Theorem odf1o1 17987

Description: An element with zero order has infinitely many multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)

Hypotheses

Ref	Expression
odf1o1.x	|- X = ( Base ` G )
odf1o1.t	|- .x. = (.g ` G )
odf1o1.o	|- O = ( od ` G )
odf1o1.k	|- K = (mrCls ` (SubGrp ` G ) )

Assertion

Ref	Expression
odf1o1	|- ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) -> ( x e. ZZ |-> ( x .x. A ) ) : ZZ -1-1-onto-> ( K ` { A } ) )

Distinct variable groups:   x, A   x, G   x, K   x, O   x, .x.   x, X

Proof of Theorem odf1o1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ x e. ZZ ) -> G e. Grp )
2		odf1o1.x	. . . . . . . 8 |- X = ( Base ` G )
3	2	subgacs 17629	. . . . . . 7 |- ( G e. Grp -> (SubGrp ` G ) e. (ACS ` X ) )
4		acsmre 16313	. . . . . . 7 |- ( (SubGrp ` G ) e. (ACS ` X ) -> (SubGrp ` G ) e. (Moore ` X ) )
5	1, 3, 4	3syl 18	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ x e. ZZ ) -> (SubGrp ` G ) e. (Moore ` X ) )
6		simpl2 1065	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ x e. ZZ ) -> A e. X )
7	6	snssd 4340	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ x e. ZZ ) -> { A } C_ X )
8		odf1o1.k	. . . . . . 7 |- K = (mrCls ` (SubGrp ` G ) )
9	8	mrccl 16271	. . . . . 6 |- ( ( (SubGrp ` G ) e. (Moore ` X ) /\ { A } C_ X ) -> ( K ` { A } ) e. (SubGrp ` G ) )
10	5, 7, 9	syl2anc 693	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ x e. ZZ ) -> ( K ` { A } ) e. (SubGrp ` G ) )
11		simpr 477	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ x e. ZZ ) -> x e. ZZ )
12	5, 8, 7	mrcssidd 16285	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ x e. ZZ ) -> { A } C_ ( K ` { A } ) )
13		snidg 4206	. . . . . . 7 |- ( A e. X -> A e. { A } )
14	6, 13	syl 17	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ x e. ZZ ) -> A e. { A } )
15	12, 14	sseldd 3604	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ x e. ZZ ) -> A e. ( K ` { A } ) )
16		odf1o1.t	. . . . . 6 |- .x. = (.g ` G )
17	16	subgmulgcl 17607	. . . . 5 |- ( ( ( K ` { A } ) e. (SubGrp ` G ) /\ x e. ZZ /\ A e. ( K ` { A } ) ) -> ( x .x. A ) e. ( K ` { A } ) )
18	10, 11, 15, 17	syl3anc 1326	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ x e. ZZ ) -> ( x .x. A ) e. ( K ` { A } ) )
19	18	ex 450	. . 3 |- ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) -> ( x e. ZZ -> ( x .x. A ) e. ( K ` { A } ) ) )
20		simpl3 1066	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( O ` A ) = 0 )
21	20	breq1d 4663	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( O ` A ) || ( x - y ) <-> 0 || ( x - y ) ) )
22		zsubcl 11419	. . . . . . . 8 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( x - y ) e. ZZ )
23	22	adantl 482	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( x - y ) e. ZZ )
24		0dvds 15002	. . . . . . 7 |- ( ( x - y ) e. ZZ -> ( 0 || ( x - y ) <-> ( x - y ) = 0 ) )
25	23, 24	syl 17	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( 0 || ( x - y ) <-> ( x - y ) = 0 ) )
26	21, 25	bitrd 268	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( O ` A ) || ( x - y ) <-> ( x - y ) = 0 ) )
27		simpl1 1064	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> G e. Grp )
28		simpl2 1065	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> A e. X )
29		simprl 794	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> x e. ZZ )
30		simprr 796	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> y e. ZZ )
