28 modqval 10350 . . . . . . . . . . . 12  |-  ( ( N  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( N  mod  D )  =  ( N  -  ( D  x.  ( |_ `  ( N  /  D
) ) ) ) )
292, 4, 6, 28syl3anc 1249 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  =  ( N  -  ( D  x.  ( |_ `  ( N  /  D ) ) ) ) )
3020nn0cnd 9256 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  CC )
31 zmulcl 9331 . . . . . . . . . . . . . 14  |-  ( ( D  e.  ZZ  /\  ( |_ `  ( N  /  D ) )  e.  ZZ )  -> 
( D  x.  ( |_ `  ( N  /  D ) ) )  e.  ZZ )
3218, 17, 31syl2an2 594 . . . . . . . . . . . . 13  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  e.  ZZ )
3332zcnd 9401 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  e.  CC )
34 zcn 9283 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
3534adantr 276 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  CC )
3630, 33, 35subexsub 8354 . . . . . . . . . . 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 147 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  =  ( N  -  ( N  mod  D ) ) )
3827, 37eqtr3d 2224 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( |_ `  ( N  /  D
) )  x.  D
)  =  ( N  -  ( N  mod  D ) ) )
39 dvds0lem 11835 . . . . . . . . 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 1252 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  ||  ( N  -  ( N  mod  D ) ) )
41 divalg2 11958 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) )
42 breq1 4021 . . . . . . . . . . 11  |-  ( z  =  ( N  mod  D )  ->  ( z  <  D  <->  ( N  mod  D )  <  D ) )
43 oveq2 5900 . . . . . . . . . . . 12  |-  ( z  =  ( N  mod  D )  ->  ( N  -  z )  =  ( N  -  ( N  mod  D ) ) )
4443breq2d 4030 . . . . . . . . . . 11  |-  ( z  =  ( N  mod  D )  ->  ( D  ||  ( N  -  z
)  <->  D  ||  ( N  -  ( N  mod  D ) ) ) )
4542, 44anbi12d 473 . . . . . . . . . 10  |-  ( z  =  ( N  mod  D )  ->  ( (
z  <  D  /\  D  ||  ( N  -  z ) )  <->  ( ( N  mod  D )  < 
D  /\  D  ||  ( N  -  ( N  mod  D ) ) ) ) )
4645riota2 5870 . . . . . . . . 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 411 . . . . . . . 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 945 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) )  =  ( N  mod  D ) )
4948eqcomd 2195 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  =  ( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z
) ) ) )
5049sneqd 3620 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  { ( N  mod  D ) }  =  {
( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) ) } )
51 snriota 5877 . . . . . 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 2225 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  { ( N  mod  D ) }  =  {
z  e.  NN0  | 
( z  <  D  /\  D  ||  ( N  -  z ) ) } )
5453eleq2d 2259 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  e.  {
( N  mod  D
) }  <->  R  e.  { z  e.  NN0  | 
( z  <  D  /\  D  ||  ( N  -  z ) ) } ) )
5513, 54bitrd 188 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( R  =  ( N  mod  D )  <-> 
R  e.  { z  e.  NN0  |  (
z  <  D  /\  D  ||  ( N  -  z ) ) } ) )
56 breq1 4021 . . . 4  |-  ( z  =  R  ->  (
z  <  D  <->  R  <  D ) )
57 oveq2 5900 . . . . 5  |-  ( z  =  R  ->  ( N  -  z )  =  ( N  -  R ) )
5857breq2d 4030 . . . 4  |-  ( z  =  R  ->  ( D  ||  ( N  -  z )  <->  D  ||  ( N  -  R )
) )
5956, 58anbi12d 473 . . 3  |-  ( z  =  R  ->  (
( z  <  D  /\  D  ||  ( N  -  z ) )  <-> 
( R  <  D  /\  D  ||  ( N  -  R ) ) ) )
6059elrab 2908 . 2  |-  ( R  e.  { z  e. 
NN0  |  ( z  <  D  /\  D  ||  ( N  -  z
) ) }  <->  ( R  e.  NN0  /\  ( R  <  D  /\  D  ||  ( N  -  R
) ) ) )
6155, 60bitrdi 196 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 104    <-> wb 105    = wceq 1364    e. wcel 2160   E!wreu 2470   {crab 2472   {csn 3607   class class class wbr 4018   ` cfv 5232   iota_crio 5847  (class class class)co 5892   CCcc 7834   0cc0 7836    x. cmul 7841    < clt 8017    - cmin 8153    / cdiv 8654   NNcn 8944   NN0cn0 9201   ZZcz 9278   QQcq 9644   |_cfl 10294    mod cmo 10348    || cdvds 11821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-mulrcl 7935  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-precex 7946  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-apti 7951  ax-pre-ltadd 7952  ax-pre-mulgt0 7953  ax-pre-mulext 7954  ax-arch 7955
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-frec 6411  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-reap 8557  df-ap 8564  df-div 8655  df-inn 8945  df-2 9003  df-n0 9202  df-z 9279  df-uz 9554  df-q 9645  df-rp 9679  df-fl 10296  df-mod 10349  df-seqfrec 10472  df-exp 10546  df-cj 10878  df-re 10879  df-im 10880  df-rsqrt 11034  df-abs 11035  df-dvds 11822
This theorem is referenced by:  divalgmodcl  11960
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