Theorem divalglemex 12689

Description: Lemma for divalg 12691. 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.

Step	Hyp	Ref	Expression
1		simpl1 1031	. . . 4 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> N e. ZZ )
2		simpl2 1032	. . . . . 6 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> D e. ZZ )
3	2	znegcld 9770	. . . . 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 )
5	2	zred 9768	. . . . . . 7 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> D e. RR )
6	5	lt0neg1d 8843	. . . . . 6 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> ( D < 0 <-> 0 < -u D ) )
7	4, 6	mpbid 147	. . . . 5 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> 0 < -u D )
8		elnnz 9654	. . . . 5 |- ( -u D e. NN <-> ( -u D e. ZZ /\ 0 < -u D ) )
9	3, 7, 8	sylanbrc 421	. . . 4 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> -u D e. NN )
10		divalglemnn 12685	. . . 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 ) ) )
11	1, 9, 10	syl2anc 415	. . 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 533	. . . . . . . 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 )
13	12	znegcld 9770	. . . . . . 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 1034	. . . . . . 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 1035	. . . . . . . 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 1068	. . . . . . . . . . 11 |- ( ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) /\ r e. ZZ ) -> D e. ZZ )
17	16	ad2antrr 492	. . . . . . . . . 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 )
18	17	zcnd 9769	. . . . . . . . 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 )
19	18	absnegd 11955	. . . . . . . 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 ) )
20	15, 19	breqtrd 4156	. . . . . . 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 1036	. . . . . . . 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 ) )
22	12	zcnd 9769	. . . . . . . . . 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 8724	. . . . . . . . . 10 |- ( ( k e. CC /\ D e. CC ) -> ( -u k x. D ) = ( k x. -u D ) )
24	22, 18, 23	syl2anc 415	. . . . . . . . 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 ) )
25	24	oveq1d 6100	. . . . . . . 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 ) )
26	21, 25	eqtr4d 2274	. . . . . . 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 6092	. . . . . . . . . . 11 |- ( q = -u k -> ( q x. D ) = ( -u k x. D ) )
28	27	oveq1d 6100	. . . . . . . . . 10 |- ( q = -u k -> ( ( q x. D ) + r ) = ( ( -u k x. D ) + r ) )
29	28	eqeq2d 2250	. . . . . . . . 9 |- ( q = -u k -> ( N = ( ( q x. D ) + r ) <-> N = ( ( -u k x. D ) + r ) ) )
30	29	3anbi3d 1359	. . . . . . . 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 ) ) ) )
31	30	rspcev 2929	. . . . . . 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 ) ) )
32	13, 14, 20, 26, 31	syl13anc 1280	. . . . . 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 ) ) )
33	32	ex 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 ) ) ) )
34	33	rexlimdva 2668	. . . 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 ) ) ) )
35	34	reximdva 2652	. . 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 ) ) ) )
36	11, 35	mpd 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 1033	. . 3 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D = 0 ) -> D =/= 0 )
39	37, 38	pm2.21ddne 2503	. 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 1031	. . 3 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ 0 < D ) -> N e. ZZ )
41		simpl2 1032	. . . 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 9654	. . . 4 |- ( D e. NN <-> ( D e. ZZ /\ 0 < D ) )
44	41, 42, 43	sylanbrc 421	. . 3 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ 0 < D ) -> D e. NN )
45		divalglemnn 12685	. . 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 ) ) )
46	40, 44, 45	syl2anc 415	. 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 9686	. . 3 |- ( D e. ZZ -> ( D < 0 \/ D = 0 \/ 0 < D ) )
48	47	3ad2ant2 1050	. 2 |- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> ( D < 0 \/ D = 0 \/ 0 < D ) )
49	36, 39, 46, 48	mpjao3dan 1348	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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ w3o 1008   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   E.wrex 2529   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   0cc0 8179   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   -ucneg 8498   NNcn 9304   ZZcz 9644   abscabs 11763

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765

This theorem is used by:  divalglemeuneg  12690