Theorem divalglemnn 11782

Description: Lemma for divalg 11788. 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 10221	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. NN0 )
2	1	nn0zd 9263	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. ZZ )
3		znq 9511	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N / D ) e. QQ )
4	3	flqcld 10154	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. ZZ )
5	1	nn0ge0d 9125	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> 0 <_ ( N mod D ) )
6		zq 9513	. . . . 5 |- ( N e. ZZ -> N e. QQ )
7	6	adantr 274	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> N e. QQ )
8		nnq 9520	. . . . 5 |- ( D e. NN -> D e. QQ )
9	8	adantl 275	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. QQ )
10		nngt0 8837	. . . . 5 |- ( D e. NN -> 0 < D )
11	10	adantl 275	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> 0 < D )
12		modqlt 10210	. . . 4 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) < D )
13	7, 9, 11, 12	syl3anc 1217	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < D )
14		nnre 8819	. . . . 5 |- ( D e. NN -> D e. RR )
15	14	adantl 275	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. RR )
16		0red 7858	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> 0 e. RR )
17	16, 15, 11	ltled 7973	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> 0 <_ D )
18	15, 17	absidd 11044	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( abs ` D ) = D )
19	13, 18	breqtrrd 3988	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < ( abs ` D ) )
20	1	nn0cnd 9124	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. CC )
21	4	zcnd 9266	. . . 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 8826	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. CC )
24	21, 23	mulcld 7877	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( |_ ` ( N / D ) ) x. D ) e. CC )
25		modqvalr 10202	. . . . . 6 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) = ( N - ( ( |_ ` ( N / D ) ) x. D ) ) )
26	7, 9, 11, 25	syl3anc 1217	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( N - ( ( |_ ` ( N / D ) ) x. D ) ) )
27	26	oveq1d 5829	. . . 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 9266	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> N e. CC )
30	29, 24	npcand 8169	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( N - ( ( |_ ` ( N / D ) ) x. D ) ) + ( ( |_ ` ( N / D ) ) x. D ) ) = N )
31	27, 30	eqtr2d 2188	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> N = ( ( N mod D ) + ( ( |_ ` ( N / D ) ) x. D ) ) )
32	20, 24, 31	comraddd 8011	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> N = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) )
33		breq2 3965	. . . 4 |- ( r = ( N mod D ) -> ( 0 <_ r <-> 0 <_ ( N mod D ) ) )
34		breq1 3964	. . . 4 |- ( r = ( N mod D ) -> ( r < ( abs ` D ) <-> ( N mod D ) < ( abs ` D ) ) )
35		oveq2 5822	. . . . 5 |- ( r = ( N mod D ) -> ( ( q x. D ) + r ) = ( ( q x. D ) + ( N mod D ) ) )
36	35	eqeq2d 2166	. . . 4 |- ( r = ( N mod D ) -> ( N = ( ( q x. D ) + r ) <-> N = ( ( q x. D ) + ( N mod D ) ) ) )
37	33, 34, 36	3anbi123d 1291	. . 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 5821	. . . . . 6 |- ( q = ( |_ ` ( N / D ) ) -> ( q x. D ) = ( ( |_ ` ( N / D ) ) x. D ) )
39	38	oveq1d 5829	. . . . 5 |- ( q = ( |_ ` ( N / D ) ) -> ( ( q x. D ) + ( N mod D ) ) = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) )
40	39	eqeq2d 2166	. . . 4 |- ( q = ( |_ ` ( N / D ) ) -> ( N = ( ( q x. D ) + ( N mod D ) ) <-> N = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) ) )
41	40	3anbi3d 1297	. . 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 2828	. 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 1229	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 963   = wceq 1332   e. wcel 2125   E.wrex 2433   class class class wbr 3961   ` cfv 5163  (class class class)co 5814   RRcr 7710   0cc0 7711   + caddc 7714   x. cmul 7716   < clt 7891   <_ cle 7892   - cmin 8025   / cdiv 8524   NNcn 8812   ZZcz 9146   QQcq 9506   |_cfl 10145   mod cmo 10199   abscabs 10874

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829  ax-arch 7830

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-recs 6242  df-frec 6328  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-inn 8813  df-2 8871  df-n0 9070  df-z 9147  df-uz 9419  df-q 9507  df-rp 9539  df-fl 10147  df-mod 10200  df-seqfrec 10323  df-exp 10397  df-cj 10719  df-re 10720  df-im 10721  df-rsqrt 10875  df-abs 10876

This theorem is referenced by:  divalglemeunn  11785  divalglemex  11786