Theorem ndvdssub 11627

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 8984	. . . . . . . 8 |- ( K e. NN -> K e. NN0 )
2		nnne0 8748	. . . . . . . 8 |- ( K e. NN -> K =/= 0 )
3	1, 2	jca 304	. . . . . . 7 |- ( K e. NN -> ( K e. NN0 /\ K =/= 0 ) )
4		df-ne 2309	. . . . . . . . . . . 12 |- ( K =/= 0 <-> -. K = 0 )
5	4	anbi2i 452	. . . . . . . . . . 11 |- ( ( K < D /\ K =/= 0 ) <-> ( K < D /\ -. K = 0 ) )
6		divalg2 11623	. . . . . . . . . . . . . . . . . . 19 |- ( ( N e. ZZ /\ D e. NN ) -> E! r e. NN0 ( r < D /\ D || ( N - r ) ) )
7		breq1 3932	. . . . . . . . . . . . . . . . . . . . 21 |- ( r = x -> ( r < D <-> x < D ) )
8		oveq2 5782	. . . . . . . . . . . . . . . . . . . . . 22 |- ( r = x -> ( N - r ) = ( N - x ) )
9	8	breq2d 3941	. . . . . . . . . . . . . . . . . . . . 21 |- ( r = x -> ( D || ( N - r ) <-> D || ( N - x ) ) )
10	7, 9	anbi12d 464	. . . . . . . . . . . . . . . . . . . 20 |- ( r = x -> ( ( r < D /\ D || ( N - r ) ) <-> ( x < D /\ D || ( N - x ) ) ) )
11	10	reu4 2878	. . . . . . . . . . . . . . . . . . 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 121	. . . . . . . . . . . . . . . . . 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 8745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( D e. NN -> 0 < D )
14	13	3ad2ant2 1003	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> 0 < D )
15		zcn 9059	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( N e. ZZ -> N e. CC )
16	15	subid1d 8062	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. ZZ -> ( N - 0 ) = N )
17	16	breq2d 3941	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. ZZ -> ( D || ( N - 0 ) <-> D || N ) )
18	17	biimpar 295	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N e. ZZ /\ D || N ) -> D || ( N - 0 ) )
19	18	3adant2 1000	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> D || ( N - 0 ) )
20	14, 19	jca 304	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( 0 < D /\ D || ( N - 0 ) ) )
21	20	3expa 1181	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) -> ( 0 < D /\ D || ( N - 0 ) ) )
22	21	anim2i 339	. . . . . . . . . . . . . . . . . . . . . 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 266	. . . . . . . . . . . . . . . . . . . . 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 8992	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 e. NN0
25		breq1 3932	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( x = 0 -> ( x < D <-> 0 < D ) )
26		oveq2 5782	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( x = 0 -> ( N - x ) = ( N - 0 ) )
27	26	breq2d 3941	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( x = 0 -> ( D || ( N - x ) <-> D || ( N - 0 ) ) )
28	25, 27	anbi12d 464	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( x = 0 -> ( ( x < D /\ D || ( N - x ) ) <-> ( 0 < D /\ D || ( N - 0 ) ) ) )
29	28	anbi2d 459	. . . . . . . . . . . . . . . . . . . . . . . 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 2149	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( x = 0 -> ( r = x <-> r = 0 ) )
31	29, 30	imbi12d 233	. . . . . . . . . . . . . . . . . . . . . . 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 2785	. . . . . . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . . . 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 2495	. . . . . . . . . . . . . . . . . 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 383	. . . . . . . . . . . . . . . . 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 2509	. . . . . . . . . . . . . . . . 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 121	. . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . 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 1178	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) )
43		breq1 3932	. . . . . . . . . . . . . . . 16 |- ( r = K -> ( r < D <-> K < D ) )
44		oveq2 5782	. . . . . . . . . . . . . . . . 17 |- ( r = K -> ( N - r ) = ( N - K ) )
45	44	breq2d 3941	. . . . . . . . . . . . . . . 16 |- ( r = K -> ( D || ( N - r ) <-> D || ( N - K ) ) )
46	43, 45	anbi12d 464	. . . . . . . . . . . . . . 15 |- ( r = K -> ( ( r < D /\ D || ( N - r ) ) <-> ( K < D /\ D || ( N - K ) ) ) )
47		eqeq1 2146	. . . . . . . . . . . . . . 15 |- ( r = K -> ( r = 0 <-> K = 0 ) )
48	46, 47	imbi12d 233	. . . . . . . . . . . . . 14 |- ( r = K -> ( ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) <-> ( ( K < D /\ D || ( N - K ) ) -> K = 0 ) ) )
49	48	rspcv 2785	. . . . . . . . . . . . 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		pm3.37 678	. . . . . . . . . . . 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 164	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN0 -> ( ( K < D /\ K =/= 0 ) -> -. D || ( N - K ) ) ) )
54	53	exp4a 363	. . . . . . . . 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 346	. . . . . . 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 252	. . . 4 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( ( K e. NN /\ K < D ) -> -. D || ( N - K ) ) )
60	59	3expia 1183	. . 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 1178	1 |- ( ( N e. ZZ /\ D e. NN /\ ( K e. NN /\ K < D ) ) -> ( D || N -> -. D || ( N - K ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2308   A.wral 2416   E.wrex 2417   E!wreu 2418   class class class wbr 3929  (class class class)co 5774   0cc0 7620   < clt 7800   - cmin 7933   NNcn 8720   NN0cn0 8977   ZZcz 9054   || cdvds 11493

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-2 8779  df-n0 8978  df-z 9055  df-uz 9327  df-q 9412  df-rp 9442  df-fl 10043  df-mod 10096  df-seqfrec 10219  df-exp 10293  df-cj 10614  df-re 10615  df-im 10616  df-rsqrt 10770  df-abs 10771  df-dvds 11494

This theorem is referenced by:  ndvdsadd  11628