Theorem divalglemex 12714

Description: Lemma for divalg 12716. 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 9781	. . . . 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 9779	. . . . . . 7 |- ( ( ( N e. ZZ /\ D e. ZZ /\ D =/= 0 ) /\ D < 0 ) -> D e. RR )
6	5	lt0neg1d 8850	. . . . . 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 9665	. . . . 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 12710	. . . 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 9781	. . . . . . 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 9780	. . . . . . . . 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 11978	. . . . . . . 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 4157	. . . . . . 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 9780	. . . . . . . . . 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 8731	. . . . . . . . . 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 6103	. . . . . . . 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 6095	. . . . . . . . . . 11 |- ( q = -u k -> ( q x. D ) = ( -u k x. D ) )
28	27	oveq1d 6103	. . . . . . . . . 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 9665	. . . 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 12710	. . 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 9697	. . 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 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   0cc0 8185   + caddc 8188   x. cmul 8190   < clt 8366   <_ cle 8367   -ucneg 8505   NNcn 9313   ZZcz 9655   abscabs 11785

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fl 10722  df-mod 10781  df-seqfrec 10906  df-exp 10997  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787

This theorem is used by:  divalglemeuneg  12715