Theorem divalglemex 12087

Description: Lemma for divalg 12089. 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 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 )
3	2	znegcld 9450	. . . . 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 9448	. . . . . . 7 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> D e. RR )
6	5	lt0neg1d 8542	. . . . . 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 9336	. . . . 5 |- ( -u D e. NN <-> ( -u D e. ZZ /\ 0 < -u D ) )
9	3, 7, 8	sylanbrc 417	. . . 4 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> -u D e. NN )
10		divalglemnn 12083	. . . 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 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 )
13	12	znegcld 9450	. . . . . . 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 )
17	16	ad2antrr 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 )
18	17	zcnd 9449	. . . . . . . . 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 11354	. . . . . . . 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 4059	. . . . . . 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 ) )
22	12	zcnd 9449	. . . . . . . . . 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 8423	. . . . . . . . . 10 |- ( ( k e. CC /\ D e. CC ) -> ( -u k x. D ) = ( k x. -u D ) )
24	22, 18, 23	syl2anc 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 ) )
25	24	oveq1d 5937	. . . . . . . 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 2232	. . . . . . 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 5929	. . . . . . . . . . 11 |- ( q = -u k -> ( q x. D ) = ( -u k x. D ) )
28	27	oveq1d 5937	. . . . . . . . . 10 |- ( q = -u k -> ( ( q x. D ) + r ) = ( ( -u k x. D ) + r ) )
29	28	eqeq2d 2208	. . . . . . . . 9 |- ( q = -u k -> ( N = ( ( q x. D ) + r ) <-> N = ( ( -u k x. D ) + r ) ) )
30	29	3anbi3d 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 ) ) ) )
31	30	rspcev 2868	. . . . . . 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 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 ) ) )
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 2614	. . . 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 2599	. . 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 1004	. . 3 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D = 0 ) -> D =/= 0 )
39	37, 38	pm2.21ddne 2450	. 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 9336	. . . 4 |- ( D e. NN <-> ( D e. ZZ /\ 0 < D ) )
44	41, 42, 43	sylanbrc 417	. . 3 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ 0 < D ) -> D e. NN )
45		divalglemnn 12083	. . 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 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 9368	. . 3 |- ( D e. ZZ -> ( D < 0 \/ D = 0 \/ 0 < D ) )
48	47	3ad2ant2 1021	. 2 |- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> ( D < 0 \/ D = 0 \/ 0 < D ) )
49	36, 39, 46, 48	mpjao3dan 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ w3o 979   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   E.wrex 2476   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   + caddc 7882   x. cmul 7884   < clt 8061   <_ cle 8062   -ucneg 8198   NNcn 8990   ZZcz 9326   abscabs 11162

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-fl 10360  df-mod 10415  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164

This theorem is referenced by:  divalglemeuneg  12088