Theorem odf1 17979

Description: The multiples of an element with infinite order form an infinite cyclic subgroup of G. (Contributed by Mario Carneiro, 14-Jan-2015.) (Revised by Mario Carneiro, 23-Sep-2015.)

Hypotheses

Ref	Expression
odf1.1	|- X = ( Base ` G )
odf1.2	|- O = ( od ` G )
odf1.3	|- .x. = (.g ` G )
odf1.4	|- F = ( x e. ZZ |-> ( x .x. A ) )

Assertion

Ref	Expression
odf1	|- ( ( G e. Grp /\ A e. X ) -> ( ( O ` A ) = 0 <-> F : ZZ -1-1-> X ) )

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

Allowed substitution hint:   F( x)

Proof of Theorem odf1

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		odf1.1	. . . . . . . 8 |- X = ( Base ` G )
2		odf1.3	. . . . . . . 8 |- .x. = (.g ` G )
3	1, 2	mulgcl 17559	. . . . . . 7 |- ( ( G e. Grp /\ x e. ZZ /\ A e. X ) -> ( x .x. A ) e. X )
4	3	3expa 1265	. . . . . 6 |- ( ( ( G e. Grp /\ x e. ZZ ) /\ A e. X ) -> ( x .x. A ) e. X )
5	4	an32s 846	. . . . 5 |- ( ( ( G e. Grp /\ A e. X ) /\ x e. ZZ ) -> ( x .x. A ) e. X )
6		odf1.4	. . . . 5 |- F = ( x e. ZZ |-> ( x .x. A ) )
7	5, 6	fmptd 6385	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> F : ZZ --> X )
8	7	adantr 481	. . 3 |- ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) -> F : ZZ --> X )
9		oveq1 6657	. . . . . . . . 9 |- ( x = y -> ( x .x. A ) = ( y .x. A ) )
10		ovex 6678	. . . . . . . . 9 |- ( x .x. A ) e. _V
11	9, 6, 10	fvmpt3i 6287	. . . . . . . 8 |- ( y e. ZZ -> ( F ` y ) = ( y .x. A ) )
12		oveq1 6657	. . . . . . . . 9 |- ( x = z -> ( x .x. A ) = ( z .x. A ) )
13	12, 6, 10	fvmpt3i 6287	. . . . . . . 8 |- ( z e. ZZ -> ( F ` z ) = ( z .x. A ) )
14	11, 13	eqeqan12d 2638	. . . . . . 7 |- ( ( y e. ZZ /\ z e. ZZ ) -> ( ( F ` y ) = ( F ` z ) <-> ( y .x. A ) = ( z .x. A ) ) )
15	14	adantl 482	. . . . . 6 |- ( ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( ( F ` y ) = ( F ` z ) <-> ( y .x. A ) = ( z .x. A ) ) )
16		simplr 792	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( O ` A ) = 0 )
17	16	breq1d 4663	. . . . . . 7 |- ( ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( ( O ` A ) || ( y - z ) <-> 0 || ( y - z ) ) )
18		odf1.2	. . . . . . . . . 10 |- O = ( od ` G )
19		eqid 2622	. . . . . . . . . 10 |- ( 0g ` G ) = ( 0g ` G )
20	1, 18, 2, 19	odcong 17968	. . . . . . . . 9 |- ( ( G e. Grp /\ A e. X /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( ( O ` A ) || ( y - z ) <-> ( y .x. A ) = ( z .x. A ) ) )
21	20	3expa 1265	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( ( O ` A ) || ( y - z ) <-> ( y .x. A ) = ( z .x. A ) ) )
22	21	adantlr 751	. . . . . . 7 |- ( ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( ( O ` A ) || ( y - z ) <-> ( y .x. A ) = ( z .x. A ) ) )
23		zsubcl 11419	. . . . . . . . 9 |- ( ( y e. ZZ /\ z e. ZZ ) -> ( y - z ) e. ZZ )
24	23	adantl 482	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( y - z ) e. ZZ )
25		0dvds 15002	. . . . . . . 8 |- ( ( y - z ) e. ZZ -> ( 0 || ( y - z ) <-> ( y - z ) = 0 ) )
26	24, 25	syl 17	. . . . . . 7 |- ( ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( 0 || ( y - z ) <-> ( y - z ) = 0 ) )
27	17, 22, 26	3bitr3d 298	. . . . . 6 |- ( ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( ( y .x. A ) = ( z .x. A ) <-> ( y - z ) = 0 ) )
