Theorem divalgmod 12613

Description: The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor (compare divalg2 12612 and divalgb 12611). This demonstration theorem justifies the use of mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.) (Revised by AV, 21-Aug-2021.)

Assertion

Ref	Expression
divalgmod	|- ( ( N e. ZZ /\ D e. NN ) -> ( R = ( N mod D ) <-> ( R e. NN0 /\ ( R < D /\ D || ( N - R ) ) ) ) )

Proof of Theorem divalgmod

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		zq 9958	. . . . . . . 8 |- ( N e. ZZ -> N e. QQ )
2	1	adantr 276	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> N e. QQ )
3		nnq 9965	. . . . . . . 8 |- ( D e. NN -> D e. QQ )
4	3	adantl 277	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> D e. QQ )
5		simpr 110	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN ) -> D e. NN )
6	5	nngt0d 9281	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> 0 < D )
7	2, 4, 6	modqcld 10690	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. QQ )
8		snidg 3718	. . . . . 6 |- ( ( N mod D ) e. QQ -> ( N mod D ) e. { ( N mod D ) } )
9	7, 8	syl 14	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. { ( N mod D ) } )
10		eleq1 2295	. . . . 5 |- ( R = ( N mod D ) -> ( R e. { ( N mod D ) } <-> ( N mod D ) e. { ( N mod D ) } ) )
11	9, 10	syl5ibrcom 157	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> ( R = ( N mod D ) -> R e. { ( N mod D ) } ) )
12		elsni 3707	. . . 4 |- ( R e. { ( N mod D ) } -> R = ( N mod D ) )
13	11, 12	impbid1 142	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( R = ( N mod D ) <-> R e. { ( N mod D ) } ) )
14		modqlt 10695	. . . . . . . . 9 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) < D )
15	2, 4, 6, 14	syl3anc 1274	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < D )
16		znq 9956	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( N / D ) e. QQ )
17	16	flqcld 10637	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. ZZ )
18		nnz 9596	. . . . . . . . . 10 |- ( D e. NN -> D e. ZZ )
19	18	adantl 277	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> D e. ZZ )
20		zmodcl 10706	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. NN0 )
21	20	nn0zd 9698	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. ZZ )
22		zsubcl 9618	. . . . . . . . . 10 |- ( ( N e. ZZ /\ ( N mod D ) e. ZZ ) -> ( N - ( N mod D ) ) e. ZZ )
23	21, 22	syldan 282	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( N - ( N mod D ) ) e. ZZ )
24		nncn 9245	. . . . . . . . . . . 12 |- ( D e. NN -> D e. CC )
25	24	adantl 277	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> D e. CC )
26	17	zcnd 9701	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. CC )
27	25, 26	mulcomd 8295	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) = ( ( |_ ` ( N / D ) ) x. D ) )
28		modqval 10686	. . . . . . . . . . . 12 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) = ( N - ( D x. ( |_ ` ( N / D ) ) ) ) )
29	2, 4, 6, 28	syl3anc 1274	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( N - ( D x. ( |_ ` ( N / D ) ) ) ) )
30	20	nn0cnd 9555	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. CC )
31		zmulcl 9631	. . . . . . . . . . . . . 14 |- ( ( D e. ZZ /\ ( |_ ` ( N / D ) ) e. ZZ ) -> ( D x. ( |_ ` ( N / D ) ) ) e. ZZ )
32	18, 17, 31	syl2an2 598	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) e. ZZ )
33	32	zcnd 9701	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) e. CC )
34		zcn 9582	. . . . . . . . . . . . 13 |- ( N e. ZZ -> N e. CC )
35	34	adantr 276	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> N e. CC )
36	30, 33, 35	subexsub 8645	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( N mod D ) = ( N - ( D x. ( |_ ` ( N / D ) ) ) ) <-> ( D x. ( |_ ` ( N / D ) ) ) = ( N - ( N mod D ) ) ) )
