Theorem difmodm1lt 42317

Description: The difference between an integer modulo a positive integer and the integer decreased by 1 modulo the same modulus is less than the modulus decreased by 1 (if the modulus is greater than 2). This theorem would not be valid for an odd A and N = 2, since ( ( A mod N ) - ( ( A - 1 ) mod N ) ) would be ( 1 - 0 ) = 1 which is not less than ( N - 1 ) = 1. (Contributed by AV, 6-Jun-2012.)

Assertion

Ref	Expression
difmodm1lt	|- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) )

Proof of Theorem difmodm1lt

Step	Hyp	Ref	Expression
1		simpl 473	. . . 4 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( A mod N ) = 1 )
2		zre 11381	. . . . . . . . 9 |- ( A e. ZZ -> A e. RR )
3	2	3ad2ant1 1082	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> A e. RR )
4		nnre 11027	. . . . . . . . 9 |- ( N e. NN -> N e. RR )
5	4	3ad2ant2 1083	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> N e. RR )
6		1lt2 11194	. . . . . . . . . . 11 |- 1 < 2
7		1red 10055	. . . . . . . . . . . . 13 |- ( N e. NN -> 1 e. RR )
8		2re 11090	. . . . . . . . . . . . . 14 |- 2 e. RR
9	8	a1i 11	. . . . . . . . . . . . 13 |- ( N e. NN -> 2 e. RR )
10	7, 9, 4	3jca 1242	. . . . . . . . . . . 12 |- ( N e. NN -> ( 1 e. RR /\ 2 e. RR /\ N e. RR ) )
11		lttr 10114	. . . . . . . . . . . 12 |- ( ( 1 e. RR /\ 2 e. RR /\ N e. RR ) -> ( ( 1 < 2 /\ 2 < N ) -> 1 < N ) )
12	10, 11	syl 17	. . . . . . . . . . 11 |- ( N e. NN -> ( ( 1 < 2 /\ 2 < N ) -> 1 < N ) )
13	6, 12	mpani 712	. . . . . . . . . 10 |- ( N e. NN -> ( 2 < N -> 1 < N ) )
14	13	a1i 11	. . . . . . . . 9 |- ( A e. ZZ -> ( N e. NN -> ( 2 < N -> 1 < N ) ) )
15	14	3imp 1256	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> 1 < N )
16	3, 5, 15	3jca 1242	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A e. RR /\ N e. RR /\ 1 < N ) )
17	16	adantl 482	. . . . . 6 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( A e. RR /\ N e. RR /\ 1 < N ) )
18		m1mod0mod1 41339	. . . . . 6 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( ( ( A - 1 ) mod N ) = 0 <-> ( A mod N ) = 1 ) )
19	17, 18	syl 17	. . . . 5 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( ( A - 1 ) mod N ) = 0 <-> ( A mod N ) = 1 ) )
20	1, 19	mpbird 247	. . . 4 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A - 1 ) mod N ) = 0 )
21	1, 20	oveq12d 6668	. . 3 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) = ( 1 - 0 ) )
22		df-2 11079	. . . . . . . . . 10 |- 2 = ( 1 + 1 )
23	22	breq1i 4660	. . . . . . . . 9 |- ( 2 < N <-> ( 1 + 1 ) < N )
24	23	biimpi 206	. . . . . . . 8 |- ( 2 < N -> ( 1 + 1 ) < N )
25	24	adantl 482	. . . . . . 7 |- ( ( N e. NN /\ 2 < N ) -> ( 1 + 1 ) < N )
26		1red 10055	. . . . . . . 8 |- ( ( N e. NN /\ 2 < N ) -> 1 e. RR )
27	4	adantr 481	. . . . . . . 8 |- ( ( N e. NN /\ 2 < N ) -> N e. RR )
28	26, 26, 27	ltaddsub2d 10628	. . . . . . 7 |- ( ( N e. NN /\ 2 < N ) -> ( ( 1 + 1 ) < N <-> 1 < ( N - 1 ) ) )
29	25, 28	mpbid 222	. . . . . 6 |- ( ( N e. NN /\ 2 < N ) -> 1 < ( N - 1 ) )
30		1m0e1 11131	. . . . . . 7 |- ( 1 - 0 ) = 1
31	30	breq1i 4660	. . . . . 6 |- ( ( 1 - 0 ) < ( N - 1 ) <-> 1 < ( N - 1 ) )
32	29, 31	sylibr 224	. . . . 5 |- ( ( N e. NN /\ 2 < N ) -> ( 1 - 0 ) < ( N - 1 ) )
33	32	3adant1 1079	. . . 4 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( 1 - 0 ) < ( N - 1 ) )
34	33	adantl 482	. . 3 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( 1 - 0 ) < ( N - 1 ) )
35	21, 34	eqbrtrd 4675	. 2 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) )
36		zmodfz 12692	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN ) -> ( A mod N ) e. ( 0 ... ( N - 1 ) ) )
37	36	3adant3 1081	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A mod N ) e. ( 0 ... ( N - 1 ) ) )
38		elfzle2 12345	. . . . . 6 |- ( ( A mod N ) e. ( 0 ... ( N - 1 ) ) -> ( A mod N ) <_ ( N - 1 ) )
39	37, 38	syl 17	. . . . 5 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A mod N ) <_ ( N - 1 ) )
40	39	adantl 482	. . . 4 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( A mod N ) <_ ( N - 1 ) )
41		nnrp 11842	. . . . . . . . 9 |- ( N e. NN -> N e. RR+ )
42	41	3ad2ant2 1083	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> N e. RR+ )
43	3, 42	modcld 12674	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A mod N ) e. RR )
44		peano2rem 10348	. . . . . . . . 9 |- ( N e. RR -> ( N - 1 ) e. RR )
45	4, 44	syl 17	. . . . . . . 8 |- ( N e. NN -> ( N - 1 ) e. RR )
46	45	3ad2ant2 1083	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( N - 1 ) e. RR )
47		peano2zm 11420	. . . . . . . . . 10 |- ( A e. ZZ -> ( A - 1 ) e. ZZ )
48	47	zred 11482	. . . . . . . . 9 |- ( A e. ZZ -> ( A - 1 ) e. RR )
49	48	3ad2ant1 1082	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A - 1 ) e. RR )
50	49, 42	modcld 12674	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A - 1 ) mod N ) e. RR )
51	43, 46, 50	3jca 1242	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A mod N ) e. RR /\ ( N - 1 ) e. RR /\ ( ( A - 1 ) mod N ) e. RR ) )
