Theorem divalgb 13707

Description: Express the Division Algorithm as stated in divalg 13706 in terms of ||. (Contributed by Paul Chapman, 31-Mar-2011.)

Assertion

Ref	Expression
divalgb	|- ((N e. ZZ /\ D e. ZZ /\ D =/= 0) -> (E!r e. ZZ E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)) <-> E!r e. NN0 (r < (abs` D) /\ D||(N - r))))

Distinct variable groups:   D,q,r   N,q,r

Proof of Theorem divalgb

Step	Hyp	Ref	Expression
1		divides 13664	. . . . . . . . . . . 12 |- ((D e. ZZ /\ (N - r) e. ZZ) -> (D||(N - r) <-> E.q e. ZZ (q x. D) = (N - r)))
2		zsubcl 7377	. . . . . . . . . . . 12 |- ((N e. ZZ /\ r e. ZZ) -> (N - r) e. ZZ)
3	1, 2	sylan2 500	. . . . . . . . . . 11 |- ((D e. ZZ /\ (N e. ZZ /\ r e. ZZ)) -> (D||(N - r) <-> E.q e. ZZ (q x. D) = (N - r)))
4	3	3impb 1063	. . . . . . . . . 10 |- ((D e. ZZ /\ N e. ZZ /\ r e. ZZ) -> (D||(N - r) <-> E.q e. ZZ (q x. D) = (N - r)))
5	4	3com12 1071	. . . . . . . . 9 |- ((N e. ZZ /\ D e. ZZ /\ r e. ZZ) -> (D||(N - r) <-> E.q e. ZZ (q x. D) = (N - r)))
6		ax-17 1317	. . . . . . . . . . 11 |- ((N e. ZZ /\ r e. ZZ /\ D e. ZZ) -> A.q(N e. ZZ /\ r e. ZZ /\ D e. ZZ))
7		subadd 6532	. . . . . . . . . . . . . . . . . . 19 |- ((N e. CC /\ r e. CC /\ (q x. D) e. CC) -> ((N - r) = (q x. D) <-> (r + (q x. D)) = N))
8		zcn 7349	. . . . . . . . . . . . . . . . . . 19 |- (N e. ZZ -> N e. CC)
9		zcn 7349	. . . . . . . . . . . . . . . . . . 19 |- (r e. ZZ -> r e. CC)
10		zmulcl 7389	. . . . . . . . . . . . . . . . . . . 20 |- ((q e. ZZ /\ D e. ZZ) -> (q x. D) e. ZZ)
11		zcn 7349	. . . . . . . . . . . . . . . . . . . 20 |- ((q x. D) e. ZZ -> (q x. D) e. CC)
12	10, 11	syl 12	. . . . . . . . . . . . . . . . . . 19 |- ((q e. ZZ /\ D e. ZZ) -> (q x. D) e. CC)
13	7, 8, 9, 12	syl3an 1139	. . . . . . . . . . . . . . . . . 18 |- ((N e. ZZ /\ r e. ZZ /\ (q e. ZZ /\ D e. ZZ)) -> ((N - r) = (q x. D) <-> (r + (q x. D)) = N))
14		addcom 6458	. . . . . . . . . . . . . . . . . . . . 21 |- ((r e. CC /\ (q x. D) e. CC) -> (r + (q x. D)) = ((q x. D) + r))
15	14, 9, 12	syl2an 503	. . . . . . . . . . . . . . . . . . . 20 |- ((r e. ZZ /\ (q e. ZZ /\ D e. ZZ)) -> (r + (q x. D)) = ((q x. D) + r))
16	15	3adant1 894	. . . . . . . . . . . . . . . . . . 19 |- ((N e. ZZ /\ r e. ZZ /\ (q e. ZZ /\ D e. ZZ)) -> (r + (q x. D)) = ((q x. D) + r))
17	16	eqeq1d 1892	. . . . . . . . . . . . . . . . . 18 |- ((N e. ZZ /\ r e. ZZ /\ (q e. ZZ /\ D e. ZZ)) -> ((r + (q x. D)) = N <-> ((q x. D) + r) = N))
18	13, 17	bitrd 587	. . . . . . . . . . . . . . . . 17 |- ((N e. ZZ /\ r e. ZZ /\ (q e. ZZ /\ D e. ZZ)) -> ((N - r) = (q x. D) <-> ((q x. D) + r) = N))
