Theorem divalgmod 12180

Description: The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor (compare divalg2 12179 and divalgb 12178). 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 9746	. . . . . . . 8 |- ( N e. ZZ -> N e. QQ )
2	1	adantr 276	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> N e. QQ )
3		nnq 9753	. . . . . . . 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 9079	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> 0 < D )
7	2, 4, 6	modqcld 10471	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. QQ )
8		snidg 3661	. . . . . 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 2267	. . . . 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 3650	. . . 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 10476	. . . . . . . . 9 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) < D )
15	2, 4, 6, 14	syl3anc 1249	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) < D )
16		znq 9744	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( N / D ) e. QQ )
17	16	flqcld 10418	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. ZZ )
18		nnz 9390	. . . . . . . . . 10 |- ( D e. NN -> D e. ZZ )
19	18	adantl 277	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> D e. ZZ )
20		zmodcl 10487	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. NN0 )
21	20	nn0zd 9492	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. ZZ )
22		zsubcl 9412	. . . . . . . . . 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 9043	. . . . . . . . . . . 12 |- ( D e. NN -> D e. CC )
25	24	adantl 277	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> D e. CC )
26	17	zcnd 9495	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. CC )
27	25, 26	mulcomd 8093	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) = ( ( |_ ` ( N / D ) ) x. D ) )
28		modqval 10467	. . . . . . . . . . . 12 |- ( ( N e. QQ /\ D e. QQ /\ 0 < D ) -> ( N mod D ) = ( N - ( D x. ( |_ ` ( N / D ) ) ) ) )
29	2, 4, 6, 28	syl3anc 1249	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( N - ( D x. ( |_ ` ( N / D ) ) ) ) )
30	20	nn0cnd 9349	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. CC )
31		zmulcl 9425	. . . . . . . . . . . . . 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 9495	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) e. CC )
34		zcn 9376	. . . . . . . . . . . . 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 8443	. . . . . . . . . . 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 2239	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( |_ ` ( N / D ) ) x. D ) = ( N - ( N mod D ) ) )
39		dvds0lem 12054	. . . . . . . . 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 1252	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN ) -> D || ( N - ( N mod D ) ) )
41		divalg2 12179	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> E! z e. NN0 ( z < D /\ D || ( N - z ) ) )
42		breq1 4046	. . . . . . . . . . 11 |- ( z = ( N mod D ) -> ( z < D <-> ( N mod D ) < D ) )
43		oveq2 5951	. . . . . . . . . . . 12 |- ( z = ( N mod D ) -> ( N - z ) = ( N - ( N mod D ) ) )
44	43	breq2d 4055	. . . . . . . . . . 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 5921	. . . . . . . . 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 945	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) = ( N mod D ) )
49	48	eqcomd 2210	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) )
50	49	sneqd 3645	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> { ( N mod D ) } = { ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) } )
51		snriota 5928	. . . . . 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 2240	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> { ( N mod D ) } = { z e. NN0 | ( z < D /\ D || ( N - z ) ) } )
54	53	eleq2d 2274	. . 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 4046	. . . 4 |- ( z = R -> ( z < D <-> R < D ) )
57		oveq2 5951	. . . . 5 |- ( z = R -> ( N - z ) = ( N - R ) )
58	57	breq2d 4055	. . . 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 2928	. 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 1372   e. wcel 2175   E!wreu 2485   {crab 2487   {csn 3632   class class class wbr 4043   ` cfv 5270   iota_crio 5897  (class class class)co 5943   CCcc 7922   0cc0 7924   x. cmul 7929   < clt 8106   - cmin 8242   / cdiv 8744   NNcn 9035   NN0cn0 9294   ZZcz 9371   QQcq 9739   |_cfl 10409   mod cmo 10465   || cdvds 12040

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-frec 6476  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fl 10411  df-mod 10466  df-seqfrec 10591  df-exp 10682  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252  df-dvds 12041

This theorem is referenced by:  divalgmodcl  12181