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Theorem divalglemex 11897
Description: Lemma for divalg 11899. 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 1000 . . . 4  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  N  e.  ZZ )
2 simpl2 1001 . . . . . 6  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  D  e.  ZZ )
32znegcld 9353 . . . . 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 9351 . . . . . . 7  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  <  0 )  ->  D  e.  RR )
65lt0neg1d 8449 . . . . . 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 9239 . . . . 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 11893 . . . 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 9353 . . . . . . 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 1003 . . . . . . 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 1004 . . . . . . . 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 1037 . . . . . . . . . . 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 9352 . . . . . . . . 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 11169 . . . . . . . 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 4026 . . . . . . 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 1005 . . . . . . . 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 9352 . . . . . . . . . 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 8331 . . . . . . . . . 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 5883 . . . . . . . 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 2213 . . . . . . 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 5875 . . . . . . . . . . 11  |-  ( q  =  -u k  ->  (
q  x.  D )  =  ( -u k  x.  D ) )
2827oveq1d 5883 . . . . . . . . . 10  |-  ( q  =  -u k  ->  (
( q  x.  D
)  +  r )  =  ( ( -u k  x.  D )  +  r ) )
2928eqeq2d 2189 . . . . . . . . 9  |-  ( q  =  -u k  ->  ( N  =  ( (
q  x.  D )  +  r )  <->  N  =  ( ( -u k  x.  D )  +  r ) ) )
30293anbi3d 1318 . . . . . . . 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 2841 . . . . . . 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 1240 . . . . . 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 2594 . . . 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 2579 . . 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 1002 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  D  =  0 )  ->  D  =/=  0
)
3937, 38pm2.21ddne 2430 . 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 1000 . . 3  |-  ( ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  /\  0  <  D )  ->  N  e.  ZZ )
41 simpl2 1001 . . . 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 9239 . . . 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 11893 . . 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 9271 . . 3  |-  ( D  e.  ZZ  ->  ( D  <  0  \/  D  =  0  \/  0  <  D ) )
48473ad2ant2 1019 . 2  |-  ( ( N  e.  ZZ  /\  D  e.  ZZ  /\  D  =/=  0 )  ->  ( D  <  0  \/  D  =  0  \/  0  <  D ) )
4936, 39, 46, 48mpjao3dan 1307 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 977    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347   E.wrex 2456   class class class wbr 4000   ` cfv 5211  (class class class)co 5868   CCcc 7787   0cc0 7789    + caddc 7792    x. cmul 7794    < clt 7969    <_ cle 7970   -ucneg 8106   NNcn 8895   ZZcz 9229   abscabs 10977
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 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907  ax-arch 7908
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 4289  df-po 4292  df-iso 4293  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606  df-inn 8896  df-2 8954  df-n0 9153  df-z 9230  df-uz 9505  df-q 9596  df-rp 9628  df-fl 10243  df-mod 10296  df-seqfrec 10419  df-exp 10493  df-cj 10822  df-re 10823  df-im 10824  df-rsqrt 10978  df-abs 10979
This theorem is referenced by:  divalglemeuneg  11898
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