Theorem ndvdssub 11820

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 9097	. . . . . . . 8 |- ( K e. NN -> K e. NN0 )
2		nnne0 8861	. . . . . . . 8 |- ( K e. NN -> K =/= 0 )
3	1, 2	jca 304	. . . . . . 7 |- ( K e. NN -> ( K e. NN0 /\ K =/= 0 ) )
4		df-ne 2328	. . . . . . . . . . . 12 |- ( K =/= 0 <-> -. K = 0 )
5	4	anbi2i 453	. . . . . . . . . . 11 |- ( ( K < D /\ K =/= 0 ) <-> ( K < D /\ -. K = 0 ) )
6		divalg2 11816	. . . . . . . . . . . . . . . . . . 19 |- ( ( N e. ZZ /\ D e. NN ) -> E! r e. NN0 ( r < D /\ D || ( N - r ) ) )
7		breq1 3968	. . . . . . . . . . . . . . . . . . . . 21 |- ( r = x -> ( r < D <-> x < D ) )
8		oveq2 5832	. . . . . . . . . . . . . . . . . . . . . 22 |- ( r = x -> ( N - r ) = ( N - x ) )
9	8	breq2d 3977	. . . . . . . . . . . . . . . . . . . . 21 |- ( r = x -> ( D || ( N - r ) <-> D || ( N - x ) ) )
10	7, 9	anbi12d 465	. . . . . . . . . . . . . . . . . . . 20 |- ( r = x -> ( ( r < D /\ D || ( N - r ) ) <-> ( x < D /\ D || ( N - x ) ) ) )
11	10	reu4 2906	. . . . . . . . . . . . . . . . . . 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 8858	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( D e. NN -> 0 < D )
14	13	3ad2ant2 1004	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> 0 < D )
15		zcn 9172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( N e. ZZ -> N e. CC )
16	15	subid1d 8175	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. ZZ -> ( N - 0 ) = N )
17	16	breq2d 3977	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1001	. . . . . . . . . . . . . . . . . . . . . . . . 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 1185	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) -> ( 0 < D /\ D || ( N - 0 ) ) )
22	21	anim2i 340	. . . . . . . . . . . . . . . . . . . . . 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 9105	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 e. NN0
25		breq1 3968	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( x = 0 -> ( x < D <-> 0 < D ) )
26		oveq2 5832	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( x = 0 -> ( N - x ) = ( N - 0 ) )
27	26	breq2d 3977	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( x = 0 -> ( D || ( N - x ) <-> D || ( N - 0 ) ) )
28	25, 27	anbi12d 465	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( x = 0 -> ( ( x < D /\ D || ( N - x ) ) <-> ( 0 < D /\ D || ( N - 0 ) ) ) )
29	28	anbi2d 460	. . . . . . . . . . . . . . . . . . . . . . . 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 2167	. . . . . . . . . . . . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . . . . . . . . . . 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 2520	. . . . . . . . . . . . . . . . . 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 384	. . . . . . . . . . . . . . . . 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 2534	. . . . . . . . . . . . . . . . 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 1182	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) )
43		breq1 3968	. . . . . . . . . . . . . . . 16 |- ( r = K -> ( r < D <-> K < D ) )
44		oveq2 5832	. . . . . . . . . . . . . . . . 17 |- ( r = K -> ( N - r ) = ( N - K ) )
45	44	breq2d 3977	. . . . . . . . . . . . . . . 16 |- ( r = K -> ( D || ( N - r ) <-> D || ( N - K ) ) )
46	43, 45	anbi12d 465	. . . . . . . . . . . . . . 15 |- ( r = K -> ( ( r < D /\ D || ( N - r ) ) <-> ( K < D /\ D || ( N - K ) ) ) )
47		eqeq1 2164	. . . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . 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		pm3.37 679	. . . . . . . . . . . 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 364	. . . . . . . . 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 347	. . . . . . 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 1187	. . 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 1182	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 963   = wceq 1335   e. wcel 2128   =/= wne 2327   A.wral 2435   E.wrex 2436   E!wreu 2437   class class class wbr 3965  (class class class)co 5824   0cc0 7732   < clt 7912   - cmin 8046   NNcn 8833   NN0cn0 9090   ZZcz 9167   || cdvds 11683

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-frec 6338  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-2 8892  df-n0 9091  df-z 9168  df-uz 9440  df-q 9529  df-rp 9561  df-fl 10169  df-mod 10222  df-seqfrec 10345  df-exp 10419  df-cj 10742  df-re 10743  df-im 10744  df-rsqrt 10898  df-abs 10899  df-dvds 11684

This theorem is referenced by:  ndvdsadd  11821