28		zcn 11382	. . . . . . . 8 |- ( y e. ZZ -> y e. CC )
29		zcn 11382	. . . . . . . 8 |- ( z e. ZZ -> z e. CC )
30		subeq0 10307	. . . . . . . 8 |- ( ( y e. CC /\ z e. CC ) -> ( ( y - z ) = 0 <-> y = z ) )
31	28, 29, 30	syl2an 494	. . . . . . 7 |- ( ( y e. ZZ /\ z e. ZZ ) -> ( ( y - z ) = 0 <-> y = z ) )
32	31	adantl 482	. . . . . 6 |- ( ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( ( y - z ) = 0 <-> y = z ) )
33	15, 27, 32	3bitrd 294	. . . . 5 |- ( ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( ( F ` y ) = ( F ` z ) <-> y = z ) )
34	33	biimpd 219	. . . 4 |- ( ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) /\ ( y e. ZZ /\ z e. ZZ ) ) -> ( ( F ` y ) = ( F ` z ) -> y = z ) )
35	34	ralrimivva 2971	. . 3 |- ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) -> A. y e. ZZ A. z e. ZZ ( ( F ` y ) = ( F ` z ) -> y = z ) )
36		dff13 6512	. . 3 |- ( F : ZZ -1-1-> X <-> ( F : ZZ --> X /\ A. y e. ZZ A. z e. ZZ ( ( F ` y ) = ( F ` z ) -> y = z ) ) )
37	8, 35, 36	sylanbrc 698	. 2 |- ( ( ( G e. Grp /\ A e. X ) /\ ( O ` A ) = 0 ) -> F : ZZ -1-1-> X )
38	1, 18, 2, 19	odid 17957	. . . . . 6 |- ( A e. X -> ( ( O ` A ) .x. A ) = ( 0g ` G ) )
39	1, 19, 2	mulg0 17546	. . . . . 6 |- ( A e. X -> ( 0 .x. A ) = ( 0g ` G ) )
40	38, 39	eqtr4d 2659	. . . . 5 |- ( A e. X -> ( ( O ` A ) .x. A ) = ( 0 .x. A ) )
41	40	ad2antlr 763	. . . 4 |- ( ( ( G e. Grp /\ A e. X ) /\ F : ZZ -1-1-> X ) -> ( ( O ` A ) .x. A ) = ( 0 .x. A ) )
42	1, 18	odcl 17955	. . . . . . 7 |- ( A e. X -> ( O ` A ) e. NN0 )
43	42	ad2antlr 763	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X ) /\ F : ZZ -1-1-> X ) -> ( O ` A ) e. NN0 )
44	43	nn0zd 11480	. . . . 5 |- ( ( ( G e. Grp /\ A e. X ) /\ F : ZZ -1-1-> X ) -> ( O ` A ) e. ZZ )
45		oveq1 6657	. . . . . 6 |- ( x = ( O ` A ) -> ( x .x. A ) = ( ( O ` A ) .x. A ) )
46	45, 6, 10	fvmpt3i 6287	. . . . 5 |- ( ( O ` A ) e. ZZ -> ( F ` ( O ` A ) ) = ( ( O ` A ) .x. A ) )
47	44, 46	syl 17	. . . 4 |- ( ( ( G e. Grp /\ A e. X ) /\ F : ZZ -1-1-> X ) -> ( F ` ( O ` A ) ) = ( ( O ` A ) .x. A ) )
48		0zd 11389	. . . . 5 |- ( ( ( G e. Grp /\ A e. X ) /\ F : ZZ -1-1-> X ) -> 0 e. ZZ )
49		oveq1 6657	. . . . . 6 |- ( x = 0 -> ( x .x. A ) = ( 0 .x. A ) )
50	49, 6, 10	fvmpt3i 6287	. . . . 5 |- ( 0 e. ZZ -> ( F ` 0 ) = ( 0 .x. A ) )
51	48, 50	syl 17	. . . 4 |- ( ( ( G e. Grp /\ A e. X ) /\ F : ZZ -1-1-> X ) -> ( F ` 0 ) = ( 0 .x. A ) )
52	41, 47, 51	3eqtr4d 2666	. . 3 |- ( ( ( G e. Grp /\ A e. X ) /\ F : ZZ -1-1-> X ) -> ( F ` ( O ` A ) ) = ( F ` 0 ) )
53		simpr 477	. . . 4 |- ( ( ( G e. Grp /\ A e. X ) /\ F : ZZ -1-1-> X ) -> F : ZZ -1-1-> X )
54		f1fveq 6519	. . . 4 |- ( ( F : ZZ -1-1-> X /\ ( ( O ` A ) e. ZZ /\ 0 e. ZZ ) ) -> ( ( F ` ( O ` A ) ) = ( F ` 0 ) <-> ( O ` A ) = 0 ) )
55	53, 44, 48, 54	syl12anc 1324	. . 3 |- ( ( ( G e. Grp /\ A e. X ) /\ F : ZZ -1-1-> X ) -> ( ( F ` ( O ` A ) ) = ( F ` 0 ) <-> ( O ` A ) = 0 ) )
56	52, 55	mpbid 222	. 2 |- ( ( ( G e. Grp /\ A e. X ) /\ F : ZZ -1-1-> X ) -> ( O ` A ) = 0 )
57	37, 56	impbida 877	1 |- ( ( G e. Grp /\ A e. X ) -> ( ( O ` A ) = 0 <-> F : ZZ -1-1-> X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   - cmin 10266   NN0cn0 11292   ZZcz 11377   || 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:  odinf  17980  odcl2  17982  zrhchr  30020