31		odf1o1.o	. . . . . . 7 |- O = ( od ` G )
32		eqid 2622	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
33	2, 31, 16, 32	odcong 17968	. . . . . 6 |- ( ( G e. Grp /\ A e. X /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( O ` A ) || ( x - y ) <-> ( x .x. A ) = ( y .x. A ) ) )
34	27, 28, 29, 30, 33	syl112anc 1330	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( O ` A ) || ( x - y ) <-> ( x .x. A ) = ( y .x. A ) ) )
35		zcn 11382	. . . . . . 7 |- ( x e. ZZ -> x e. CC )
36		zcn 11382	. . . . . . 7 |- ( y e. ZZ -> y e. CC )
37		subeq0 10307	. . . . . . 7 |- ( ( x e. CC /\ y e. CC ) -> ( ( x - y ) = 0 <-> x = y ) )
38	35, 36, 37	syl2an 494	. . . . . 6 |- ( ( x e. ZZ /\ y e. ZZ ) -> ( ( x - y ) = 0 <-> x = y ) )
39	38	adantl 482	. . . . 5 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x - y ) = 0 <-> x = y ) )
40	26, 34, 39	3bitr3d 298	. . . 4 |- ( ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( x .x. A ) = ( y .x. A ) <-> x = y ) )
41	40	ex 450	. . 3 |- ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) -> ( ( x e. ZZ /\ y e. ZZ ) -> ( ( x .x. A ) = ( y .x. A ) <-> x = y ) ) )
42	19, 41	dom2lem 7995	. 2 |- ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) -> ( x e. ZZ |-> ( x .x. A ) ) : ZZ -1-1-> ( K ` { A } ) )
43		f1f 6101	. . . 4 |- ( ( x e. ZZ |-> ( x .x. A ) ) : ZZ -1-1-> ( K ` { A } ) -> ( x e. ZZ |-> ( x .x. A ) ) : ZZ --> ( K ` { A } ) )
44	42, 43	syl 17	. . 3 |- ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) -> ( x e. ZZ |-> ( x .x. A ) ) : ZZ --> ( K ` { A } ) )
45		eqid 2622	. . . . . 6 |- ( x e. ZZ |-> ( x .x. A ) ) = ( x e. ZZ |-> ( x .x. A ) )
46	2, 16, 45, 8	cycsubg2 17631	. . . . 5 |- ( ( G e. Grp /\ A e. X ) -> ( K ` { A } ) = ran ( x e. ZZ |-> ( x .x. A ) ) )
47	46	3adant3 1081	. . . 4 |- ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) -> ( K ` { A } ) = ran ( x e. ZZ |-> ( x .x. A ) ) )
48	47	eqcomd 2628	. . 3 |- ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) -> ran ( x e. ZZ |-> ( x .x. A ) ) = ( K ` { A } ) )
49		dffo2 6119	. . 3 |- ( ( x e. ZZ |-> ( x .x. A ) ) : ZZ -onto-> ( K ` { A } ) <-> ( ( x e. ZZ |-> ( x .x. A ) ) : ZZ --> ( K ` { A } ) /\ ran ( x e. ZZ |-> ( x .x. A ) ) = ( K ` { A } ) ) )
50	44, 48, 49	sylanbrc 698	. 2 |- ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) -> ( x e. ZZ |-> ( x .x. A ) ) : ZZ -onto-> ( K ` { A } ) )
51		df-f1o 5895	. 2 |- ( ( x e. ZZ |-> ( x .x. A ) ) : ZZ -1-1-onto-> ( K ` { A } ) <-> ( ( x e. ZZ |-> ( x .x. A ) ) : ZZ -1-1-> ( K ` { A } ) /\ ( x e. ZZ |-> ( x .x. A ) ) : ZZ -onto-> ( K ` { A } ) ) )
52	42, 50, 51	sylanbrc 698	1 |- ( ( G e. Grp /\ A e. X /\ ( O ` A ) = 0 ) -> ( x e. ZZ |-> ( x .x. A ) ) : ZZ -1-1-onto-> ( K ` { A } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   {csn 4177   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   -->wf 5884   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   - cmin 10266   ZZcz 11377   || cdvds 14983   Basecbs 15857   0gc0g 16100  Moorecmre 16242  mrClscmrc 16243  ACScacs 16245   Grpcgrp 17422  ._gcmg 17540  SubGrpcsubg 17588   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-int 4476  df-iun 4522  df-iin 4523  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-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-od 17948

This theorem is referenced by:  odhash  17989