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Theorem divalgmod 11915
Description: The result of the  mod operator satisfies the requirements for the remainder  R in the division algorithm for a positive divisor (compare divalg2 11914 and divalgb 11913). 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 9615 . . . . . . . 8  |-  ( N  e.  ZZ  ->  N  e.  QQ )
21adantr 276 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  QQ )
3 nnq 9622 . . . . . . . 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 8952 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  0  <  D )
72, 4, 6modqcld 10314 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  QQ )
8 snidg 3620 . . . . . 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 2240 . . . . 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 3609 . . . 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 10319 . . . . . . . . 9  |-  ( ( N  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( N  mod  D )  < 
D )
152, 4, 6, 14syl3anc 1238 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  <  D )
16 znq 9613 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  /  D
)  e.  QQ )
1716flqcld 10263 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( |_ `  ( N  /  D ) )  e.  ZZ )
18 nnz 9261 . . . . . . . . . 10  |-  ( D  e.  NN  ->  D  e.  ZZ )
1918adantl 277 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  ZZ )
20 zmodcl 10330 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  NN0 )
2120nn0zd 9362 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  ZZ )
22 zsubcl 9283 . . . . . . . . . 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 8916 . . . . . . . . . . . 12  |-  ( D  e.  NN  ->  D  e.  CC )
2524adantl 277 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  CC )
2617zcnd 9365 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( |_ `  ( N  /  D ) )  e.  CC )
2725, 26mulcomd 7969 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  =  ( ( |_
`  ( N  /  D ) )  x.  D ) )
28 modqval 10310 . . . . . . . . . . . 12  |-  ( ( N  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( N  mod  D )  =  ( N  -  ( D  x.  ( |_ `  ( N  /  D
) ) ) ) )
292, 4, 6, 28syl3anc 1238 . . . . . . . . . . 11  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  =  ( N  -  ( D  x.  ( |_ `  ( N  /  D ) ) ) ) )
3020nn0cnd 9220 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  CC )
31 zmulcl 9295 . . . . . . . . . . . . . 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 9365 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( D  x.  ( |_ `  ( N  /  D ) ) )  e.  CC )
34 zcn 9247 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
3534adantr 276 . . . . . . . . . . . 12  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  CC )
3630, 33, 35subexsub 8319 . . . . . . . . . . 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 2212 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( |_ `  ( N  /  D
) )  x.  D
)  =  ( N  -  ( N  mod  D ) ) )
39 dvds0lem 11792 . . . . . . . . 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 1241 . . . . . . . 8  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  ||  ( N  -  ( N  mod  D ) ) )
41 divalg2 11914 . . . . . . . . 9  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E! z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) )
42 breq1 4003 . . . . . . . . . . 11  |-  ( z  =  ( N  mod  D )  ->  ( z  <  D  <->  ( N  mod  D )  <  D ) )
43 oveq2 5877 . . . . . . . . . . . 12  |-  ( z  =  ( N  mod  D )  ->  ( N  -  z )  =  ( N  -  ( N  mod  D ) ) )
4443breq2d 4012 . . . . . . . . . . 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 5847 . . . . . . . . 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 943 . . . . . . 7  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) )  =  ( N  mod  D ) )
4948eqcomd 2183 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  =  ( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z
) ) ) )
5049sneqd 3604 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  { ( N  mod  D ) }  =  {
( iota_ z  e.  NN0  ( z  <  D  /\  D  ||  ( N  -  z ) ) ) } )
51 snriota 5854 . . . . . 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 2213 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  { ( N  mod  D ) }  =  {
z  e.  NN0  | 
( z  <  D  /\  D  ||  ( N  -  z ) ) } )
5453eleq2d 2247 . . 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 4003 . . . 4  |-  ( z  =  R  ->  (
z  <  D  <->  R  <  D ) )
57 oveq2 5877 . . . . 5  |-  ( z  =  R  ->  ( N  -  z )  =  ( N  -  R ) )
5857breq2d 4012 . . . 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 2893 . 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 1353    e. wcel 2148   E!wreu 2457   {crab 2459   {csn 3591   class class class wbr 4000   ` cfv 5212   iota_crio 5824  (class class class)co 5869   CCcc 7800   0cc0 7802    x. cmul 7807    < clt 7982    - cmin 8118    / cdiv 8618   NNcn 8908   NN0cn0 9165   ZZcz 9242   QQcq 9608   |_cfl 10254    mod cmo 10308    || cdvds 11778
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779
This theorem is referenced by:  divalgmodcl  11916
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