Theorem ndvdssub 12616

Description: Corollary of the division algorithm. If an integer D greater than 1 divides N, then it does not divide any of N - 1, N - 2... N - ( D - 1 ). (Contributed by Paul Chapman, 31-Mar-2011.)

Assertion

Ref	Expression
ndvdssub	|- ( ( N e. ZZ /\ D e. NN /\ ( K e. NN /\ K < D ) ) -> ( D || N -> -. D || ( N - K ) ) )

Proof of Theorem ndvdssub

Dummy variables r x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnn0 9503	. . . . . . . 8 |- ( K e. NN -> K e. NN0 )
2		nnne0 9265	. . . . . . . 8 |- ( K e. NN -> K =/= 0 )
3	1, 2	jca 306	. . . . . . 7 |- ( K e. NN -> ( K e. NN0 /\ K =/= 0 ) )
4		df-ne 2413	. . . . . . . . . . . 12 |- ( K =/= 0 <-> -. K = 0 )
5	4	anbi2i 457	. . . . . . . . . . 11 |- ( ( K < D /\ K =/= 0 ) <-> ( K < D /\ -. K = 0 ) )
6		divalg2 12612	. . . . . . . . . . . . . . . . . . 19 |- ( ( N e. ZZ /\ D e. NN ) -> E! r e. NN0 ( r < D /\ D || ( N - r ) ) )
7		breq1 4112	. . . . . . . . . . . . . . . . . . . . 21 |- ( r = x -> ( r < D <-> x < D ) )
8		oveq2 6058	. . . . . . . . . . . . . . . . . . . . . 22 |- ( r = x -> ( N - r ) = ( N - x ) )
9	8	breq2d 4121	. . . . . . . . . . . . . . . . . . . . 21 |- ( r = x -> ( D || ( N - r ) <-> D || ( N - x ) ) )
10	7, 9	anbi12d 473	. . . . . . . . . . . . . . . . . . . 20 |- ( r = x -> ( ( r < D /\ D || ( N - r ) ) <-> ( x < D /\ D || ( N - x ) ) ) )
11	10	reu4 3011	. . . . . . . . . . . . . . . . . . 19 |- ( E! r e. NN0 ( r < D /\ D || ( N - r ) ) <-> ( E. r e. NN0 ( r < D /\ D || ( N - r ) ) /\ A. r e. NN0 A. x e. NN0 ( ( ( r < D /\ D || ( N - r ) ) /\ ( x < D /\ D || ( N - x ) ) ) -> r = x ) ) )
12	6, 11	sylib 122	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. ZZ /\ D e. NN ) -> ( E. r e. NN0 ( r < D /\ D || ( N - r ) ) /\ A. r e. NN0 A. x e. NN0 ( ( ( r < D /\ D || ( N - r ) ) /\ ( x < D /\ D || ( N - x ) ) ) -> r = x ) ) )
13		nngt0 9262	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( D e. NN -> 0 < D )
14	13	3ad2ant2 1046	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> 0 < D )
15		zcn 9582	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( N e. ZZ -> N e. CC )
16	15	subid1d 8573	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. ZZ -> ( N - 0 ) = N )
17	16	breq2d 4121	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. ZZ -> ( D || ( N - 0 ) <-> D || N ) )
18	17	biimpar 297	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N e. ZZ /\ D || N ) -> D || ( N - 0 ) )
19	18	3adant2 1043	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> D || ( N - 0 ) )
20	14, 19	jca 306	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( 0 < D /\ D || ( N - 0 ) ) )
21	20	3expa 1230	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) -> ( 0 < D /\ D || ( N - 0 ) ) )
22	21	anim2i 342	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( r < D /\ D || ( N - r ) ) /\ ( ( N e. ZZ /\ D e. NN ) /\ D || N ) ) -> ( ( r < D /\ D || ( N - r ) ) /\ ( 0 < D /\ D || ( N - 0 ) ) ) )
23	22	ancoms 268	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) /\ ( r < D /\ D || ( N - r ) ) ) -> ( ( r < D /\ D || ( N - r ) ) /\ ( 0 < D /\ D || ( N - 0 ) ) ) )
24		0nn0 9511	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 e. NN0
25		breq1 4112	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( x = 0 -> ( x < D <-> 0 < D ) )
26		oveq2 6058	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( x = 0 -> ( N - x ) = ( N - 0 ) )
27	26	breq2d 4121	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( x = 0 -> ( D || ( N - x ) <-> D || ( N - 0 ) ) )
28	25, 27	anbi12d 473	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( x = 0 -> ( ( x < D /\ D || ( N - x ) ) <-> ( 0 < D /\ D || ( N - 0 ) ) ) )
29	28	anbi2d 464	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( x = 0 -> ( ( ( r < D /\ D || ( N - r ) ) /\ ( x < D /\ D || ( N - x ) ) ) <-> ( ( r < D /\ D || ( N - r ) ) /\ ( 0 < D /\ D || ( N - 0 ) ) ) ) )
30		eqeq2 2242	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( x = 0 -> ( r = x <-> r = 0 ) )
31	29, 30	imbi12d 234	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( x = 0 -> ( ( ( ( r < D /\ D || ( N - r ) ) /\ ( x < D /\ D || ( N - x ) ) ) -> r = x ) <-> ( ( ( r < D /\ D || ( N - r ) ) /\ ( 0 < D /\ D || ( N - 0 ) ) ) -> r = 0 ) ) )
32	31	rspcv 2917	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 0 e. NN0 -> ( A. x e. NN0 ( ( ( r < D /\ D || ( N - r ) ) /\ ( x < D /\ D || ( N - x ) ) ) -> r = x ) -> ( ( ( r < D /\ D || ( N - r ) ) /\ ( 0 < D /\ D || ( N - 0 ) ) ) -> r = 0 ) ) )
33	24, 32	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 |- ( A. x e. NN0 ( ( ( r < D /\ D || ( N - r ) ) /\ ( x < D /\ D || ( N - x ) ) ) -> r = x ) -> ( ( ( r < D /\ D || ( N - r ) ) /\ ( 0 < D /\ D || ( N - 0 ) ) ) -> r = 0 ) )
