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Theorem divalglemnn 11925
Description: Lemma for divalg 11931. Existence for a positive denominator. (Contributed by Jim Kingdon, 30-Nov-2021.)
Assertion
Ref Expression
divalglemnn  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  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 divalglemnn
StepHypRef Expression
1 zmodcl 10346 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  NN0 )
21nn0zd 9375 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  ZZ )
3 znq 9626 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  /  D
)  e.  QQ )
43flqcld 10279 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( |_ `  ( N  /  D ) )  e.  ZZ )
51nn0ge0d 9234 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  0  <_  ( N  mod  D ) )
6 zq 9628 . . . . 5  |-  ( N  e.  ZZ  ->  N  e.  QQ )
76adantr 276 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  QQ )
8 nnq 9635 . . . . 5  |-  ( D  e.  NN  ->  D  e.  QQ )
98adantl 277 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  QQ )
10 nngt0 8946 . . . . 5  |-  ( D  e.  NN  ->  0  <  D )
1110adantl 277 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  0  <  D )
12 modqlt 10335 . . . 4  |-  ( ( N  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( N  mod  D )  < 
D )
137, 9, 11, 12syl3anc 1238 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  <  D )
14 nnre 8928 . . . . 5  |-  ( D  e.  NN  ->  D  e.  RR )
1514adantl 277 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  RR )
16 0red 7960 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  0  e.  RR )
1716, 15, 11ltled 8078 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  0  <_  D )
1815, 17absidd 11178 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( abs `  D
)  =  D )
1913, 18breqtrrd 4033 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  <  ( abs `  D ) )
201nn0cnd 9233 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  e.  CC )
214zcnd 9378 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( |_ `  ( N  /  D ) )  e.  CC )
22 simpr 110 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  NN )
2322nncnd 8935 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  D  e.  CC )
2421, 23mulcld 7980 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( |_ `  ( N  /  D
) )  x.  D
)  e.  CC )
25 modqvalr 10327 . . . . . 6  |-  ( ( N  e.  QQ  /\  D  e.  QQ  /\  0  <  D )  ->  ( N  mod  D )  =  ( N  -  (
( |_ `  ( N  /  D ) )  x.  D ) ) )
267, 9, 11, 25syl3anc 1238 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( N  mod  D
)  =  ( N  -  ( ( |_
`  ( N  /  D ) )  x.  D ) ) )
2726oveq1d 5892 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( N  mod  D )  +  ( ( |_ `  ( N  /  D ) )  x.  D ) )  =  ( ( N  -  ( ( |_
`  ( N  /  D ) )  x.  D ) )  +  ( ( |_ `  ( N  /  D
) )  x.  D
) ) )
28 simpl 109 . . . . . 6  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  ZZ )
2928zcnd 9378 . . . . 5  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  e.  CC )
3029, 24npcand 8274 . . . 4  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  ( ( N  -  ( ( |_ `  ( N  /  D
) )  x.  D
) )  +  ( ( |_ `  ( N  /  D ) )  x.  D ) )  =  N )
3127, 30eqtr2d 2211 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  =  ( ( N  mod  D )  +  ( ( |_
`  ( N  /  D ) )  x.  D ) ) )
3220, 24, 31comraddd 8116 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  N  =  ( ( ( |_ `  ( N  /  D ) )  x.  D )  +  ( N  mod  D
) ) )
33 breq2 4009 . . . 4  |-  ( r  =  ( N  mod  D )  ->  ( 0  <_  r  <->  0  <_  ( N  mod  D ) ) )
34 breq1 4008 . . . 4  |-  ( r  =  ( N  mod  D )  ->  ( r  <  ( abs `  D
)  <->  ( N  mod  D )  <  ( abs `  D ) ) )
35 oveq2 5885 . . . . 5  |-  ( r  =  ( N  mod  D )  ->  ( (
q  x.  D )  +  r )  =  ( ( q  x.  D )  +  ( N  mod  D ) ) )
3635eqeq2d 2189 . . . 4  |-  ( r  =  ( N  mod  D )  ->  ( N  =  ( ( q  x.  D )  +  r )  <->  N  =  ( ( q  x.  D )  +  ( N  mod  D ) ) ) )
3733, 34, 363anbi123d 1312 . . 3  |-  ( r  =  ( N  mod  D )  ->  ( (
0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  r ) )  <->  ( 0  <_  ( N  mod  D )  /\  ( N  mod  D )  < 
( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  ( N  mod  D ) ) ) ) )
38 oveq1 5884 . . . . . 6  |-  ( q  =  ( |_ `  ( N  /  D
) )  ->  (
q  x.  D )  =  ( ( |_
`  ( N  /  D ) )  x.  D ) )
3938oveq1d 5892 . . . . 5  |-  ( q  =  ( |_ `  ( N  /  D
) )  ->  (
( q  x.  D
)  +  ( N  mod  D ) )  =  ( ( ( |_ `  ( N  /  D ) )  x.  D )  +  ( N  mod  D
) ) )
4039eqeq2d 2189 . . . 4  |-  ( q  =  ( |_ `  ( N  /  D
) )  ->  ( N  =  ( (
q  x.  D )  +  ( N  mod  D ) )  <->  N  =  ( ( ( |_
`  ( N  /  D ) )  x.  D )  +  ( N  mod  D ) ) ) )
41403anbi3d 1318 . . 3  |-  ( q  =  ( |_ `  ( N  /  D
) )  ->  (
( 0  <_  ( N  mod  D )  /\  ( N  mod  D )  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  ( N  mod  D ) ) )  <->  ( 0  <_  ( N  mod  D )  /\  ( N  mod  D )  < 
( abs `  D
)  /\  N  =  ( ( ( |_
`  ( N  /  D ) )  x.  D )  +  ( N  mod  D ) ) ) ) )
4237, 41rspc2ev 2858 . 2  |-  ( ( ( N  mod  D
)  e.  ZZ  /\  ( |_ `  ( N  /  D ) )  e.  ZZ  /\  (
0  <_  ( N  mod  D )  /\  ( N  mod  D )  < 
( abs `  D
)  /\  N  =  ( ( ( |_
`  ( N  /  D ) )  x.  D )  +  ( N  mod  D ) ) ) )  ->  E. r  e.  ZZ  E. q  e.  ZZ  (
0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  r ) ) )
432, 4, 5, 19, 32, 42syl113anc 1250 1  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  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 104    /\ w3a 978    = wceq 1353    e. wcel 2148   E.wrex 2456   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   RRcr 7812   0cc0 7813    + caddc 7816    x. cmul 7818    < clt 7994    <_ cle 7995    - cmin 8130    / cdiv 8631   NNcn 8921   ZZcz 9255   QQcq 9621   |_cfl 10270    mod cmo 10324   abscabs 11008
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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-fl 10272  df-mod 10325  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010
This theorem is referenced by:  divalglemeunn  11928  divalglemex  11929
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