Theorem divalgmod 13709

Description: The result of the mod operator satisfies the requirements for the remainder r in the Division Algorithm for a positive divisor (compare divalg2 13708 and divalgb 13707). This demonstration theorem justifies the use of mod to yield an explicit remainder from this point forward. (Contributed by Paul Chapman, 31-Mar-2011.)

Assertion

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

Distinct variable groups:   D,r   N,r

Proof of Theorem divalgmod

Step	Hyp	Ref	Expression
1		zmodcl 7511	. . . . . . . . 9 |- ((N e. ZZ /\ D e. NN) -> (N mod D) e. NN0)
2		divalg2 13708	. . . . . . . . 9 |- ((N e. ZZ /\ D e. NN) -> E!z e. NN0 (z < D /\ D||(N - z)))
3		breq1 3341	. . . . . . . . . . 11 |- (z = (N mod D) -> (z < D <-> (N mod D) < D))
4		opreq2 4890	. . . . . . . . . . . 12 |- (z = (N mod D) -> (N - z) = (N - (N mod D)))
5	4	breq2d 3350	. . . . . . . . . . 11 |- (z = (N mod D) -> (D||(N - z) <-> D||(N - (N mod D))))
6	3, 5	anbi12d 690	. . . . . . . . . 10 |- (z = (N mod D) -> ((z < D /\ D||(N - z)) <-> ((N mod D) < D /\ D||(N - (N mod D)))))
7	6	reuuni2 3811	. . . . . . . . 9 |- (((N mod D) e. NN0 /\ E!z e. NN0 (z < D /\ D||(N - z))) -> (((N mod D) < D /\ D||(N - (N mod D))) <-> U.{z e. NN0 | (z < D /\ D||(N - z))} = (N mod D)))
8	1, 2, 7	syl11anc 524	. . . . . . . 8 |- ((N e. ZZ /\ D e. NN) -> (((N mod D) < D /\ D||(N - (N mod D))) <-> U.{z e. NN0 | (z < D /\ D||(N - z))} = (N mod D)))
9		modlt 7504	. . . . . . . . 9 |- ((N e. RR /\ D e. RR+) -> (N mod D) < D)
10		zre 7348	. . . . . . . . 9 |- (N e. ZZ -> N e. RR)
11		nnrp 7238	. . . . . . . . 9 |- (D e. NN -> D e. RR+)
12	9, 10, 11	syl2an 503	. . . . . . . 8 |- ((N e. ZZ /\ D e. NN) -> (N mod D) < D)
13		redivcl 6978	. . . . . . . . . . . 12 |- ((N e. RR /\ D e. RR /\ D =/= 0) -> (N / D) e. RR)
14		nnre 7112	. . . . . . . . . . . 12 |- (D e. NN -> D e. RR)
15		nnne0 7132	. . . . . . . . . . . 12 |- (D e. NN -> D =/= 0)
16	13, 10, 14, 15	syl3an 1139	. . . . . . . . . . 11 |- ((N e. ZZ /\ D e. NN /\ D e. NN) -> (N / D) e. RR)
17	16	3anidm23 1156	. . . . . . . . . 10 |- ((N e. ZZ /\ D e. NN) -> (N / D) e. RR)
18		flcl 7465	. . . . . . . . . 10 |- ((N / D) e. RR -> (|_` (N / D)) e. ZZ)
19	17, 18	syl 12	. . . . . . . . 9 |- ((N e. ZZ /\ D e. NN) -> (|_` (N / D)) e. ZZ)
20		nnz 7362	. . . . . . . . . 10 |- (D e. NN -> D e. ZZ)
21	20	adantl 424	. . . . . . . . 9 |- ((N e. ZZ /\ D e. NN) -> D e. ZZ)
22		simpl 346	. . . . . . . . . 10 |- ((N e. ZZ /\ D e. NN) -> N e. ZZ)
23		nn0z 7363	. . . . . . . . . . 11 |- ((N mod D) e. NN0 -> (N mod D) e. ZZ)
24	1, 23	syl 12	. . . . . . . . . 10 |- ((N e. ZZ /\ D e. NN) -> (N mod D) e. ZZ)
