Theorem difmodm1lt 40378

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 459	. . . 4 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( A mod N ) = 1 )
2		zre 10941	. . . . . . . . 9 |- ( A e. ZZ -> A e. RR )
3	2	3ad2ant1 1029	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> A e. RR )
4		nnre 10616	. . . . . . . . 9 |- ( N e. NN -> N e. RR )
5	4	3ad2ant2 1030	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> N e. RR )
6		1lt2 10776	. . . . . . . . . . 11 |- 1 < 2
7		1red 9658	. . . . . . . . . . . . 13 |- ( N e. NN -> 1 e. RR )
8		2re 10679	. . . . . . . . . . . . . 14 |- 2 e. RR
9	8	a1i 11	. . . . . . . . . . . . 13 |- ( N e. NN -> 2 e. RR )
10	7, 9, 4	3jca 1188	. . . . . . . . . . . 12 |- ( N e. NN -> ( 1 e. RR /\ 2 e. RR /\ N e. RR ) )
11		lttr 9710	. . . . . . . . . . . 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 682	. . . . . . . . . 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 1202	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> 1 < N )
16	3, 5, 15	3jca 1188	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A e. RR /\ N e. RR /\ 1 < N ) )
17	16	adantl 468	. . . . . 6 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( A e. RR /\ N e. RR /\ 1 < N ) )
18		m1mod0mod1 38723	. . . . . 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 236	. . . 4 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A - 1 ) mod N ) = 0 )
21	1, 20	oveq12d 6308	. . 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 10668	. . . . . . . . . 10 |- 2 = ( 1 + 1 )
23	22	breq1i 4409	. . . . . . . . 9 |- ( 2 < N <-> ( 1 + 1 ) < N )
24	23	biimpi 198	. . . . . . . 8 |- ( 2 < N -> ( 1 + 1 ) < N )
25	24	adantl 468	. . . . . . 7 |- ( ( N e. NN /\ 2 < N ) -> ( 1 + 1 ) < N )
26		1red 9658	. . . . . . . 8 |- ( ( N e. NN /\ 2 < N ) -> 1 e. RR )
27	4	adantr 467	. . . . . . . 8 |- ( ( N e. NN /\ 2 < N ) -> N e. RR )
28	26, 26, 27	ltaddsub2d 10214	. . . . . . 7 |- ( ( N e. NN /\ 2 < N ) -> ( ( 1 + 1 ) < N <-> 1 < ( N - 1 ) ) )
29	25, 28	mpbid 214	. . . . . 6 |- ( ( N e. NN /\ 2 < N ) -> 1 < ( N - 1 ) )
30		1m0e1 10720	. . . . . . 7 |- ( 1 - 0 ) = 1
31	30	breq1i 4409	. . . . . 6 |- ( ( 1 - 0 ) < ( N - 1 ) <-> 1 < ( N - 1 ) )
32	29, 31	sylibr 216	. . . . 5 |- ( ( N e. NN /\ 2 < N ) -> ( 1 - 0 ) < ( N - 1 ) )
33	32	3adant1 1026	. . . 4 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( 1 - 0 ) < ( N - 1 ) )
34	33	adantl 468	. . 3 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( 1 - 0 ) < ( N - 1 ) )
35	21, 34	eqbrtrd 4423	. 2 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) )
36		zmodfz 12118	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN ) -> ( A mod N ) e. ( 0 ... ( N - 1 ) ) )
37	36	3adant3 1028	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A mod N ) e. ( 0 ... ( N - 1 ) ) )
38		elfzle2 11803	. . . . . 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 468	. . . 4 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( A mod N ) <_ ( N - 1 ) )
41		nnrp 11311	. . . . . . . . 9 |- ( N e. NN -> N e. RR+ )
42	41	3ad2ant2 1030	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> N e. RR+ )
43	3, 42	modcld 12102	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A mod N ) e. RR )
44		peano2rem 9941	. . . . . . . . 9 |- ( N e. RR -> ( N - 1 ) e. RR )
45	4, 44	syl 17	. . . . . . . 8 |- ( N e. NN -> ( N - 1 ) e. RR )
46	45	3ad2ant2 1030	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( N - 1 ) e. RR )
47		peano2zm 10980	. . . . . . . . . 10 |- ( A e. ZZ -> ( A - 1 ) e. ZZ )
48	47	zred 11040	. . . . . . . . 9 |- ( A e. ZZ -> ( A - 1 ) e. RR )
49	48	3ad2ant1 1029	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A - 1 ) e. RR )
50	49, 42	modcld 12102	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A - 1 ) mod N ) e. RR )
51	43, 46, 50	3jca 1188	. . . . . 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 468	. . . . 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 10108	. . . . 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 214	. . 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 535	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A - 1 ) e. RR /\ N e. RR+ ) )
57	56	adantl 468	. . . . . . 7 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A - 1 ) e. RR /\ N e. RR+ ) )
58		modge0 12106	. . . . . . 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 205	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A mod N ) = 1 <-> ( ( A - 1 ) mod N ) = 0 ) )
62	61	notbid 296	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( -. ( A mod N ) = 1 <-> -. ( ( A - 1 ) mod N ) = 0 ) )
63	62	biimpac 489	. . . . . . 7 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> -. ( ( A - 1 ) mod N ) = 0 )
64	63	neqned 2631	. . . . . 6 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A - 1 ) mod N ) =/= 0 )
65	59, 64	jca 535	. . . . 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 9644	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> 0 e. RR )
67	66, 50	jca 535	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( 0 e. RR /\ ( ( A - 1 ) mod N ) e. RR ) )
68	67	adantl 468	. . . . . 6 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( 0 e. RR /\ ( ( A - 1 ) mod N ) e. RR ) )
69		ltlen 9735	. . . . . 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 236	. . . 4 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> 0 < ( ( A - 1 ) mod N ) )
72	50, 46	jca 535	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( ( A - 1 ) mod N ) e. RR /\ ( N - 1 ) e. RR ) )
73	72	adantl 468	. . . . 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 10106	. . . . 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 214	. . 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 10047	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) e. RR )
78	46, 50	resubcld 10047	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) e. RR )
79	77, 78, 46	3jca 1188	. . . . 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 468	. . . 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 9724	. . . 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 685	. 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 799	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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   class class class wbr 4402  (class class class)co 6290   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   < clt 9675   <_ cle 9676   - cmin 9860   NNcn 10609   2c2 10659   ZZcz 10937   RR+crp 11302   ...cfz 11784   mod cmo 12096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fl 12028  df-mod 12097

This theorem is referenced by: (None)