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Theorem divalglemex 12063
Description: Lemma for divalg 12065. 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 1002 . . . 4  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  N  e.  ZZ )
2 simpl2 1003 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  D  e.  ZZ )
32znegcld 9441 . . . . 5  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  -u D  e.  ZZ )
4 simpr 110 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  D  <  0 )
52zred 9439 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  D  e.  RR )
65lt0neg1d 8534 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  -> 
( D  <  0  <->  0  <  -u D ) )
74, 6mpbid 147 . . . . 5  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  -> 
0  <  -u D )
8 elnnz 9327 . . . . 5  |-  ( -u D  e.  NN  <->  ( -u D  e.  ZZ  /\  0  <  -u D ) )
93, 7, 8sylanbrc 417 . . . 4  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  -u D  e.  NN )
10 divalglemnn 12059 . . . 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 411 . . 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 528 . . . . . . . 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 9441 . . . . . . 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 1005 . . . . . . 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 1006 . . . . . . . 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 1039 . . . . . . . . . . 11  |-  ( ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  /\  r  e.  ZZ )  ->  D  e.  ZZ )
1716ad2antrr 488 . . . . . . . . . 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 9440 . . . . . . . . 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 11333 . . . . . . . 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 4055 . . . . . . 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 1007 . . . . . . . 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 9440 . . . . . . . . . 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 8416 . . . . . . . . . 10  |-  ( ( k  e.  CC  /\  D  e.  CC )  ->  ( -u k  x.  D )  =  ( k  x.  -u D
) )
2422, 18, 23syl2anc 411 . . . . . . . . 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 5933 . . . . . . . 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 2229 . . . . . . 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 5925 . . . . . . . . . . 11  |-  ( q  =  -u k  ->  (
q  x.  D )  =  ( -u k  x.  D ) )
2827oveq1d 5933 . . . . . . . . . 10  |-  ( q  =  -u k  ->  (
( q  x.  D
)  +  r )  =  ( ( -u k  x.  D )  +  r ) )
2928eqeq2d 2205 . . . . . . . . 9  |-  ( q  =  -u k  ->  ( N  =  ( (
q  x.  D )  +  r )  <->  N  =  ( ( -u k  x.  D )  +  r ) ) )
30293anbi3d 1329 . . . . . . . 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 2864 . . . . . . 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 1251 . . . . . 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 115 . . . . 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 2611 . . . 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 2596 . . 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 110 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  =  0 )  ->  D  =  0 )
38 simpl3 1004 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  =  0 )  ->  D  =/=  0
)
3937, 38pm2.21ddne 2447 . 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 1002 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  0  <  D )  ->  N  e.  ZZ )
41 simpl2 1003 . . . 4  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  0  <  D )  ->  D  e.  ZZ )
42 simpr 110 . . . 4  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  0  <  D )  -> 
0  <  D )
43 elnnz 9327 . . . 4  |-  ( D  e.  NN  <->  ( D  e.  ZZ  /\  0  < 
D ) )
4441, 42, 43sylanbrc 417 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  0  <  D )  ->  D  e.  NN )
45 divalglemnn 12059 . . 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 411 . 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 9359 . . 3  |-  ( D  e.  ZZ  ->  ( D  <  0  \/  D  =  0  \/  0  <  D ) )
48473ad2ant2 1021 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  ->  ( D  <  0  \/  D  =  0  \/  0  <  D ) )
4936, 39, 46, 48mpjao3dan 1318 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 104    \/ w3o 979    /\ w3a 980    = wceq 1364    e. wcel 2164    =/= wne 2364   E.wrex 2473   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   0cc0 7872    + caddc 7875    x. cmul 7877    < clt 8054    <_ cle 8055   -ucneg 8191   NNcn 8982   ZZcz 9317   abscabs 11141
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 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991
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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-frec 6444  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fl 10339  df-mod 10394  df-seqfrec 10519  df-exp 10610  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143
This theorem is referenced by:  divalglemeuneg  12064
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