Theorem divalgmod 11915

Description: The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor (compare divalg2 11914 and divalgb 11913). 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 9615	. . . . . . . 8 |- ( N e. ZZ -> N e. QQ )
2	1	adantr 276	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> N e. QQ )
3		nnq 9622	. . . . . . . 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 8952	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> 0 < D )
7	2, 4, 6	modqcld 10314	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. QQ )
8		snidg 3620	. . . . . 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 2240	. . . . 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 3609	. . . 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 10319	. . . . . . . . 9 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) < D )
15	2, 4, 6, 14	syl3anc 1238	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < D )
16		znq 9613	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( N / D ) e. QQ )
17	16	flqcld 10263	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. ZZ )
18		nnz 9261	. . . . . . . . . 10 |- ( D e. NN -> D e. ZZ )
19	18	adantl 277	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> D e. ZZ )
20		zmodcl 10330	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. NN0 )
21	20	nn0zd 9362	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. ZZ )
22		zsubcl 9283	. . . . . . . . . 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 8916	. . . . . . . . . . . 12 |- ( D e. NN -> D e. CC )
25	24	adantl 277	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> D e. CC )
26	17	zcnd 9365	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. CC )
27	25, 26	mulcomd 7969	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) = ( ( |_ ` ( N / D ) ) x. D ) )
28		modqval 10310	. . . . . . . . . . . 12 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) = ( N - ( D x. ( |_ ` ( N / D ) ) ) ) )
29	2, 4, 6, 28	syl3anc 1238	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( N - ( D x. ( |_ ` ( N / D ) ) ) ) )
30	20	nn0cnd 9220	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. CC )
31		zmulcl 9295	. . . . . . . . . . . . . 14 |- ( ( D e. ZZ /\ ( |_ ` ( N / D ) ) e. ZZ ) -> ( D x. ( |_ ` ( N / D ) ) ) e. ZZ )
32	18, 17, 31	syl2an2 594	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) e. ZZ )
33	32	zcnd 9365	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) e. CC )
34		zcn 9247	. . . . . . . . . . . . 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 8319	. . . . . . . . . . 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 2212	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( |_ ` ( N / D ) ) x. D ) = ( N - ( N mod D ) ) )
39		dvds0lem 11792	. . . . . . . . 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 1241	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN ) -> D || ( N - ( N mod D ) ) )
41		divalg2 11914	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> E! z e. NN0 ( z < D /\ D || ( N - z ) ) )
42		breq1 4003	. . . . . . . . . . 11 |- ( z = ( N mod D ) -> ( z < D <-> ( N mod D ) < D ) )
43		oveq2 5877	. . . . . . . . . . . 12 |- ( z = ( N mod D ) -> ( N - z ) = ( N - ( N mod D ) ) )
44	43	breq2d 4012	. . . . . . . . . . 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 5847	. . . . . . . . 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 943	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) = ( N mod D ) )
49	48	eqcomd 2183	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) )
50	49	sneqd 3604	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> { ( N mod D ) } = { ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) } )
51		snriota 5854	. . . . . 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 2213	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> { ( N mod D ) } = { z e. NN0 | ( z < D /\ D || ( N - z ) ) } )
54	53	eleq2d 2247	. . 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 4003	. . . 4 |- ( z = R -> ( z < D <-> R < D ) )
57		oveq2 5877	. . . . 5 |- ( z = R -> ( N - z ) = ( N - R ) )
58	57	breq2d 4012	. . . 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 2893	. 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 1353   e. wcel 2148   E!wreu 2457   {crab 2459   {csn 3591   class class class wbr 4000   ` cfv 5212   iota_crio 5824  (class class class)co 5869   CCcc 7800   0cc0 7802   x. cmul 7807   < clt 7982   - cmin 8118   / cdiv 8618   NNcn 8908   NN0cn0 9165   ZZcz 9242   QQcq 9608   |_cfl 10254   mod cmo 10308   || cdvds 11778

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779

This theorem is referenced by:  divalgmodcl  11916