Theorem ndvdssub 15133

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 11299	. . . . . . . 8 |- ( K e. NN -> K e. NN0 )
2		nnne0 11053	. . . . . . . 8 |- ( K e. NN -> K =/= 0 )
3	1, 2	jca 554	. . . . . . 7 |- ( K e. NN -> ( K e. NN0 /\ K =/= 0 ) )
4		df-ne 2795	. . . . . . . . . . . 12 |- ( K =/= 0 <-> -. K = 0 )
5	4	anbi2i 730	. . . . . . . . . . 11 |- ( ( K < D /\ K =/= 0 ) <-> ( K < D /\ -. K = 0 ) )
6		divalg2 15128	. . . . . . . . . . . . . . . . . . 19 |- ( ( N e. ZZ /\ D e. NN ) -> E! r e. NN0 ( r < D /\ D || ( N - r ) ) )
7		breq1 4656	. . . . . . . . . . . . . . . . . . . . 21 |- ( r = x -> ( r < D <-> x < D ) )
8		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . 22 |- ( r = x -> ( N - r ) = ( N - x ) )
9	8	breq2d 4665	. . . . . . . . . . . . . . . . . . . . 21 |- ( r = x -> ( D || ( N - r ) <-> D || ( N - x ) ) )
10	7, 9	anbi12d 747	. . . . . . . . . . . . . . . . . . . 20 |- ( r = x -> ( ( r < D /\ D || ( N - r ) ) <-> ( x < D /\ D || ( N - x ) ) ) )
11	10	reu4 3400	. . . . . . . . . . . . . . . . . . 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 208	. . . . . . . . . . . . . . . . . 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 11049	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( D e. NN -> 0 < D )
14	13	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> 0 < D )
15		zcn 11382	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( N e. ZZ -> N e. CC )
16	15	subid1d 10381	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. ZZ -> ( N - 0 ) = N )
17	16	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. ZZ -> ( D || ( N - 0 ) <-> D || N ) )
18	17	biimpar 502	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N e. ZZ /\ D || N ) -> D || ( N - 0 ) )
19	18	3adant2 1080	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> D || ( N - 0 ) )
20	14, 19	jca 554	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( 0 < D /\ D || ( N - 0 ) ) )
21	20	3expa 1265	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( N e. ZZ /\ D e. NN ) /\ D || N ) -> ( 0 < D /\ D || ( N - 0 ) ) )
22	21	anim2i 593	. . . . . . . . . . . . . . . . . . . . . 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 469	. . . . . . . . . . . . . . . . . . . . 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 11307	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 e. NN0
25		breq1 4656	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( x = 0 -> ( x < D <-> 0 < D ) )
26		oveq2 6658	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( x = 0 -> ( N - x ) = ( N - 0 ) )
27	26	breq2d 4665	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( x = 0 -> ( D || ( N - x ) <-> D || ( N - 0 ) ) )
28	25, 27	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( x = 0 -> ( ( x < D /\ D || ( N - x ) ) <-> ( 0 < D /\ D || ( N - 0 ) ) ) )
29	28	anbi2d 740	. . . . . . . . . . . . . . . . . . . . . . . 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 2633	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( x = 0 -> ( r = x <-> r = 0 ) )
31	29, 30	imbi12d 334	. . . . . . . . . . . . . . . . . . . . . . 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 3305	. . . . . . . . . . . . . . . . . . . . . 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 34	. . . . . . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . . . . . 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 2952	. . . . . . . . . . . . . . . . . 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 658	. . . . . . . . . . . . . . . . 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 2960	. . . . . . . . . . . . . . . . 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 208	. . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . . 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 52	. . . . . . . . . . . . . 14 |- ( ( N e. ZZ /\ D e. NN ) -> ( D || N -> A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) ) )
42	41	3impia 1261	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) )
43		breq1 4656	. . . . . . . . . . . . . . . 16 |- ( r = K -> ( r < D <-> K < D ) )
44		oveq2 6658	. . . . . . . . . . . . . . . . 17 |- ( r = K -> ( N - r ) = ( N - K ) )
45	44	breq2d 4665	. . . . . . . . . . . . . . . 16 |- ( r = K -> ( D || ( N - r ) <-> D || ( N - K ) ) )
46	43, 45	anbi12d 747	. . . . . . . . . . . . . . 15 |- ( r = K -> ( ( r < D /\ D || ( N - r ) ) <-> ( K < D /\ D || ( N - K ) ) ) )
47		eqeq1 2626	. . . . . . . . . . . . . . 15 |- ( r = K -> ( r = 0 <-> K = 0 ) )
48	46, 47	imbi12d 334	. . . . . . . . . . . . . 14 |- ( r = K -> ( ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) <-> ( ( K < D /\ D || ( N - K ) ) -> K = 0 ) ) )
49	48	rspcv 3305	. . . . . . . . . . . . 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 31	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN0 -> ( ( K < D /\ D || ( N - K ) ) -> K = 0 ) ) )
51		pm4.14 602	. . . . . . . . . . . 12 |- ( ( ( K < D /\ D || ( N - K ) ) -> K = 0 ) <-> ( ( K < D /\ -. K = 0 ) -> -. D || ( N - K ) ) )
52	50, 51	syl6ib 241	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN0 -> ( ( K < D /\ -. K = 0 ) -> -. D || ( N - K ) ) ) )
53	5, 52	syl7bi 245	. . . . . . . . . 10 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN0 -> ( ( K < D /\ K =/= 0 ) -> -. D || ( N - K ) ) ) )
54	53	exp4a 633	. . . . . . . . 9 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN0 -> ( K < D -> ( K =/= 0 -> -. D || ( N - K ) ) ) ) )
55	54	com23 86	. . . . . . . 8 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K < D -> ( K e. NN0 -> ( K =/= 0 -> -. D || ( N - K ) ) ) ) )
56	55	imp4a 614	. . . . . . 7 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K < D -> ( ( K e. NN0 /\ K =/= 0 ) -> -. D || ( N - K ) ) ) )
57	3, 56	syl7 74	. . . . . 6 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K < D -> ( K e. NN -> -. D || ( N - K ) ) ) )
58	57	com23 86	. . . . 5 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( K e. NN -> ( K < D -> -. D || ( N - K ) ) ) )
59	58	impd 447	. . . 4 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> ( ( K e. NN /\ K < D ) -> -. D || ( N - K ) ) )
60	59	3expia 1267	. . 3 |- ( ( N e. ZZ /\ D e. NN ) -> ( D || N -> ( ( K e. NN /\ K < D ) -> -. D || ( N - K ) ) ) )
61	60	com23 86	. 2 |- ( ( N e. ZZ /\ D e. NN ) -> ( ( K e. NN /\ K < D ) -> ( D || N -> -. D || ( N - K ) ) ) )
62	61	3impia 1261	1 |- ( ( N e. ZZ /\ D e. NN /\ ( K e. NN /\ K < D ) ) -> ( D || N -> -. D || ( N - K ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   class class class wbr 4653  (class class class)co 6650   0cc0 9936   < clt 10074   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984

This theorem is referenced by:  ndvdsadd  15134