19		eqcom 1886	. . . . . . . . . . . . . . . . . 18 |- ((N - r) = (q x. D) <-> (q x. D) = (N - r))
20		eqcom 1886	. . . . . . . . . . . . . . . . . 18 |- (((q x. D) + r) = N <-> N = ((q x. D) + r))
21	19, 20	bibi12i 672	. . . . . . . . . . . . . . . . 17 |- (((N - r) = (q x. D) <-> ((q x. D) + r) = N) <-> ((q x. D) = (N - r) <-> N = ((q x. D) + r)))
22	18, 21	sylib 215	. . . . . . . . . . . . . . . 16 |- ((N e. ZZ /\ r e. ZZ /\ (q e. ZZ /\ D e. ZZ)) -> ((q x. D) = (N - r) <-> N = ((q x. D) + r)))
23	22	3expia 1069	. . . . . . . . . . . . . . 15 |- ((N e. ZZ /\ r e. ZZ) -> ((q e. ZZ /\ D e. ZZ) -> ((q x. D) = (N - r) <-> N = ((q x. D) + r))))
24	23	exp3a 405	. . . . . . . . . . . . . 14 |- ((N e. ZZ /\ r e. ZZ) -> (q e. ZZ -> (D e. ZZ -> ((q x. D) = (N - r) <-> N = ((q x. D) + r)))))
25	24	com23 36	. . . . . . . . . . . . 13 |- ((N e. ZZ /\ r e. ZZ) -> (D e. ZZ -> (q e. ZZ -> ((q x. D) = (N - r) <-> N = ((q x. D) + r)))))
26	25	3impia 1064	. . . . . . . . . . . 12 |- ((N e. ZZ /\ r e. ZZ /\ D e. ZZ) -> (q e. ZZ -> ((q x. D) = (N - r) <-> N = ((q x. D) + r))))
27	26	imp 377	. . . . . . . . . . 11 |- (((N e. ZZ /\ r e. ZZ /\ D e. ZZ) /\ q e. ZZ) -> ((q x. D) = (N - r) <-> N = ((q x. D) + r)))
28	6, 27	rexbida 2118	. . . . . . . . . 10 |- ((N e. ZZ /\ r e. ZZ /\ D e. ZZ) -> (E.q e. ZZ (q x. D) = (N - r) <-> E.q e. ZZ N = ((q x. D) + r)))
29	28	3com23 1074	. . . . . . . . 9 |- ((N e. ZZ /\ D e. ZZ /\ r e. ZZ) -> (E.q e. ZZ (q x. D) = (N - r) <-> E.q e. ZZ N = ((q x. D) + r)))
30	5, 29	bitrd 587	. . . . . . . 8 |- ((N e. ZZ /\ D e. ZZ /\ r e. ZZ) -> (D||(N - r) <-> E.q e. ZZ N = ((q x. D) + r)))
31	30	anbi2d 678	. . . . . . 7 |- ((N e. ZZ /\ D e. ZZ /\ r e. ZZ) -> (((0 <_ r /\ r < (abs` D)) /\ D||(N - r)) <-> ((0 <_ r /\ r < (abs` D)) /\ E.q e. ZZ N = ((q x. D) + r))))
32		df-3an 860	. . . . . . . . 9 |- ((0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)) <-> ((0 <_ r /\ r < (abs` D)) /\ N = ((q x. D) + r)))
33	32	rexbii 2128	. . . . . . . 8 |- (E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)) <-> E.q e. ZZ ((0 <_ r /\ r < (abs` D)) /\ N = ((q x. D) + r)))
34		r19.42v 2237	. . . . . . . 8 |- (E.q e. ZZ ((0 <_ r /\ r < (abs` D)) /\ N = ((q x. D) + r)) <-> ((0 <_ r /\ r < (abs` D)) /\ E.q e. ZZ N = ((q x. D) + r)))
35	33, 34	bitri 190	. . . . . . 7 |- (E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)) <-> ((0 <_ r /\ r < (abs` D)) /\ E.q e. ZZ N = ((q x. D) + r)))
36	31, 35	syl6rbbr 598	. . . . . 6 |- ((N e. ZZ /\ D e. ZZ /\ r e. ZZ) -> (E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)) <-> ((0 <_ r /\ r < (abs` D)) /\ D||(N - r))))
