Theorem oddvds 17966

Description: The only multiples of A that are equal to the identity are the multiples of the order of A. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)

Hypotheses

Ref	Expression
odcl.1	|- X = ( Base ` G )
odcl.2	|- O = ( od ` G )
odid.3	|- .x. = (.g ` G )
odid.4	|- .0. = ( 0g ` G )

Assertion

Ref	Expression
oddvds	|- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )

Proof of Theorem oddvds

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( O ` A ) e. NN )
2		simpl3 1066	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> N e. ZZ )
3		dvdsval3 14987	. . . 4 |- ( ( ( O ` A ) e. NN /\ N e. ZZ ) -> ( ( O ` A ) || N <-> ( N mod ( O ` A ) ) = 0 ) )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( O ` A ) || N <-> ( N mod ( O ` A ) ) = 0 ) )
5		simpl2 1065	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> A e. X )
6		odcl.1	. . . . . . 7 |- X = ( Base ` G )
7		odid.4	. . . . . . 7 |- .0. = ( 0g ` G )
8		odid.3	. . . . . . 7 |- .x. = (.g ` G )
9	6, 7, 8	mulg0 17546	. . . . . 6 |- ( A e. X -> ( 0 .x. A ) = .0. )
10	5, 9	syl 17	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( 0 .x. A ) = .0. )
11		oveq1 6657	. . . . . 6 |- ( ( N mod ( O ` A ) ) = 0 -> ( ( N mod ( O ` A ) ) .x. A ) = ( 0 .x. A ) )
12	11	eqeq1d 2624	. . . . 5 |- ( ( N mod ( O ` A ) ) = 0 -> ( ( ( N mod ( O ` A ) ) .x. A ) = .0. <-> ( 0 .x. A ) = .0. ) )
13	10, 12	syl5ibrcom 237	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) = 0 -> ( ( N mod ( O ` A ) ) .x. A ) = .0. ) )
14	2	zred 11482	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> N e. RR )
15	1	nnrpd 11870	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( O ` A ) e. RR+ )
16		modlt 12679	. . . . . . . 8 |- ( ( N e. RR /\ ( O ` A ) e. RR+ ) -> ( N mod ( O ` A ) ) < ( O ` A ) )
17	14, 15, 16	syl2anc 693	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( N mod ( O ` A ) ) < ( O ` A ) )
18	2, 1	zmodcld 12691	. . . . . . . . 9 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( N mod ( O ` A ) ) e. NN0 )
19	18	nn0red 11352	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( N mod ( O ` A ) ) e. RR )
20	1	nnred 11035	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( O ` A ) e. RR )
21	19, 20	ltnled 10184	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) < ( O ` A ) <-> -. ( O ` A ) <_ ( N mod ( O ` A ) ) ) )
22	17, 21	mpbid 222	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> -. ( O ` A ) <_ ( N mod ( O ` A ) ) )
23		odcl.2	. . . . . . . . . . . 12 |- O = ( od ` G )
24	6, 23, 8, 7	odlem2 17958	. . . . . . . . . . 11 |- ( ( A e. X /\ ( N mod ( O ` A ) ) e. NN /\ ( ( N mod ( O ` A ) ) .x. A ) = .0. ) -> ( O ` A ) e. ( 1 ... ( N mod ( O ` A ) ) ) )
25		elfzle2 12345	. . . . . . . . . . 11 |- ( ( O ` A ) e. ( 1 ... ( N mod ( O ` A ) ) ) -> ( O ` A ) <_ ( N mod ( O ` A ) ) )
26	24, 25	syl 17	. . . . . . . . . 10 |- ( ( A e. X /\ ( N mod ( O ` A ) ) e. NN /\ ( ( N mod ( O ` A ) ) .x. A ) = .0. ) -> ( O ` A ) <_ ( N mod ( O ` A ) ) )
27	26	3com23 1271	. . . . . . . . 9 |- ( ( A e. X /\ ( ( N mod ( O ` A ) ) .x. A ) = .0. /\ ( N mod ( O ` A ) ) e. NN ) -> ( O ` A ) <_ ( N mod ( O ` A ) ) )
28	27	3expia 1267	. . . . . . . 8 |- ( ( A e. X /\ ( ( N mod ( O ` A ) ) .x. A ) = .0. ) -> ( ( N mod ( O ` A ) ) e. NN -> ( O ` A ) <_ ( N mod ( O ` A ) ) ) )
29	28	con3d 148	. . . . . . 7 |- ( ( A e. X /\ ( ( N mod ( O ` A ) ) .x. A ) = .0. ) -> ( -. ( O ` A ) <_ ( N mod ( O ` A ) ) -> -. ( N mod ( O ` A ) ) e. NN ) )
30	29	impancom 456	. . . . . 6 |- ( ( A e. X /\ -. ( O ` A ) <_ ( N mod ( O ` A ) ) ) -> ( ( ( N mod ( O ` A ) ) .x. A ) = .0. -> -. ( N mod ( O ` A ) ) e. NN ) )
31	5, 22, 30	syl2anc 693	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( ( N mod ( O ` A ) ) .x. A ) = .0. -> -. ( N mod ( O ` A ) ) e. NN ) )
32		elnn0 11294	. . . . . . 7 |- ( ( N mod ( O ` A ) ) e. NN0 <-> ( ( N mod ( O ` A ) ) e. NN \/ ( N mod ( O ` A ) ) = 0 ) )
