Theorem divalgmod 12094

Description: The result of the mod operator satisfies the requirements for the remainder R in the division algorithm for a positive divisor (compare divalg2 12093 and divalgb 12092). 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 9702	. . . . . . . 8 |- ( N e. ZZ -> N e. QQ )
2	1	adantr 276	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> N e. QQ )
3		nnq 9709	. . . . . . . 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 9036	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN ) -> 0 < D )
7	2, 4, 6	modqcld 10422	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. QQ )
8		snidg 3652	. . . . . 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 2259	. . . . 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 3641	. . . 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 10427	. . . . . . . . 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 9700	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( N / D ) e. QQ )
17	16	flqcld 10369	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. ZZ )
18		nnz 9347	. . . . . . . . . 10 |- ( D e. NN -> D e. ZZ )
19	18	adantl 277	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> D e. ZZ )
20		zmodcl 10438	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. NN0 )
21	20	nn0zd 9448	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. ZZ )
22		zsubcl 9369	. . . . . . . . . 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 9000	. . . . . . . . . . . 12 |- ( D e. NN -> D e. CC )
25	24	adantl 277	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> D e. CC )
26	17	zcnd 9451	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN ) -> ( |_ ` ( N / D ) ) e. CC )
27	25, 26	mulcomd 8050	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) = ( ( |_ ` ( N / D ) ) x. D ) )
28		modqval 10418	. . . . . . . . . . . 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 9306	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) e. CC )
31		zmulcl 9381	. . . . . . . . . . . . . 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 9451	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN ) -> ( D x. ( |_ ` ( N / D ) ) ) e. CC )
34		zcn 9333	. . . . . . . . . . . . 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 8400	. . . . . . . . . . 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 2231	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( |_ ` ( N / D ) ) x. D ) = ( N - ( N mod D ) ) )
39		dvds0lem 11968	. . . . . . . . 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 12093	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN ) -> E! z e. NN0 ( z < D /\ D || ( N - z ) ) )
42		breq1 4037	. . . . . . . . . . 11 |- ( z = ( N mod D ) -> ( z < D <-> ( N mod D ) < D ) )
43		oveq2 5931	. . . . . . . . . . . 12 |- ( z = ( N mod D ) -> ( N - z ) = ( N - ( N mod D ) ) )
44	43	breq2d 4046	. . . . . . . . . . 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 5901	. . . . . . . . 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 2202	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN ) -> ( N mod D ) = ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) )
50	49	sneqd 3636	. . . . 5 |- ( ( N e. ZZ /\ D e. NN ) -> { ( N mod D ) } = { ( iota_ z e. NN0 ( z < D /\ D || ( N - z ) ) ) } )
51		snriota 5908	. . . . . 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 2232	. . . 4 |- ( ( N e. ZZ /\ D e. NN ) -> { ( N mod D ) } = { z e. NN0 | ( z < D /\ D || ( N - z ) ) } )
54	53	eleq2d 2266	. . 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 4037	. . . 4 |- ( z = R -> ( z < D <-> R < D ) )
57		oveq2 5931	. . . . 5 |- ( z = R -> ( N - z ) = ( N - R ) )
58	57	breq2d 4046	. . . 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 2920	. 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 1364   e. wcel 2167   E!wreu 2477   {crab 2479   {csn 3623   class class class wbr 4034   ` cfv 5259   iota_crio 5877  (class class class)co 5923   CCcc 7879   0cc0 7881   x. cmul 7886   < clt 8063   - cmin 8199   / cdiv 8701   NNcn 8992   NN0cn0 9251   ZZcz 9328   QQcq 9695   |_cfl 10360   mod cmo 10416   || cdvds 11954

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-mulrcl 7980  ax-addcom 7981  ax-mulcom 7982  ax-addass 7983  ax-mulass 7984  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-1rid 7988  ax-0id 7989  ax-rnegex 7990  ax-precex 7991  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997  ax-pre-mulgt0 7998  ax-pre-mulext 7999  ax-arch 8000

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-recs 6364  df-frec 6450  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-reap 8604  df-ap 8611  df-div 8702  df-inn 8993  df-2 9051  df-n0 9252  df-z 9329  df-uz 9604  df-q 9696  df-rp 9731  df-fl 10362  df-mod 10417  df-seqfrec 10542  df-exp 10633  df-cj 11009  df-re 11010  df-im 11011  df-rsqrt 11165  df-abs 11166  df-dvds 11955

This theorem is referenced by:  divalgmodcl  12095