37		anass 487	. . . . . 6 |- (((0 <_ r /\ r < (abs` D)) /\ D||(N - r)) <-> (0 <_ r /\ (r < (abs` D) /\ D||(N - r))))
38	36, 37	syl6bb 595	. . . . 5 |- ((N e. ZZ /\ D e. ZZ /\ r e. ZZ) -> (E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)) <-> (0 <_ r /\ (r < (abs` D) /\ D||(N - r)))))
39	38	3expa 1067	. . . 4 |- (((N e. ZZ /\ D e. ZZ) /\ r e. ZZ) -> (E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)) <-> (0 <_ r /\ (r < (abs` D) /\ D||(N - r)))))
40	39	reubidva 2259	. . 3 |- ((N e. ZZ /\ D e. ZZ) -> (E!r e. ZZ E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)) <-> E!r e. ZZ (0 <_ r /\ (r < (abs` D) /\ D||(N - r)))))
41		elnn0z 7356	. . . . . . 7 |- (r e. NN0 <-> (r e. ZZ /\ 0 <_ r))
42	41	anbi1i 539	. . . . . 6 |- ((r e. NN0 /\ (r < (abs` D) /\ D||(N - r))) <-> ((r e. ZZ /\ 0 <_ r) /\ (r < (abs` D) /\ D||(N - r))))
43		anass 487	. . . . . 6 |- (((r e. ZZ /\ 0 <_ r) /\ (r < (abs` D) /\ D||(N - r))) <-> (r e. ZZ /\ (0 <_ r /\ (r < (abs` D) /\ D||(N - r)))))
44	42, 43	bitri 190	. . . . 5 |- ((r e. NN0 /\ (r < (abs` D) /\ D||(N - r))) <-> (r e. ZZ /\ (0 <_ r /\ (r < (abs` D) /\ D||(N - r)))))
45	44	eubii 1780	. . . 4 |- (E!r(r e. NN0 /\ (r < (abs` D) /\ D||(N - r))) <-> E!r(r e. ZZ /\ (0 <_ r /\ (r < (abs` D) /\ D||(N - r)))))
46		df-reu 2111	. . . 4 |- (E!r e. NN0 (r < (abs` D) /\ D||(N - r)) <-> E!r(r e. NN0 /\ (r < (abs` D) /\ D||(N - r))))
47		df-reu 2111	. . . 4 |- (E!r e. ZZ (0 <_ r /\ (r < (abs` D) /\ D||(N - r))) <-> E!r(r e. ZZ /\ (0 <_ r /\ (r < (abs` D) /\ D||(N - r)))))
48	45, 46, 47	3bitr4ri 201	. . 3 |- (E!r e. ZZ (0 <_ r /\ (r < (abs` D) /\ D||(N - r))) <-> E!r e. NN0 (r < (abs` D) /\ D||(N - r)))
49	40, 48	syl6bb 595	. 2 |- ((N e. ZZ /\ D e. ZZ) -> (E!r e. ZZ E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)) <-> E!r e. NN0 (r < (abs` D) /\ D||(N - r))))
50	49	3adant3 896	1 |- ((N e. ZZ /\ D e. ZZ /\ D =/= 0) -> (E!r e. ZZ E.q e. ZZ (0 <_ r /\ r < (abs` D) /\ N = ((q x. D) + r)) <-> E!r e. NN0 (r < (abs` D) /\ D||(N - r))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E!weu 1771   =/= wne 2017  E.wrex 2106  E!wreu 2107   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386   + caddc 6389   x. cmul 6391   - cmin 6445   <_ cle 6448  NN0cn0 6450  ZZcz 6451   < clt 6653  abscabs 8000  ||cdivides 13662

This theorem is referenced by:  divalg2 13708

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-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-n 7108  df-n0 7309  df-z 7345  df-divides 13663

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