Theorem m1mod0mod1 41339

Description: An integer decreased by 1 is 0 modulo a positive integer iff the integer is 1 modulo the same modulus. (Contributed by AV, 6-Jun-2020.)

Assertion

Ref	Expression
m1mod0mod1	|- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( ( ( A - 1 ) mod N ) = 0 <-> ( A mod N ) = 1 ) )

Proof of Theorem m1mod0mod1

Step	Hyp	Ref	Expression
1		recn 10026	. . . . . . 7 |- ( A e. RR -> A e. CC )
2		npcan1 10455	. . . . . . . 8 |- ( A e. CC -> ( ( A - 1 ) + 1 ) = A )
3	2	eqcomd 2628	. . . . . . 7 |- ( A e. CC -> A = ( ( A - 1 ) + 1 ) )
4	1, 3	syl 17	. . . . . 6 |- ( A e. RR -> A = ( ( A - 1 ) + 1 ) )
5	4	3ad2ant1 1082	. . . . 5 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> A = ( ( A - 1 ) + 1 ) )
6	5	adantr 481	. . . 4 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> A = ( ( A - 1 ) + 1 ) )
7	6	oveq1d 6665	. . 3 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> ( A mod N ) = ( ( ( A - 1 ) + 1 ) mod N ) )
8		simpr 477	. . . . . 6 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> ( ( A - 1 ) mod N ) = 0 )
9		1mod 12702	. . . . . . . 8 |- ( ( N e. RR /\ 1 < N ) -> ( 1 mod N ) = 1 )
10	9	3adant1 1079	. . . . . . 7 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( 1 mod N ) = 1 )
11	10	adantr 481	. . . . . 6 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> ( 1 mod N ) = 1 )
12	8, 11	oveq12d 6668	. . . . 5 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> ( ( ( A - 1 ) mod N ) + ( 1 mod N ) ) = ( 0 + 1 ) )
13	12	oveq1d 6665	. . . 4 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> ( ( ( ( A - 1 ) mod N ) + ( 1 mod N ) ) mod N ) = ( ( 0 + 1 ) mod N ) )
14		peano2rem 10348	. . . . . . . 8 |- ( A e. RR -> ( A - 1 ) e. RR )
15	14	3ad2ant1 1082	. . . . . . 7 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( A - 1 ) e. RR )
16		1red 10055	. . . . . . 7 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> 1 e. RR )
17		simpl 473	. . . . . . . . 9 |- ( ( N e. RR /\ 1 < N ) -> N e. RR )
18		0lt1 10550	. . . . . . . . . . 11 |- 0 < 1
19		0re 10040	. . . . . . . . . . . 12 |- 0 e. RR
20		1re 10039	. . . . . . . . . . . 12 |- 1 e. RR
21		lttr 10114	. . . . . . . . . . . 12 |- ( ( 0 e. RR /\ 1 e. RR /\ N e. RR ) -> ( ( 0 < 1 /\ 1 < N ) -> 0 < N ) )
22	19, 20, 21	mp3an12 1414	. . . . . . . . . . 11 |- ( N e. RR -> ( ( 0 < 1 /\ 1 < N ) -> 0 < N ) )
23	18, 22	mpani 712	. . . . . . . . . 10 |- ( N e. RR -> ( 1 < N -> 0 < N ) )
24	23	imp 445	. . . . . . . . 9 |- ( ( N e. RR /\ 1 < N ) -> 0 < N )
25	17, 24	elrpd 11869	. . . . . . . 8 |- ( ( N e. RR /\ 1 < N ) -> N e. RR+ )
26	25	3adant1 1079	. . . . . . 7 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> N e. RR+ )
27	15, 16, 26	3jca 1242	. . . . . 6 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( ( A - 1 ) e. RR /\ 1 e. RR /\ N e. RR+ ) )
28	27	adantr 481	. . . . 5 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> ( ( A - 1 ) e. RR /\ 1 e. RR /\ N e. RR+ ) )
29		modaddabs 12708	. . . . 5 |- ( ( ( A - 1 ) e. RR /\ 1 e. RR /\ N e. RR+ ) -> ( ( ( ( A - 1 ) mod N ) + ( 1 mod N ) ) mod N ) = ( ( ( A - 1 ) + 1 ) mod N ) )
