Theorem divalglemnn 11877

Description: Lemma for divalg 11883. Existence for a positive denominator. (Contributed by Jim Kingdon, 30-Nov-2021.)

Assertion

Ref	Expression
divalglemnn	|- ( ( N e. ZZ /\ D e. NN ) -> 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 divalglemnn

Step	Hyp	Ref	Expression
1		zmodcl 10300	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. NN0 )
2	1	nn0zd 9332	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. ZZ )
3		znq 9583	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N / D ) e. QQ )
4	3	flqcld 10233	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. ZZ )
5	1	nn0ge0d 9191	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> 0 <_ ( N mod D ) )
6		zq 9585	. . . . 5 |- ( N e. ZZ -> N e. QQ )
7	6	adantr 274	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> N e. QQ )
8		nnq 9592	. . . . 5 |- ( D e. NN -> D e. QQ )
9	8	adantl 275	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. QQ )
10		nngt0 8903	. . . . 5 |- ( D e. NN -> 0 < D )
11	10	adantl 275	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> 0 < D )
12		modqlt 10289	. . . 4 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) < D )
13	7, 9, 11, 12	syl3anc 1233	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < D )
14		nnre 8885	. . . . 5 |- ( D e. NN -> D e. RR )
15	14	adantl 275	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. RR )
16		0red 7921	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> 0 e. RR )
17	16, 15, 11	ltled 8038	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> 0 <_ D )
18	15, 17	absidd 11131	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( abs ` D ) = D )
19	13, 18	breqtrrd 4017	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < ( abs ` D ) )
20	1	nn0cnd 9190	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. CC )
21	4	zcnd 9335	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. CC )
22		simpr 109	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> D e. NN )
23	22	nncnd 8892	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. CC )
24	21, 23	mulcld 7940	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( |_ ` ( N / D ) ) x. D ) e. CC )
25		modqvalr 10281	. . . . . 6 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) = ( N - ( ( |_ ` ( N / D ) ) x. D ) ) )
26	7, 9, 11, 25	syl3anc 1233	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( N - ( ( |_ ` ( N / D ) ) x. D ) ) )
27	26	oveq1d 5868	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( N mod D ) + ( ( |_ ` ( N / D ) ) x. D ) ) = ( ( N - ( ( |_ ` ( N / D ) ) x. D ) ) + ( ( |_ ` ( N / D ) ) x. D ) ) )
28		simpl 108	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> N e. ZZ )
29	28	zcnd 9335	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> N e. CC )
30	29, 24	npcand 8234	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( N - ( ( |_ ` ( N / D ) ) x. D ) ) + ( ( |_ ` ( N / D ) ) x. D ) ) = N )
31	27, 30	eqtr2d 2204	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> N = ( ( N mod D ) + ( ( |_ ` ( N / D ) ) x. D ) ) )
32	20, 24, 31	comraddd 8076	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> N = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) )
33		breq2 3993	. . . 4 |- ( r = ( N mod D ) -> ( 0 <_ r <-> 0 <_ ( N mod D ) ) )
34		breq1 3992	. . . 4 |- ( r = ( N mod D ) -> ( r < ( abs ` D ) <-> ( N mod D ) < ( abs ` D ) ) )
35		oveq2 5861	. . . . 5 |- ( r = ( N mod D ) -> ( ( q x. D ) + r ) = ( ( q x. D ) + ( N mod D ) ) )
36	35	eqeq2d 2182	. . . 4 |- ( r = ( N mod D ) -> ( N = ( ( q x. D ) + r ) <-> N = ( ( q x. D ) + ( N mod D ) ) ) )
37	33, 34, 36	3anbi123d 1307	. . 3 |- ( r = ( N mod D ) -> ( ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) <-> ( 0 <_ ( N mod D ) /\ ( N mod D ) < ( abs ` D ) /\ N = ( ( q x. D ) + ( N mod D ) ) ) ) )
38		oveq1 5860	. . . . . 6 |- ( q = ( |_ ` ( N / D ) ) -> ( q x. D ) = ( ( |_ ` ( N / D ) ) x. D ) )
39	38	oveq1d 5868	. . . . 5 |- ( q = ( |_ ` ( N / D ) ) -> ( ( q x. D ) + ( N mod D ) ) = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) )
40	39	eqeq2d 2182	. . . 4 |- ( q = ( |_ ` ( N / D ) ) -> ( N = ( ( q x. D ) + ( N mod D ) ) <-> N = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) ) )
41	40	3anbi3d 1313	. . 3 |- ( q = ( |_ ` ( N / D ) ) -> ( ( 0 <_ ( N mod D ) /\ ( N mod D ) < ( abs ` D ) /\ N = ( ( q x. D ) + ( N mod D ) ) ) <-> ( 0 <_ ( N mod D ) /\ ( N mod D ) < ( abs ` D ) /\ N = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) ) ) )
42	37, 41	rspc2ev 2849	. 2 |- ( ( ( N mod D ) e. ZZ /\ ( |_ ` ( N / D ) ) e. ZZ /\ ( 0 <_ ( N mod D ) /\ ( N mod D ) < ( abs ` D ) /\ N = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) ) ) -> E. r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )
43	2, 4, 5, 19, 32, 42	syl113anc 1245	1 |- ( ( N e. ZZ /\ D e. NN ) -> E. r e. ZZ E. q e. ZZ ( 0 <_ r /\ r < ( abs ` D ) /\ N = ( ( q x. D ) + r ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   RRcr 7773   0cc0 7774   + caddc 7777   x. cmul 7779   < clt 7954   <_ cle 7955   - cmin 8090   / cdiv 8589   NNcn 8878   ZZcz 9212   QQcq 9578   |_cfl 10224   mod cmo 10278   abscabs 10961

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963

This theorem is referenced by:  divalglemeunn  11880  divalglemex  11881