Theorem divalgmod 12453

Description: The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor (compare divalg2 12452 and divalgb 12451). 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 9833	. . . . . . . 8 |- ( N e. ZZ -> N e. QQ )
2	1	adantr 276	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> N e. QQ )
3		nnq 9840	. . . . . . . 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 9165	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> 0 < D )
7	2, 4, 6	modqcld 10562	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. QQ )
8		snidg 3695	. . . . . 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 2292	. . . . 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 3684	. . . 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 10567	. . . . . . . . 9 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) < D )
15	2, 4, 6, 14	syl3anc 1271	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < D )
16		znq 9831	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( N / D ) e. QQ )
17	16	flqcld 10509	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. ZZ )
18		nnz 9476	. . . . . . . . . 10 |- ( D e. NN -> D e. ZZ )
19	18	adantl 277	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> D e. ZZ )
20		zmodcl 10578	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. NN0 )
21	20	nn0zd 9578	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. ZZ )
22		zsubcl 9498	. . . . . . . . . 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 9129	. . . . . . . . . . . 12 |- ( D e. NN -> D e. CC )
25	24	adantl 277	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> D e. CC )
26	17	zcnd 9581	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. CC )
27	25, 26	mulcomd 8179	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) = ( ( |_ ` ( N / D ) ) x. D ) )
28		modqval 10558	. . . . . . . . . . . 12 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) = ( N - ( D x. ( |_ ` ( N / D ) ) ) ) )
29	2, 4, 6, 28	syl3anc 1271	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( N - ( D x. ( |_ ` ( N / D ) ) ) ) )
30	20	nn0cnd 9435	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. CC )
31		zmulcl 9511	. . . . . . . . . . . . . 14 |- ( ( D e. ZZ /\ ( |_ ` ( N / D ) ) e. ZZ ) -> ( D x. ( |_ ` ( N / D ) ) ) e. ZZ )
32	18, 17, 31	syl2an2 596	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) e. ZZ )
33	32	zcnd 9581	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) e. CC )
34		zcn 9462	. . . . . . . . . . . . 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 8529	. . . . . . . . . . 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 2264	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( |_ ` ( N / D ) ) x. D ) = ( N - ( N mod D ) ) )
39		dvds0lem 12327	. . . . . . . . 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 1274	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN ) -> D || ( N - ( N mod D ) ) )
41		divalg2 12452	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> E! z e. NN0 ( z < D /\ D || ( N - z ) ) )
42		breq1 4086	. . . . . . . . . . 11 |- ( z = ( N mod D ) -> ( z < D <-> ( N mod D ) < D ) )
43		oveq2 6015	. . . . . . . . . . . 12 |- ( z = ( N mod D ) -> ( N - z ) = ( N - ( N mod D ) ) )
44	43	breq2d 4095	. . . . . . . . . . 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 5984	. . . . . . . . 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 949	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) = ( N mod D ) )
49	48	eqcomd 2235	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) )
50	49	sneqd 3679	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> { ( N mod D ) } = { ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) } )
51		snriota 5992	. . . . . 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 2265	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> { ( N mod D ) } = { z e. NN0 | ( z < D /\ D || ( N - z ) ) } )
54	53	eleq2d 2299	. . 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 4086	. . . 4 |- ( z = R -> ( z < D <-> R < D ) )
57		oveq2 6015	. . . . 5 |- ( z = R -> ( N - z ) = ( N - R ) )
58	57	breq2d 4095	. . . 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 2959	. 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 1395   e. wcel 2200   E!wreu 2510   {crab 2512   {csn 3666   class class class wbr 4083   ` cfv 5318   iota_crio 5959  (class class class)co 6007   CCcc 8008   0cc0 8010   x. cmul 8015   < clt 8192   - cmin 8328   / cdiv 8830   NNcn 9121   NN0cn0 9380   ZZcz 9457   QQcq 9826   |_cfl 10500   mod cmo 10556   || cdvds 12313

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128  ax-arch 8129

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-2 9180  df-n0 9381  df-z 9458  df-uz 9734  df-q 9827  df-rp 9862  df-fl 10502  df-mod 10557  df-seqfrec 10682  df-exp 10773  df-cj 11368  df-re 11369  df-im 11370  df-rsqrt 11524  df-abs 11525  df-dvds 12314

This theorem is referenced by:  divalgmodcl  12454