Theorem divalglemex 12104

Description: Lemma for divalg 12106. 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 9467	. . . . 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 9465	. . . . . . 7 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> D e. RR )
6	5	lt0neg1d 8559	. . . . . 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 9353	. . . . 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 12100	. . . 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 9467	. . . . . . 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 9466	. . . . . . . . 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 11371	. . . . . . . 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 4060	. . . . . . 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 9466	. . . . . . . . . 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 8440	. . . . . . . . . 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 5940	. . . . . . . 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 5932	. . . . . . . . . . 11 |- ( q = -u k -> ( q x. D ) = ( -u k x. D ) )
28	27	oveq1d 5940	. . . . . . . . . 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 9353	. . . 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 12100	. . 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 9385	. . 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 4034   ` cfv 5259  (class class class)co 5925   CCcc 7894   0cc0 7896   + caddc 7899   x. cmul 7901   < clt 8078   <_ cle 8079   -ucneg 8215   NNcn 9007   ZZcz 9343   abscabs 11179

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015

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 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fl 10377  df-mod 10432  df-seqfrec 10557  df-exp 10648  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181

This theorem is referenced by:  divalglemeuneg  12105