30	28, 29	syl 17	. . . 4 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> ( ( ( ( A - 1 ) mod N ) + ( 1 mod N ) ) mod N ) = ( ( ( A - 1 ) + 1 ) mod N ) )
31		0p1e1 11132	. . . . . . . 8 |- ( 0 + 1 ) = 1
32	31	oveq1i 6660	. . . . . . 7 |- ( ( 0 + 1 ) mod N ) = ( 1 mod N )
33	32, 9	syl5eq 2668	. . . . . 6 |- ( ( N e. RR /\ 1 < N ) -> ( ( 0 + 1 ) mod N ) = 1 )
34	33	3adant1 1079	. . . . 5 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( ( 0 + 1 ) mod N ) = 1 )
35	34	adantr 481	. . . 4 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> ( ( 0 + 1 ) mod N ) = 1 )
36	13, 30, 35	3eqtr3d 2664	. . 3 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> ( ( ( A - 1 ) + 1 ) mod N ) = 1 )
37	7, 36	eqtrd 2656	. 2 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( ( A - 1 ) mod N ) = 0 ) -> ( A mod N ) = 1 )
38		simpr 477	. . . . . 6 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( A mod N ) = 1 ) -> ( A mod N ) = 1 )
39	38	eqcomd 2628	. . . . 5 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( A mod N ) = 1 ) -> 1 = ( A mod N ) )
40	39	oveq2d 6666	. . . 4 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( A mod N ) = 1 ) -> ( A - 1 ) = ( A - ( A mod N ) ) )
41	40	oveq1d 6665	. . 3 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( A mod N ) = 1 ) -> ( ( A - 1 ) mod N ) = ( ( A - ( A mod N ) ) mod N ) )
42		simp1 1061	. . . . . . . . 9 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> A e. RR )
43	42, 26	modcld 12674	. . . . . . . 8 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( A mod N ) e. RR )
44	43	recnd 10068	. . . . . . 7 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( A mod N ) e. CC )
45	44	subidd 10380	. . . . . 6 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( ( A mod N ) - ( A mod N ) ) = 0 )
46	45	oveq1d 6665	. . . . 5 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( ( ( A mod N ) - ( A mod N ) ) mod N ) = ( 0 mod N ) )
47		modsubmod 12728	. . . . . 6 |- ( ( A e. RR /\ ( A mod N ) e. RR /\ N e. RR+ ) -> ( ( ( A mod N ) - ( A mod N ) ) mod N ) = ( ( A - ( A mod N ) ) mod N ) )
48	42, 43, 26, 47	syl3anc 1326	. . . . 5 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( ( ( A mod N ) - ( A mod N ) ) mod N ) = ( ( A - ( A mod N ) ) mod N ) )
49		0mod 12701	. . . . . 6 |- ( N e. RR+ -> ( 0 mod N ) = 0 )
50	26, 49	syl 17	. . . . 5 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( 0 mod N ) = 0 )
51	46, 48, 50	3eqtr3d 2664	. . . 4 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( ( A - ( A mod N ) ) mod N ) = 0 )
52	51	adantr 481	. . 3 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( A mod N ) = 1 ) -> ( ( A - ( A mod N ) ) mod N ) = 0 )
53	41, 52	eqtrd 2656	. 2 |- ( ( ( A e. RR /\ N e. RR /\ 1 < N ) /\ ( A mod N ) = 1 ) -> ( ( A - 1 ) mod N ) = 0 )
54	37, 53	impbida 877	1 |- ( ( A e. RR /\ N e. RR /\ 1 < N ) -> ( ( ( A - 1 ) mod N ) = 0 <-> ( A mod N ) = 1 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   RR+crp 11832   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-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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fl 12593  df-mod 12669

This theorem is referenced by:  dfodd4  41571  difmodm1lt  42317