34	23, 33	syl5 32	. . . . . . . . . . . . . . . . . . . 20 |- ( A. x e. NN0 ( ( ( r < D /\ D || ( N - r ) ) /\ ( x < D /\ D || ( N - x ) ) ) -> r = x ) -> ( ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) /\ ( r < D /\ D || ( N - r ) ) ) -> r = 0 ) )
35	34	expd 258	. . . . . . . . . . . . . . . . . . 19 |- ( A. x e. NN0 ( ( ( r < D /\ D || ( N - r ) ) /\ ( x < D /\ D || ( N - x ) ) ) -> r = x ) -> ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) -> ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) ) )
36	35	ralimi 2605	. . . . . . . . . . . . . . . . . 18 |- ( A. r e. NN0 A. x e. NN0 ( ( ( r < D /\ D || ( N - r ) ) /\ ( x < D /\ D || ( N - x ) ) ) -> r = x ) -> A. r e. NN0 ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) -> ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) ) )
37	12, 36	simpl2im 386	. . . . . . . . . . . . . . . . 17 |- ( ( N e. ZZ /\ D e. NN ) -> A. r e. NN0 ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) -> ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) ) )
38		r19.21v 2619	. . . . . . . . . . . . . . . . 17 |- ( A. r e. NN0 ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) -> ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) ) <-> ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) -> A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) ) )
39	37, 38	sylib 122	. . . . . . . . . . . . . . . 16 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) -> A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) ) )
40	39	expd 258	. . . . . . . . . . . . . . 15 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( N e. ZZ /\ D e. NN ) -> ( D || N -> A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) ) ) )
41	40	pm2.43i 49	. . . . . . . . . . . . . 14 |- ( ( N e. ZZ /\ D e. NN ) -> ( D || N -> A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) ) )
42	41	3impia 1227	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) )
43		breq1 4112	. . . . . . . . . . . . . . . 16 |- ( r = K -> ( r < D <-> K < D ) )
44		oveq2 6058	. . . . . . . . . . . . . . . . 17 |- ( r = K -> ( N - r ) = ( N - K ) )
45	44	breq2d 4121	. . . . . . . . . . . . . . . 16 |- ( r = K -> ( D || ( N - r ) <-> D || ( N - K ) ) )
46	43, 45	anbi12d 473	. . . . . . . . . . . . . . 15 |- ( r = K -> ( ( r < D /\ D || ( N - r ) ) <-> ( K < D /\ D || ( N - K ) ) ) )
47		eqeq1 2239	. . . . . . . . . . . . . . 15 |- ( r = K -> ( r = 0 <-> K = 0 ) )
48	46, 47	imbi12d 234	. . . . . . . . . . . . . 14 |- ( r = K -> ( ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) <-> ( ( K < D /\ D || ( N - K ) ) -> K = 0 ) ) )
49	48	rspcv 2917	. . . . . . . . . . . . 13 |- ( K e. NN0 -> ( A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) -> ( ( K < D /\ D || ( N - K ) ) -> K = 0 ) ) )
50	42, 49	syl5com 29	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN0 -> ( ( K < D /\ D || ( N - K ) ) -> K = 0 ) ) )
51		pm3.37 696	. . . . . . . . . . . 12 |- ( ( ( K < D /\ D || ( N - K ) ) -> K = 0 ) -> ( ( K < D /\ -. K = 0 ) -> -. D || ( N - K ) ) )
52	50, 51	syl6 33	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN0 -> ( ( K < D /\ -. K = 0 ) -> -. D || ( N - K ) ) ) )
53	5, 52	syl7bi 165	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN0 -> ( ( K < D /\ K =/= 0 ) -> -. D || ( N - K ) ) ) )
54	53	exp4a 366	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN0 -> ( K < D -> ( K =/= 0 -> -. D || ( N - K ) ) ) ) )
55	54	com23 78	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K < D -> ( K e. NN0 -> ( K =/= 0 -> -. D || ( N - K ) ) ) ) )
56	55	imp4a 349	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K < D -> ( ( K e. NN0 /\ K =/= 0 ) -> -. D || ( N - K ) ) ) )
57	3, 56	syl7 69	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K < D -> ( K e. NN -> -. D || ( N - K ) ) ) )
58	57	com23 78	. . . . 5 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN -> ( K < D -> -. D || ( N - K ) ) ) )
59	58	impd 254	. . . 4 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( ( K e. NN /\ K < D ) -> -. D || ( N - K ) ) )
60	59	3expia 1232	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( D || N -> ( ( K e. NN /\ K < D ) -> -. D || ( N - K ) ) ) )
61	60	com23 78	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( K e. NN /\ K < D ) -> ( D || N -> -. D || ( N - K ) ) ) )
62	61	3impia 1227	1 |- ( ( N e. ZZ /\ D e. NN /\ ( K e. NN /\ K < D ) ) -> ( D || N -> -. D || ( N - K ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520   E.wrex 2521   E!wreu 2522   class class class wbr 4109  (class class class)co 6050   0cc0 8127   < clt 8308   - cmin 8444   NNcn 9237   NN0cn0 9496   ZZcz 9577   || cdvds 12473

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474

This theorem is referenced by:  ndvdsadd  12617