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Theorem divalglemex 11653
Description: Lemma for divalg 11655. The quotient and remainder exist. (Contributed by Jim Kingdon, 30-Nov-2021.)
Assertion
Ref Expression
divalglemex  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  ->  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 divalglemex
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 simpl1 985 . . . 4  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  N  e.  ZZ )
2 simpl2 986 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  D  e.  ZZ )
32znegcld 9198 . . . . 5  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  -u D  e.  ZZ )
4 simpr 109 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  D  <  0 )
52zred 9196 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  D  e.  RR )
65lt0neg1d 8300 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  -> 
( D  <  0  <->  0  <  -u D ) )
74, 6mpbid 146 . . . . 5  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  -> 
0  <  -u D )
8 elnnz 9087 . . . . 5  |-  ( -u D  e.  NN  <->  ( -u D  e.  ZZ  /\  0  <  -u D ) )
93, 7, 8sylanbrc 414 . . . 4  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  -u D  e.  NN )
10 divalglemnn 11649 . . . 4  |-  ( ( N  e.  ZZ  /\  -u D  e.  NN )  ->  E. r  e.  ZZ  E. k  e.  ZZ  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )
111, 9, 10syl2anc 409 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  E. r  e.  ZZ  E. k  e.  ZZ  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )
12 simplr 520 . . . . . . . 8  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  -> 
k  e.  ZZ )
1312znegcld 9198 . . . . . . 7  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  ->  -u k  e.  ZZ )
14 simpr1 988 . . . . . . 7  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  -> 
0  <_  r )
15 simpr2 989 . . . . . . . 8  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  -> 
r  <  ( abs `  -u D ) )
16 simpll2 1022 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  ->  D  e.  ZZ )
1716ad2antrr 480 . . . . . . . . . 10  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  ->  D  e.  ZZ )
1817zcnd 9197 . . . . . . . . 9  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  ->  D  e.  CC )
1918absnegd 10992 . . . . . . . 8  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  -> 
( abs `  -u D
)  =  ( abs `  D ) )
2015, 19breqtrd 3961 . . . . . . 7  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  -> 
r  <  ( abs `  D ) )
21 simpr3 990 . . . . . . . 8  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  ->  N  =  ( (
k  x.  -u D
)  +  r ) )
2212zcnd 9197 . . . . . . . . . 10  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  -> 
k  e.  CC )
23 mulneg12 8182 . . . . . . . . . 10  |-  ( ( k  e.  CC  /\  D  e.  CC )  ->  ( -u k  x.  D )  =  ( k  x.  -u D
) )
2422, 18, 23syl2anc 409 . . . . . . . . 9  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  -> 
( -u k  x.  D
)  =  ( k  x.  -u D ) )
2524oveq1d 5796 . . . . . . . 8  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  -> 
( ( -u k  x.  D )  +  r )  =  ( ( k  x.  -u D
)  +  r ) )
2621, 25eqtr4d 2176 . . . . . . 7  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  ->  N  =  ( ( -u k  x.  D )  +  r ) )
27 oveq1 5788 . . . . . . . . . . 11  |-  ( q  =  -u k  ->  (
q  x.  D )  =  ( -u k  x.  D ) )
2827oveq1d 5796 . . . . . . . . . 10  |-  ( q  =  -u k  ->  (
( q  x.  D
)  +  r )  =  ( ( -u k  x.  D )  +  r ) )
2928eqeq2d 2152 . . . . . . . . 9  |-  ( q  =  -u k  ->  ( N  =  ( (
q  x.  D )  +  r )  <->  N  =  ( ( -u k  x.  D )  +  r ) ) )
30293anbi3d 1297 . . . . . . . 8  |-  ( q  =  -u k  ->  (
( 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r ) )  <->  ( 0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( -u k  x.  D )  +  r ) ) ) )
3130rspcev 2792 . . . . . . 7  |-  ( (
-u k  e.  ZZ  /\  ( 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( -u k  x.  D )  +  r ) ) )  ->  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) ) )
3213, 14, 20, 26, 31syl13anc 1219 . . . . . 6  |-  ( ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  /\  (
0  <_  r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) ) )  ->  E. q  e.  ZZ  ( 0  <_  r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D )  +  r ) ) )
3332ex 114 . . . . 5  |-  ( ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  /\  k  e.  ZZ )  ->  (
( 0  <_  r  /\  r  <  ( abs `  -u D )  /\  N  =  ( (
k  x.  -u D
)  +  r ) )  ->  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) ) ) )
3433rexlimdva 2552 . . . 4  |-  ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  ->  ( E. k  e.  ZZ  ( 0  <_  r  /\  r  <  ( abs `  -u D )  /\  N  =  ( (
k  x.  -u D
)  +  r ) )  ->  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) ) ) )
3534reximdva 2537 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  -> 
( E. r  e.  ZZ  E. k  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  -u D
)  /\  N  =  ( ( k  x.  -u D )  +  r ) )  ->  E. r  e.  ZZ  E. q  e.  ZZ  ( 0  <_ 
r  /\  r  <  ( abs `  D )  /\  N  =  ( ( q  x.  D
)  +  r ) ) ) )
3611, 35mpd 13 . 2  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  E. r  e.  ZZ  E. q  e.  ZZ  (
0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  r ) ) )
37 simpr 109 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  =  0 )  ->  D  =  0 )
38 simpl3 987 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  =  0 )  ->  D  =/=  0
)
3937, 38pm2.21ddne 2392 . 2  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  =  0 )  ->  E. r  e.  ZZ  E. q  e.  ZZ  (
0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  r ) ) )
40 simpl1 985 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  0  <  D )  ->  N  e.  ZZ )
41 simpl2 986 . . . 4  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  0  <  D )  ->  D  e.  ZZ )
42 simpr 109 . . . 4  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  0  <  D )  -> 
0  <  D )
43 elnnz 9087 . . . 4  |-  ( D  e.  NN  <->  ( D  e.  ZZ  /\  0  < 
D ) )
4441, 42, 43sylanbrc 414 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  0  <  D )  ->  D  e.  NN )
45 divalglemnn 11649 . . 3  |-  ( ( N  e.  ZZ  /\  D  e.  NN )  ->  E. r  e.  ZZ  E. q  e.  ZZ  (
0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  r ) ) )
4640, 44, 45syl2anc 409 . 2  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  0  <  D )  ->  E. r  e.  ZZ  E. q  e.  ZZ  (
0  <_  r  /\  r  <  ( abs `  D
)  /\  N  =  ( ( q  x.  D )  +  r ) ) )
47 ztri3or0 9119 . . 3  |-  ( D  e.  ZZ  ->  ( D  <  0  \/  D  =  0  \/  0  <  D ) )
48473ad2ant2 1004 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  ->  ( D  <  0  \/  D  =  0  \/  0  <  D ) )
4936, 39, 46, 48mpjao3dan 1286 1  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  ->  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 103    \/ w3o 962    /\ w3a 963    = wceq 1332    e. wcel 1481    =/= wne 2309   E.wrex 2418   class class class wbr 3936   ` cfv 5130  (class class class)co 5781   CCcc 7641   0cc0 7643    + caddc 7646    x. cmul 7648    < clt 7823    <_ cle 7824   -ucneg 7957   NNcn 8743   ZZcz 9077   abscabs 10800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-n0 9001  df-z 9078  df-uz 9350  df-q 9438  df-rp 9470  df-fl 10073  df-mod 10126  df-seqfrec 10249  df-exp 10323  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802
This theorem is referenced by:  divalglemeuneg  11654
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