Theorem difmodm1lt 40833

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 464	. . . 4 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( A mod N ) = 1 )
2		zre 10965	. . . . . . . . 9 |- ( A e. ZZ -> A e. RR )
3	2	3ad2ant1 1051	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> A e. RR )
4		nnre 10638	. . . . . . . . 9 |- ( N e. NN -> N e. RR )
5	4	3ad2ant2 1052	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> N e. RR )
6		1lt2 10799	. . . . . . . . . . 11 |- 1 < 2
7		1red 9676	. . . . . . . . . . . . 13 |- ( N e. NN -> 1 e. RR )
8		2re 10701	. . . . . . . . . . . . . 14 |- 2 e. RR
9	8	a1i 11	. . . . . . . . . . . . 13 |- ( N e. NN -> 2 e. RR )
10	7, 9, 4	3jca 1210	. . . . . . . . . . . 12 |- ( N e. NN -> ( 1 e. RR /\ 2 e. RR /\ N e. RR ) )
11		lttr 9728	. . . . . . . . . . . 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 690	. . . . . . . . . 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 1224	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> 1 < N )
16	3, 5, 15	3jca 1210	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A e. RR /\ N e. RR /\ 1 < N ) )
17	16	adantl 473	. . . . . 6 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( A e. RR /\ N e. RR /\ 1 < N ) )
18		m1mod0mod1 38868	. . . . . 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 240	. . . 4 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A - 1 ) mod N ) = 0 )
21	1, 20	oveq12d 6326	. . 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 10690	. . . . . . . . . 10 |- 2 = ( 1 + 1 )
23	22	breq1i 4402	. . . . . . . . 9 |- ( 2 < N <-> ( 1 + 1 ) < N )
24	23	biimpi 199	. . . . . . . 8 |- ( 2 < N -> ( 1 + 1 ) < N )
25	24	adantl 473	. . . . . . 7 |- ( ( N e. NN /\ 2 < N ) -> ( 1 + 1 ) < N )
26		1red 9676	. . . . . . . 8 |- ( ( N e. NN /\ 2 < N ) -> 1 e. RR )
27	4	adantr 472	. . . . . . . 8 |- ( ( N e. NN /\ 2 < N ) -> N e. RR )
28	26, 26, 27	ltaddsub2d 10235	. . . . . . 7 |- ( ( N e. NN /\ 2 < N ) -> ( ( 1 + 1 ) < N <-> 1 < ( N - 1 ) ) )
29	25, 28	mpbid 215	. . . . . 6 |- ( ( N e. NN /\ 2 < N ) -> 1 < ( N - 1 ) )
30		1m0e1 10742	. . . . . . 7 |- ( 1 - 0 ) = 1
31	30	breq1i 4402	. . . . . 6 |- ( ( 1 - 0 ) < ( N - 1 ) <-> 1 < ( N - 1 ) )
32	29, 31	sylibr 217	. . . . 5 |- ( ( N e. NN /\ 2 < N ) -> ( 1 - 0 ) < ( N - 1 ) )
33	32	3adant1 1048	. . . 4 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( 1 - 0 ) < ( N - 1 ) )
34	33	adantl 473	. . 3 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( 1 - 0 ) < ( N - 1 ) )
35	21, 34	eqbrtrd 4416	. 2 |- ( ( ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) < ( N - 1 ) )
36		zmodfz 12151	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN ) -> ( A mod N ) e. ( 0 ... ( N - 1 ) ) )
37	36	3adant3 1050	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A mod N ) e. ( 0 ... ( N - 1 ) ) )
38		elfzle2 11829	. . . . . 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 473	. . . 4 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( A mod N ) <_ ( N - 1 ) )
41		nnrp 11334	. . . . . . . . 9 |- ( N e. NN -> N e. RR+ )
42	41	3ad2ant2 1052	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> N e. RR+ )
43	3, 42	modcld 12135	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A mod N ) e. RR )
44		peano2rem 9961	. . . . . . . . 9 |- ( N e. RR -> ( N - 1 ) e. RR )
45	4, 44	syl 17	. . . . . . . 8 |- ( N e. NN -> ( N - 1 ) e. RR )
46	45	3ad2ant2 1052	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( N - 1 ) e. RR )
47		peano2zm 11004	. . . . . . . . . 10 |- ( A e. ZZ -> ( A - 1 ) e. ZZ )
48	47	zred 11063	. . . . . . . . 9 |- ( A e. ZZ -> ( A - 1 ) e. RR )
49	48	3ad2ant1 1051	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( A - 1 ) e. RR )
50	49, 42	modcld 12135	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A - 1 ) mod N ) e. RR )
51	43, 46, 50	3jca 1210	. . . . . 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 473	. . . . 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 10129	. . . . 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 215	. . 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 541	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A - 1 ) e. RR /\ N e. RR+ ) )
57	56	adantl 473	. . . . . . 7 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A - 1 ) e. RR /\ N e. RR+ ) )
58		modge0 12139	. . . . . . 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 206	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A mod N ) = 1 <-> ( ( A - 1 ) mod N ) = 0 ) )
62	61	notbid 301	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( -. ( A mod N ) = 1 <-> -. ( ( A - 1 ) mod N ) = 0 ) )
63	62	biimpac 494	. . . . . . 7 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> -. ( ( A - 1 ) mod N ) = 0 )
64	63	neqned 2650	. . . . . 6 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( ( A - 1 ) mod N ) =/= 0 )
65	59, 64	jca 541	. . . . 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 9662	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> 0 e. RR )
67	66, 50	jca 541	. . . . . . 7 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( 0 e. RR /\ ( ( A - 1 ) mod N ) e. RR ) )
68	67	adantl 473	. . . . . 6 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> ( 0 e. RR /\ ( ( A - 1 ) mod N ) e. RR ) )
69		ltlen 9753	. . . . . 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 240	. . . 4 |- ( ( -. ( A mod N ) = 1 /\ ( A e. ZZ /\ N e. NN /\ 2 < N ) ) -> 0 < ( ( A - 1 ) mod N ) )
72	50, 46	jca 541	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( ( A - 1 ) mod N ) e. RR /\ ( N - 1 ) e. RR ) )
73	72	adantl 473	. . . . 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 10127	. . . . 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 215	. . 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 10068	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( A mod N ) - ( ( A - 1 ) mod N ) ) e. RR )
78	46, 50	resubcld 10068	. . . . . 6 |- ( ( A e. ZZ /\ N e. NN /\ 2 < N ) -> ( ( N - 1 ) - ( ( A - 1 ) mod N ) ) e. RR )
79	77, 78, 46	3jca 1210	. . . . 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 473	. . . 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 9742	. . . 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 693	. 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 807	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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   class class class wbr 4395  (class class class)co 6308   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   < clt 9693   <_ cle 9694   - cmin 9880   NNcn 10631   2c2 10681   ZZcz 10961   RR+crp 11325   ...cfz 11810   mod cmo 12129

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-inf 7975  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-fl 12061  df-mod 12130

This theorem is referenced by: (None)