Theorem ndvdssub 12490

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 9408	. . . . . . . 8 |- ( K e. NN -> K e. NN0 )
2		nnne0 9170	. . . . . . . 8 |- ( K e. NN -> K =/= 0 )
3	1, 2	jca 306	. . . . . . 7 |- ( K e. NN -> ( K e. NN0 /\ K =/= 0 ) )
4		df-ne 2403	. . . . . . . . . . . 12 |- ( K =/= 0 <-> -. K = 0 )
5	4	anbi2i 457	. . . . . . . . . . 11 |- ( ( K < D /\ K =/= 0 ) <-> ( K < D /\ -. K = 0 ) )
6		divalg2 12486	. . . . . . . . . . . . . . . . . . 19 |- ( ( N e. ZZ /\ D e. NN ) -> E! r e. NN0 ( r < D /\ D || ( N - r ) ) )
7		breq1 4091	. . . . . . . . . . . . . . . . . . . . 21 |- ( r = x -> ( r < D <-> x < D ) )
8		oveq2 6025	. . . . . . . . . . . . . . . . . . . . . 22 |- ( r = x -> ( N - r ) = ( N - x ) )
9	8	breq2d 4100	. . . . . . . . . . . . . . . . . . . . 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 3000	. . . . . . . . . . . . . . . . . . 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 9167	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( D e. NN -> 0 < D )
14	13	3ad2ant2 1045	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> 0 < D )
15		zcn 9483	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ( N e. ZZ -> N e. CC )
16	15	subid1d 8478	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. ZZ -> ( N - 0 ) = N )
17	16	breq2d 4100	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1042	. . . . . . . . . . . . . . . . . . . . . . . . 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 1229	. . . . . . . . . . . . . . . . . . . . . . 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 9416	. . . . . . . . . . . . . . . . . . . . . 22 |- 0 e. NN0
25		breq1 4091	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( x = 0 -> ( x < D <-> 0 < D ) )
26		oveq2 6025	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( x = 0 -> ( N - x ) = ( N - 0 ) )
27	26	breq2d 4100	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2241	. . . . . . . . . . . . . . . . . . . . . . . 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 2906	. . . . . . . . . . . . . . . . . . . . . 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 2595	. . . . . . . . . . . . . . . . . 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 2609	. . . . . . . . . . . . . . . . 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 1226	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ D e. NN /\ D || N ) -> A. r e. NN0 ( ( r < D /\ D || ( N - r ) ) -> r = 0 ) )
43		breq1 4091	. . . . . . . . . . . . . . . 16 |- ( r = K -> ( r < D <-> K < D ) )
44		oveq2 6025	. . . . . . . . . . . . . . . . 17 |- ( r = K -> ( N - r ) = ( N - K ) )
45	44	breq2d 4100	. . . . . . . . . . . . . . . 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 2238	. . . . . . . . . . . . . . 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 2906	. . . . . . . . . . . . 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		pm3.37 695	. . . . . . . . . . . 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 1231	. . 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 1226	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 1004   = wceq 1397   e. wcel 2202   =/= wne 2402   A.wral 2510   E.wrex 2511   E!wreu 2512   class class class wbr 4088  (class class class)co 6017   0cc0 8031   < clt 8213   - cmin 8349   NNcn 9142   NN0cn0 9401   ZZcz 9478   || cdvds 12347

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348

This theorem is referenced by:  ndvdsadd  12491