37	29, 36	mpbid 147	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) = ( N - ( N mod D ) ) )
38	27, 37	eqtr3d 2267	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( |_ ` ( N / D ) ) x. D ) = ( N - ( N mod D ) ) )
39		dvds0lem 12487	. . . . . . . . 9 |- ( ( ( ( |_ ` ( N / D ) ) e. ZZ /\ D e. ZZ /\ ( N - ( N mod D ) ) e. ZZ ) /\ ( ( |_ ` ( N / D ) ) x. D ) = ( N - ( N mod D ) ) ) -> D || ( N - ( N mod D ) ) )
40	17, 19, 23, 38, 39	syl31anc 1277	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN ) -> D || ( N - ( N mod D ) ) )
41		divalg2 12612	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> E! z e. NN0 ( z < D /\ D || ( N - z ) ) )
42		breq1 4112	. . . . . . . . . . 11 |- ( z = ( N mod D ) -> ( z < D <-> ( N mod D ) < D ) )
43		oveq2 6058	. . . . . . . . . . . 12 |- ( z = ( N mod D ) -> ( N - z ) = ( N - ( N mod D ) ) )
44	43	breq2d 4121	. . . . . . . . . . 11 |- ( z = ( N mod D ) -> ( D || ( N - z ) <-> D || ( N - ( N mod D ) ) ) )
45	42, 44	anbi12d 473	. . . . . . . . . 10 |- ( z = ( N mod D ) -> ( ( z < D /\ D || ( N - z ) ) <-> ( ( N mod D ) < D /\ D || ( N - ( N mod D ) ) ) ) )
46	45	riota2 6027	. . . . . . . . 9 |- ( ( ( N mod D ) e. NN0 /\ E! z e. NN0 ( z < D /\ D || ( N - z ) ) ) -> ( ( ( N mod D ) < D /\ D || ( N - ( N mod D ) ) ) <-> ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) = ( N mod D ) ) )
47	20, 41, 46	syl2anc 411	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( ( N mod D ) < D /\ D || ( N - ( N mod D ) ) ) <-> ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) = ( N mod D ) ) )
48	15, 40, 47	mpbi2and 952	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) = ( N mod D ) )
49	48	eqcomd 2238	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) )
50	49	sneqd 3702	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> { ( N mod D ) } = { ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) } )
51		snriota 6035	. . . . . 6 |- ( E! z e. NN0 ( z < D /\ D || ( N - z ) ) -> { z e. NN0 | ( z < D /\ D || ( N - z ) ) } = { ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) } )
52	41, 51	syl 14	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> { z e. NN0 | ( z < D /\ D || ( N - z ) ) } = { ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) } )
53	50, 52	eqtr4d 2268	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> { ( N mod D ) } = { z e. NN0 | ( z < D /\ D || ( N - z ) ) } )
54	53	eleq2d 2302	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( R e. { ( N mod D ) } <-> R e. { z e. NN0 | ( z < D /\ D || ( N - z ) ) } ) )
55	13, 54	bitrd 188	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( R = ( N mod D ) <-> R e. { z e. NN0 | ( z < D /\ D || ( N - z ) ) } ) )
56		breq1 4112	. . . 4 |- ( z = R -> ( z < D <-> R < D ) )
57		oveq2 6058	. . . . 5 |- ( z = R -> ( N - z ) = ( N - R ) )
58	57	breq2d 4121	. . . 4 |- ( z = R -> ( D || ( N - z ) <-> D || ( N - R ) ) )
59	56, 58	anbi12d 473	. . 3 |- ( z = R -> ( ( z < D /\ D || ( N - z ) ) <-> ( R < D /\ D || ( N - R ) ) ) )
60	59	elrab 2973	. 2 |- ( R e. { z e. NN0 | ( z < D /\ D || ( N - z ) ) } <-> ( R e. NN0 /\ ( R < D /\ D || ( N - R ) ) ) )
61	55, 60	bitrdi 196	1 |- ( ( N e. ZZ /\ D e. NN ) -> ( R = ( N mod D ) <-> ( R e. NN0 /\ ( R < D /\ D || ( N - R ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   E!wreu 2522   {crab 2524   {csn 3689   class class class wbr 4109   ` cfv 5352   iota_crio 6002  (class class class)co 6050   CCcc 8125   0cc0 8127   x. cmul 8132   < clt 8308   - cmin 8444   / cdiv 8946   NNcn 9237   NN0cn0 9496   ZZcz 9577   QQcq 9951   |_cfl 10628   mod cmo 10684   || cdvds 12473

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474

This theorem is referenced by:  divalgmodcl  12614