Theorem divalglemnn 12469

Description: Lemma for divalg 12475. 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 10596	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. NN0 )
2	1	nn0zd 9590	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. ZZ )
3		znq 9848	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N / D ) e. QQ )
4	3	flqcld 10527	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. ZZ )
5	1	nn0ge0d 9448	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> 0 <_ ( N mod D ) )
6		zq 9850	. . . . 5 |- ( N e. ZZ -> N e. QQ )
7	6	adantr 276	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> N e. QQ )
8		nnq 9857	. . . . 5 |- ( D e. NN -> D e. QQ )
9	8	adantl 277	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. QQ )
10		nngt0 9158	. . . . 5 |- ( D e. NN -> 0 < D )
11	10	adantl 277	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> 0 < D )
12		modqlt 10585	. . . 4 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) < D )
13	7, 9, 11, 12	syl3anc 1271	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < D )
14		nnre 9140	. . . . 5 |- ( D e. NN -> D e. RR )
15	14	adantl 277	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. RR )
16		0red 8170	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> 0 e. RR )
17	16, 15, 11	ltled 8288	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> 0 <_ D )
18	15, 17	absidd 11718	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( abs ` D ) = D )
19	13, 18	breqtrrd 4114	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < ( abs ` D ) )
20	1	nn0cnd 9447	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. CC )
21	4	zcnd 9593	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. CC )
22		simpr 110	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> D e. NN )
23	22	nncnd 9147	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. CC )
24	21, 23	mulcld 8190	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( |_ ` ( N / D ) ) x. D ) e. CC )
25		modqvalr 10577	. . . . . 6 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) = ( N - ( ( |_ ` ( N / D ) ) x. D ) ) )
26	7, 9, 11, 25	syl3anc 1271	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( N - ( ( |_ ` ( N / D ) ) x. D ) ) )
27	26	oveq1d 6028	. . . 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 109	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> N e. ZZ )
29	28	zcnd 9593	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> N e. CC )
30	29, 24	npcand 8484	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( N - ( ( |_ ` ( N / D ) ) x. D ) ) + ( ( |_ ` ( N / D ) ) x. D ) ) = N )
31	27, 30	eqtr2d 2263	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> N = ( ( N mod D ) + ( ( |_ ` ( N / D ) ) x. D ) ) )
32	20, 24, 31	comraddd 8326	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> N = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) )
33		breq2 4090	. . . 4 |- ( r = ( N mod D ) -> ( 0 <_ r <-> 0 <_ ( N mod D ) ) )
34		breq1 4089	. . . 4 |- ( r = ( N mod D ) -> ( r < ( abs ` D ) <-> ( N mod D ) < ( abs ` D ) ) )
35		oveq2 6021	. . . . 5 |- ( r = ( N mod D ) -> ( ( q x. D ) + r ) = ( ( q x. D ) + ( N mod D ) ) )
36	35	eqeq2d 2241	. . . 4 |- ( r = ( N mod D ) -> ( N = ( ( q x. D ) + r ) <-> N = ( ( q x. D ) + ( N mod D ) ) ) )
37	33, 34, 36	3anbi123d 1346	. . 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 6020	. . . . . 6 |- ( q = ( |_ ` ( N / D ) ) -> ( q x. D ) = ( ( |_ ` ( N / D ) ) x. D ) )
39	38	oveq1d 6028	. . . . 5 |- ( q = ( |_ ` ( N / D ) ) -> ( ( q x. D ) + ( N mod D ) ) = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) )
40	39	eqeq2d 2241	. . . 4 |- ( q = ( |_ ` ( N / D ) ) -> ( N = ( ( q x. D ) + ( N mod D ) ) <-> N = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) ) )
41	40	3anbi3d 1352	. . 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 2923	. 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 1283	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 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   RRcr 8021   0cc0 8022   + caddc 8025   x. cmul 8027   < clt 8204   <_ cle 8205   - cmin 8340   / cdiv 8842   NNcn 9133   ZZcz 9469   QQcq 9843   |_cfl 10518   mod cmo 10574   abscabs 11548

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139  ax-pre-mulext 8140  ax-arch 8141

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-div 8843  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746  df-q 9844  df-rp 9879  df-fl 10520  df-mod 10575  df-seqfrec 10700  df-exp 10791  df-cj 11393  df-re 11394  df-im 11395  df-rsqrt 11549  df-abs 11550

This theorem is referenced by:  divalglemeunn  12472  divalglemex  12473