33	18, 32	sylib 208	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) e. NN \/ ( N mod ( O ` A ) ) = 0 ) )
34	33	ord 392	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( -. ( N mod ( O ` A ) ) e. NN -> ( N mod ( O ` A ) ) = 0 ) )
35	31, 34	syld 47	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( ( N mod ( O ` A ) ) .x. A ) = .0. -> ( N mod ( O ` A ) ) = 0 ) )
36	13, 35	impbid 202	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) = 0 <-> ( ( N mod ( O ` A ) ) .x. A ) = .0. ) )
37	6, 23, 8, 7	odmod 17965	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( N mod ( O ` A ) ) .x. A ) = ( N .x. A ) )
38	37	eqeq1d 2624	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( ( N mod ( O ` A ) ) .x. A ) = .0. <-> ( N .x. A ) = .0. ) )
39	4, 36, 38	3bitrd 294	. 2 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) e. NN ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )
40		simpr 477	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( O ` A ) = 0 )
41	40	breq1d 4663	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( ( O ` A ) || N <-> 0 || N ) )
42		simpl3 1066	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> N e. ZZ )
43		0dvds 15002	. . . 4 |- ( N e. ZZ -> ( 0 || N <-> N = 0 ) )
44	42, 43	syl 17	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( 0 || N <-> N = 0 ) )
45		simpl2 1065	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> A e. X )
46	45, 9	syl 17	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( 0 .x. A ) = .0. )
47		oveq1 6657	. . . . . 6 |- ( N = 0 -> ( N .x. A ) = ( 0 .x. A ) )
48	47	eqeq1d 2624	. . . . 5 |- ( N = 0 -> ( ( N .x. A ) = .0. <-> ( 0 .x. A ) = .0. ) )
49	46, 48	syl5ibrcom 237	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( N = 0 -> ( N .x. A ) = .0. ) )
50	6, 23, 8, 7	odnncl 17964	. . . . . . . . 9 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N =/= 0 /\ ( N .x. A ) = .0. ) ) -> ( O ` A ) e. NN )
51	50	nnne0d 11065	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N =/= 0 /\ ( N .x. A ) = .0. ) ) -> ( O ` A ) =/= 0 )
52	51	expr 643	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ N =/= 0 ) -> ( ( N .x. A ) = .0. -> ( O ` A ) =/= 0 ) )
53	52	impancom 456	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N .x. A ) = .0. ) -> ( N =/= 0 -> ( O ` A ) =/= 0 ) )
54	53	necon4d 2818	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( N .x. A ) = .0. ) -> ( ( O ` A ) = 0 -> N = 0 ) )
55	54	impancom 456	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( ( N .x. A ) = .0. -> N = 0 ) )
56	49, 55	impbid 202	. . 3 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( N = 0 <-> ( N .x. A ) = .0. ) )
57	41, 44, 56	3bitrd 294	. 2 |- ( ( ( G e. Grp /\ A e. X /\ N e. ZZ ) /\ ( O ` A ) = 0 ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )
58	6, 23	odcl 17955	. . . 4 |- ( A e. X -> ( O ` A ) e. NN0 )
59	58	3ad2ant2 1083	. . 3 |- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( O ` A ) e. NN0 )
60		elnn0 11294	. . 3 |- ( ( O ` A ) e. NN0 <-> ( ( O ` A ) e. NN \/ ( O ` A ) = 0 ) )
61	59, 60	sylib 208	. 2 |- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( ( O ` A ) e. NN \/ ( O ` A ) = 0 ) )
62	39, 57, 61	mpjaodan 827	1 |- ( ( G e. Grp /\ A e. X /\ N e. ZZ ) -> ( ( O ` A ) || N <-> ( N .x. A ) = .0. ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZcz 11377   RR+crp 11832   ...cfz 12326   mod cmo 12668   || cdvds 14983   Basecbs 15857   0gc0g 16100   Grpcgrp 17422  ._gcmg 17540   odcod 17944

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-od 17948

This theorem is referenced by:  oddvdsi  17967  odcong  17968  odeq  17969  odmulgid  17971  odbezout  17975  gexdvds2  18000  gexod  18001  gexcl3  18002  odadd1  18251  odadd2  18252  oddvdssubg  18258  pgpfac1lem3a  18475  chrdvds  19876  dchrfi  24980  dchrabs  24985  dchrptlem2  24990  idomodle  37774