Theorem divalglemnn 12542

Description: Lemma for divalg 12548. 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 10652	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. NN0 )
2	1	nn0zd 9644	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. ZZ )
3		znq 9902	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N / D ) e. QQ )
4	3	flqcld 10583	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. ZZ )
5	1	nn0ge0d 9502	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> 0 <_ ( N mod D ) )
6		zq 9904	. . . . 5 |- ( N e. ZZ -> N e. QQ )
7	6	adantr 276	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> N e. QQ )
8		nnq 9911	. . . . 5 |- ( D e. NN -> D e. QQ )
9	8	adantl 277	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. QQ )
10		nngt0 9210	. . . . 5 |- ( D e. NN -> 0 < D )
11	10	adantl 277	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> 0 < D )
12		modqlt 10641	. . . 4 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) < D )
13	7, 9, 11, 12	syl3anc 1274	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < D )
14		nnre 9192	. . . . 5 |- ( D e. NN -> D e. RR )
15	14	adantl 277	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. RR )
16		0red 8223	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> 0 e. RR )
17	16, 15, 11	ltled 8340	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> 0 <_ D )
18	15, 17	absidd 11790	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( abs ` D ) = D )
19	13, 18	breqtrrd 4121	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < ( abs ` D ) )
20	1	nn0cnd 9501	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. CC )
21	4	zcnd 9647	. . . 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 9199	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> D e. CC )
24	21, 23	mulcld 8242	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( |_ ` ( N / D ) ) x. D ) e. CC )
25		modqvalr 10633	. . . . . 6 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) = ( N - ( ( |_ ` ( N / D ) ) x. D ) ) )
26	7, 9, 11, 25	syl3anc 1274	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( N - ( ( |_ ` ( N / D ) ) x. D ) ) )
27	26	oveq1d 6043	. . . 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 9647	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> N e. CC )
30	29, 24	npcand 8536	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( N - ( ( |_ ` ( N / D ) ) x. D ) ) + ( ( |_ ` ( N / D ) ) x. D ) ) = N )
31	27, 30	eqtr2d 2265	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> N = ( ( N mod D ) + ( ( |_ ` ( N / D ) ) x. D ) ) )
32	20, 24, 31	comraddd 8378	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> N = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) )
33		breq2 4097	. . . 4 |- ( r = ( N mod D ) -> ( 0 <_ r <-> 0 <_ ( N mod D ) ) )
34		breq1 4096	. . . 4 |- ( r = ( N mod D ) -> ( r < ( abs ` D ) <-> ( N mod D ) < ( abs ` D ) ) )
35		oveq2 6036	. . . . 5 |- ( r = ( N mod D ) -> ( ( q x. D ) + r ) = ( ( q x. D ) + ( N mod D ) ) )
36	35	eqeq2d 2243	. . . 4 |- ( r = ( N mod D ) -> ( N = ( ( q x. D ) + r ) <-> N = ( ( q x. D ) + ( N mod D ) ) ) )
37	33, 34, 36	3anbi123d 1349	. . 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 6035	. . . . . 6 |- ( q = ( |_ ` ( N / D ) ) -> ( q x. D ) = ( ( |_ ` ( N / D ) ) x. D ) )
39	38	oveq1d 6043	. . . . 5 |- ( q = ( |_ ` ( N / D ) ) -> ( ( q x. D ) + ( N mod D ) ) = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) )
40	39	eqeq2d 2243	. . . 4 |- ( q = ( |_ ` ( N / D ) ) -> ( N = ( ( q x. D ) + ( N mod D ) ) <-> N = ( ( ( |_ ` ( N / D ) ) x. D ) + ( N mod D ) ) ) )
41	40	3anbi3d 1355	. . 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 2926	. 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 1286	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 1005   = wceq 1398   e. wcel 2202   E.wrex 2512   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8074   0cc0 8075   + caddc 8078   x. cmul 8080   < clt 8256   <_ cle 8257   - cmin 8392   / cdiv 8894   NNcn 9185   ZZcz 9523   QQcq 9897   |_cfl 10574   mod cmo 10630   abscabs 11620

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622

This theorem is referenced by:  divalglemeunn  12545  divalglemex  12546