25		zsubcl 7377	. . . . . . . . . 10 |- ((N e. ZZ /\ (N mod D) e. ZZ) -> (N - (N mod D)) e. ZZ)
26	22, 24, 25	syl11anc 524	. . . . . . . . 9 |- ((N e. ZZ /\ D e. NN) -> (N - (N mod D)) e. ZZ)
27	14	recnd 6468	. . . . . . . . . . . 12 |- (D e. NN -> D e. CC)
28	27	adantl 424	. . . . . . . . . . 11 |- ((N e. ZZ /\ D e. NN) -> D e. CC)
29		zcn 7349	. . . . . . . . . . . 12 |- ((|_` (N / D)) e. ZZ -> (|_` (N / D)) e. CC)
30	19, 29	syl 12	. . . . . . . . . . 11 |- ((N e. ZZ /\ D e. NN) -> (|_` (N / D)) e. CC)
31		mulcom 6459	. . . . . . . . . . 11 |- ((D e. CC /\ (|_` (N / D)) e. CC) -> (D x. (|_` (N / D))) = ((|_` (N / D)) x. D))
32	28, 30, 31	syl11anc 524	. . . . . . . . . 10 |- ((N e. ZZ /\ D e. NN) -> (D x. (|_` (N / D))) = ((|_` (N / D)) x. D))
33		modval 7501	. . . . . . . . . . . 12 |- ((N e. RR /\ D e. RR+) -> (N mod D) = (N - (D x. (|_` (N / D)))))
34	33, 10, 11	syl2an 503	. . . . . . . . . . 11 |- ((N e. ZZ /\ D e. NN) -> (N mod D) = (N - (D x. (|_` (N / D)))))
35	10	recnd 6468	. . . . . . . . . . . . . 14 |- (N e. ZZ -> N e. CC)
36	35	adantr 425	. . . . . . . . . . . . 13 |- ((N e. ZZ /\ D e. NN) -> N e. CC)
37		zmulcl 7389	. . . . . . . . . . . . . . . . 17 |- ((D e. ZZ /\ (|_` (N / D)) e. ZZ) -> (D x. (|_` (N / D))) e. ZZ)
38	37, 20, 19	syl2an 503	. . . . . . . . . . . . . . . 16 |- ((D e. NN /\ (N e. ZZ /\ D e. NN)) -> (D x. (|_` (N / D))) e. ZZ)
39	38	an1s 544	. . . . . . . . . . . . . . 15 |- ((N e. ZZ /\ (D e. NN /\ D e. NN)) -> (D x. (|_` (N / D))) e. ZZ)
40	39	anabsan2 563	. . . . . . . . . . . . . 14 |- ((N e. ZZ /\ D e. NN) -> (D x. (|_` (N / D))) e. ZZ)
41		zcn 7349	. . . . . . . . . . . . . 14 |- ((D x. (|_` (N / D))) e. ZZ -> (D x. (|_` (N / D))) e. CC)
42	40, 41	syl 12	. . . . . . . . . . . . 13 |- ((N e. ZZ /\ D e. NN) -> (D x. (|_` (N / D))) e. CC)
43		nn0cn 7318	. . . . . . . . . . . . . 14 |- ((N mod D) e. NN0 -> (N mod D) e. CC)
44	1, 43	syl 12	. . . . . . . . . . . . 13 |- ((N e. ZZ /\ D e. NN) -> (N mod D) e. CC)
45		subsub23 6533	. . . . . . . . . . . . 13 |- ((N e. CC /\ (D x. (|_` (N / D))) e. CC /\ (N mod D) e. CC) -> ((N - (D x. (|_` (N / D)))) = (N mod D) <-> (N - (N mod D)) = (D x. (|_` (N / D)))))
46	36, 42, 44, 45	syl111anc 1100	. . . . . . . . . . . 12 |- ((N e. ZZ /\ D e. NN) -> ((N - (D x. (|_` (N / D)))) = (N mod D) <-> (N - (N mod D)) = (D x. (|_` (N / D)))))
47		eqcom 1886	. . . . . . . . . . . . 13 |- ((N - (D x. (|_` (N / D)))) = (N mod D) <-> (N mod D) = (N - (D x. (|_` (N / D)))))
48		eqcom 1886	. . . . . . . . . . . . 13 |- ((N - (N mod D)) = (D x. (|_` (N / D))) <-> (D x. (|_` (N / D))) = (N - (N mod D)))
49	47, 48	bibi12i 672	. . . . . . . . . . . 12 |- (((N - (D x. (|_` (N / D)))) = (N mod D) <-> (N - (N mod D)) = (D x. (|_` (N / D)))) <-> ((N mod D) = (N - (D x. (|_` (N / D)))) <-> (D x. (|_` (N / D))) = (N - (N mod D))))
