Theorem divalglemex 11619

Description: Lemma for divalg 11621. 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 984	. . . 4 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> N e. ZZ )
2		simpl2 985	. . . . . 6 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> D e. ZZ )
3	2	znegcld 9175	. . . . 5 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> -u D e. ZZ )
4		simpr 109	. . . . . 6 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> D < 0 )
5	2	zred 9173	. . . . . . 7 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> D e. RR )
6	5	lt0neg1d 8277	. . . . . 6 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> ( D < 0 <-> 0 < -u D ) )
7	4, 6	mpbid 146	. . . . 5 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> 0 < -u D )
8		elnnz 9064	. . . . 5 |- ( -u D e. NN <-> ( -u D e. ZZ /\ 0 < -u D ) )
9	3, 7, 8	sylanbrc 413	. . . 4 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> -u D e. NN )
10		divalglemnn 11615	. . . 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 408	. . 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 519	. . . . . . . 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 9175	. . . . . . 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 987	. . . . . . 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 988	. . . . . . . 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 1021	. . . . . . . . . . 11 |- ( ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) /\ r e. ZZ ) -> D e. ZZ )
17	16	ad2antrr 479	. . . . . . . . . 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 9174	. . . . . . . . 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 10961	. . . . . . . 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 3954	. . . . . . 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 989	. . . . . . . 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 9174	. . . . . . . . . 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 8159	. . . . . . . . . 10 |- ( ( k e. CC /\ D e. CC ) -> ( -u k x. D ) = ( k x. -u D ) )
24	22, 18, 23	syl2anc 408	. . . . . . . . 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 5789	. . . . . . . 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 2175	. . . . . . 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 5781	. . . . . . . . . . 11 |- ( q = -u k -> ( q x. D ) = ( -u k x. D ) )
28	27	oveq1d 5789	. . . . . . . . . 10 |- ( q = -u k -> ( ( q x. D ) + r ) = ( ( -u k x. D ) + r ) )
29	28	eqeq2d 2151	. . . . . . . . 9 |- ( q = -u k -> ( N = ( ( q x. D ) + r ) <-> N = ( ( -u k x. D ) + r ) ) )
30	29	3anbi3d 1296	. . . . . . . 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 2789	. . . . . . 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 1218	. . . . . 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 114	. . . . 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 2549	. . . 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 2534	. . 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 109	. . 3 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D = 0 ) -> D = 0 )
38		simpl3 986	. . 3 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D = 0 ) -> D =/= 0 )
39	37, 38	pm2.21ddne 2391	. 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 984	. . 3 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ 0 < D ) -> N e. ZZ )
41		simpl2 985	. . . 4 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ 0 < D ) -> D e. ZZ )
42		simpr 109	. . . 4 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ 0 < D ) -> 0 < D )
43		elnnz 9064	. . . 4 |- ( D e. NN <-> ( D e. ZZ /\ 0 < D ) )
44	41, 42, 43	sylanbrc 413	. . 3 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ 0 < D ) -> D e. NN )
45		divalglemnn 11615	. . 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 408	. 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 9096	. . 3 |- ( D e. ZZ -> ( D < 0 \/ D = 0 \/ 0 < D ) )
48	47	3ad2ant2 1003	. 2 |- ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) -> ( D < 0 \/ D = 0 \/ 0 < D ) )
49	36, 39, 46, 48	mpjao3dan 1285	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 103   \/ w3o 961   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2308   E.wrex 2417   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   0cc0 7620   + caddc 7623   x. cmul 7625   < clt 7800   <_ cle 7801   -ucneg 7934   NNcn 8720   ZZcz 9054   abscabs 10769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771

This theorem is referenced by:  divalglemeuneg  11620