52	51	adantl 482	. . . . 5 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A mod N ) e. RR /\ ( N - 1 ) e. RR /\ ( ( A - 1 ) mod N ) e. RR ) )
53		lesub1 10522	. . . . 5 |- ( ( ( A mod N ) e. RR /\ ( N - 1 ) e. RR /\ ( ( A - 1 ) mod N ) e. RR ) -> ( ( A mod N ) <_ ( N - 1 ) <-> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) <_ ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) ) )
54	52, 53	syl 17	. . . 4 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A mod N ) <_ ( N - 1 ) <-> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) <_ ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) ) )
55	40, 54	mpbid 222	. . 3 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) <_ ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) )
56	49, 42	jca 554	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A - 1 ) e. RR /\ N e. RR+ ) )
57	56	adantl 482	. . . . . . 7 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A - 1 ) e. RR /\ N e. RR+ ) )
58		modge0 12678	. . . . . . 7 |- ( ( ( A - 1 ) e. RR /\ N e. RR+ ) -> 0 <_ ( ( A - 1 ) mod N ) )
59	57, 58	syl 17	. . . . . 6 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> 0 <_ ( ( A - 1 ) mod N ) )
60	16, 18	syl 17	. . . . . . . . . 10 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( ( A - 1 ) mod N ) = 0 <-> ( A mod N ) = 1 ) )
61	60	bicomd 213	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A mod N ) = 1 <-> ( ( A - 1 ) mod N ) = 0 ) )
62	61	notbid 308	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( -. ( A mod N ) = 1 <-> -. ( ( A - 1 ) mod N ) = 0 ) )
63	62	biimpac 503	. . . . . . 7 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> -. ( ( A - 1 ) mod N ) = 0 )
64	63	neqned 2801	. . . . . 6 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A - 1 ) mod N ) =/= 0 )
65	59, 64	jca 554	. . . . 5 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( 0 <_ ( ( A - 1 ) mod N ) /\ ( ( A - 1 ) mod N ) =/= 0 ) )
66		0red 10041	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> 0 e. RR )
67	66, 50	jca 554	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( 0 e. RR /\ ( ( A - 1 ) mod N ) e. RR ) )
68	67	adantl 482	. . . . . 6 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( 0 e. RR /\ ( ( A - 1 ) mod N ) e. RR ) )
69		ltlen 10138	. . . . . 6 |- ( ( 0 e. RR /\ ( ( A - 1 ) mod N ) e. RR ) -> ( 0 < ( ( A - 1 ) mod N ) <-> ( 0 <_ ( ( A - 1 ) mod N ) /\ ( ( A - 1 ) mod N ) =/= 0 ) ) )
70	68, 69	syl 17	. . . . 5 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( 0 < ( ( A - 1 ) mod N ) <-> ( 0 <_ ( ( A - 1 ) mod N ) /\ ( ( A - 1 ) mod N ) =/= 0 ) ) )
71	65, 70	mpbird 247	. . . 4 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> 0 < ( ( A - 1 ) mod N ) )
72	50, 46	jca 554	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( ( A - 1 ) mod N ) e. RR /\ ( N - 1 ) e. RR ) )
73	72	adantl 482	. . . . 5 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( ( A - 1 ) mod N ) e. RR /\ ( N - 1 ) e. RR ) )
74		ltsubpos 10520	. . . . 5 |- ( ( ( ( A - 1 ) mod N ) e. RR /\ ( N - 1 ) e. RR ) -> ( 0 < ( ( A - 1 ) mod N ) <-> ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) ) )
75	73, 74	syl 17	. . . 4 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( 0 < ( ( A - 1 ) mod N ) <-> ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) ) )
76	71, 75	mpbid 222	. . 3 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) )
77	43, 50	resubcld 10458	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) e. RR )
78	46, 50	resubcld 10458	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) e. RR )
79	77, 78, 46	3jca 1242	. . . . 5 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( ( A mod N ) - ( ( A - 1 ) mod N ) ) e. RR /\ ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) e. RR /\ ( N - 1 ) e. RR ) )
80	79	adantl 482	. . . 4 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( ( A mod N ) - ( ( A - 1 ) mod N ) ) e. RR /\ ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) e. RR /\ ( N - 1 ) e. RR ) )
81		lelttr 10128	. . . 4 |- ( ( ( ( A mod N ) - ( ( A - 1 ) mod N ) ) e. RR /\ ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) e. RR /\ ( N - 1 ) e. RR ) -> ( ( ( ( A mod N ) - ( ( A - 1 ) mod N ) ) <_ ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) /\ ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) ) )
82	80, 81	syl 17	. . 3 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( ( ( A mod N ) - ( ( A - 1 ) mod N ) ) <_ ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) /\ ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) ) )
83	55, 76, 82	mp2and 715	. 2 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) )
84	35, 83	pm2.61ian 831	1 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   ZZcz 11377   RR+crp 11832   ...cfz 12326   mod cmo 12668

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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669

This theorem is referenced by: (None)