50	46, 49	sylib 215	. . . . . . . . . . 11 |- ((N e. ZZ /\ D e. NN) -> ((N mod D) = (N - (D x. (|_` (N / D)))) <-> (D x. (|_` (N / D))) = (N - (N mod D))))
51	34, 50	mpbid 212	. . . . . . . . . 10 |- ((N e. ZZ /\ D e. NN) -> (D x. (|_` (N / D))) = (N - (N mod D)))
52	32, 51	eqtr3d 1927	. . . . . . . . 9 |- ((N e. ZZ /\ D e. NN) -> ((|_` (N / D)) x. D) = (N - (N mod D)))
53		dvds0lem 13665	. . . . . . . . 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)))
54	19, 21, 26, 52, 53	syl31anc 1103	. . . . . . . 8 |- ((N e. ZZ /\ D e. NN) -> D||(N - (N mod D)))
55	8, 12, 54	mpbi2and 801	. . . . . . 7 |- ((N e. ZZ /\ D e. NN) -> U.{z e. NN0 | (z < D /\ D||(N - z))} = (N mod D))
56	55	eqcomd 1889	. . . . . 6 |- ((N e. ZZ /\ D e. NN) -> (N mod D) = U.{z e. NN0 | (z < D /\ D||(N - z))})
57	56	sneqd 3056	. . . . 5 |- ((N e. ZZ /\ D e. NN) -> {(N mod D)} = {U.{z e. NN0 | (z < D /\ D||(N - z))}})
58		reuunisn 3822	. . . . . 6 |- (E!z e. NN0 (z < D /\ D||(N - z)) -> {z e. NN0 | (z < D /\ D||(N - z))} = {U.{z e. NN0 | (z < D /\ D||(N - z))}})
59	2, 58	syl 12	. . . . 5 |- ((N e. ZZ /\ D e. NN) -> {z e. NN0 | (z < D /\ D||(N - z))} = {U.{z e. NN0 | (z < D /\ D||(N - z))}})
60	57, 59	eqtr4d 1928	. . . 4 |- ((N e. ZZ /\ D e. NN) -> {(N mod D)} = {z e. NN0 | (z < D /\ D||(N - z))})
61	60	eleq2d 1964	. . 3 |- ((N e. ZZ /\ D e. NN) -> (r e. {(N mod D)} <-> r e. {z e. NN0 | (z < D /\ D||(N - z))}))
62		breq1 3341	. . . . 5 |- (z = r -> (z < D <-> r < D))
63		opreq2 4890	. . . . . 6 |- (z = r -> (N - z) = (N - r))
64	63	breq2d 3350	. . . . 5 |- (z = r -> (D||(N - z) <-> D||(N - r)))
65	62, 64	anbi12d 690	. . . 4 |- (z = r -> ((z < D /\ D||(N - z)) <-> (r < D /\ D||(N - r))))
66	65	elrab 2414	. . 3 |- (r e. {z e. NN0 | (z < D /\ D||(N - z))} <-> (r e. NN0 /\ (r < D /\ D||(N - r))))
67	61, 66	syl6bb 595	. 2 |- ((N e. ZZ /\ D e. NN) -> (r e. {(N mod D)} <-> (r e. NN0 /\ (r < D /\ D||(N - r)))))
68		elsn 3058	. 2 |- (r e. {(N mod D)} <-> r = (N mod D))
69	67, 68	syl5bbr 593	1 |- ((N e. ZZ /\ D e. NN) -> (r = (N mod D) <-> (r e. NN0 /\ (r < D /\ D||(N - r)))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  E!wreu 2107  {crab 2108  {csn 3044  U.cuni 3177   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   x. cmul 6391   - cmin 6445   / cdiv 6447  NNcn 6449  NN0cn0 6450  ZZcz 6451  RR+crp 6453   < clt 6653  |_cfl 7462   mod cmo 7499  ||cdivides 13662

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-mod 7500  df-uz 7587  df-fz 7638  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-divides 13663

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