Theorem cycsubgcl 17620

Description: The set of integer powers of an element A of a group forms a subgroup containing A, called the cyclic group generated by the element A. (Contributed by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
cycsubg.x	|- X = ( Base ` G )
cycsubg.t	|- .x. = (.g ` G )
cycsubg.f	|- F = ( x e. ZZ |-> ( x .x. A ) )

Assertion

Ref	Expression
cycsubgcl	|- ( ( G e. Grp /\ A e. X ) -> ( ran F e. (SubGrp ` G ) /\ A e. ran F ) )

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

Allowed substitution hint:   F( x)

Proof of Theorem cycsubgcl

Dummy variables m n u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cycsubg.x	. . . . . . . 8 |- X = ( Base ` G )
2		cycsubg.t	. . . . . . . 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		cycsubg.f	. . . . 5 |- F = ( x e. ZZ |-> ( x .x. A ) )
7	5, 6	fmptd 6385	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> F : ZZ --> X )
8		frn 6053	. . . 4 |- ( F : ZZ --> X -> ran F C_ X )
9	7, 8	syl 17	. . 3 |- ( ( G e. Grp /\ A e. X ) -> ran F C_ X )
10		1z 11407	. . . . . . 7 |- 1 e. ZZ
11		oveq1 6657	. . . . . . . 8 |- ( x = 1 -> ( x .x. A ) = ( 1 .x. A ) )
12		ovex 6678	. . . . . . . 8 |- ( 1 .x. A ) e. _V
13	11, 6, 12	fvmpt 6282	. . . . . . 7 |- ( 1 e. ZZ -> ( F ` 1 ) = ( 1 .x. A ) )
14	10, 13	ax-mp 5	. . . . . 6 |- ( F ` 1 ) = ( 1 .x. A )
15	1, 2	mulg1 17548	. . . . . . 7 |- ( A e. X -> ( 1 .x. A ) = A )
16	15	adantl 482	. . . . . 6 |- ( ( G e. Grp /\ A e. X ) -> ( 1 .x. A ) = A )
17	14, 16	syl5eq 2668	. . . . 5 |- ( ( G e. Grp /\ A e. X ) -> ( F ` 1 ) = A )
18		ffn 6045	. . . . . . 7 |- ( F : ZZ --> X -> F Fn ZZ )
19	7, 18	syl 17	. . . . . 6 |- ( ( G e. Grp /\ A e. X ) -> F Fn ZZ )
20		fnfvelrn 6356	. . . . . 6 |- ( ( F Fn ZZ /\ 1 e. ZZ ) -> ( F ` 1 ) e. ran F )
21	19, 10, 20	sylancl 694	. . . . 5 |- ( ( G e. Grp /\ A e. X ) -> ( F ` 1 ) e. ran F )
22	17, 21	eqeltrrd 2702	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> A e. ran F )
23		ne0i 3921	. . . 4 |- ( A e. ran F -> ran F =/= (/) )
24	22, 23	syl 17	. . 3 |- ( ( G e. Grp /\ A e. X ) -> ran F =/= (/) )
25		df-3an 1039	. . . . . . . . . . . . 13 |- ( ( m e. ZZ /\ n e. ZZ /\ A e. X ) <-> ( ( m e. ZZ /\ n e. ZZ ) /\ A e. X ) )
26		eqid 2622	. . . . . . . . . . . . . 14 |- ( +g ` G ) = ( +g ` G )
27	1, 2, 26	mulgdir 17573	. . . . . . . . . . . . 13 |- ( ( G e. Grp /\ ( m e. ZZ /\ n e. ZZ /\ A e. X ) ) -> ( ( m + n ) .x. A ) = ( ( m .x. A ) ( +g ` G ) ( n .x. A ) ) )
28	25, 27	sylan2br 493	. . . . . . . . . . . 12 |- ( ( G e. Grp /\ ( ( m e. ZZ /\ n e. ZZ ) /\ A e. X ) ) -> ( ( m + n ) .x. A ) = ( ( m .x. A ) ( +g ` G ) ( n .x. A ) ) )
29	28	anass1rs 849	. . . . . . . . . . 11 |- ( ( ( G e. Grp /\ A e. X ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( ( m + n ) .x. A ) = ( ( m .x. A ) ( +g ` G ) ( n .x. A ) ) )
30		zaddcl 11417	. . . . . . . . . . . . 13 |- ( ( m e. ZZ /\ n e. ZZ ) -> ( m + n ) e. ZZ )
31	30	adantl 482	. . . . . . . . . . . 12 |- ( ( ( G e. Grp /\ A e. X ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( m + n ) e. ZZ )
32		oveq1 6657	. . . . . . . . . . . . 13 |- ( x = ( m + n ) -> ( x .x. A ) = ( ( m + n ) .x. A ) )
33		ovex 6678	. . . . . . . . . . . . 13 |- ( ( m + n ) .x. A ) e. _V
34	32, 6, 33	fvmpt 6282	. . . . . . . . . . . 12 |- ( ( m + n ) e. ZZ -> ( F ` ( m + n ) ) = ( ( m + n ) .x. A ) )
35	31, 34	syl 17	. . . . . . . . . . 11 |- ( ( ( G e. Grp /\ A e. X ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( F ` ( m + n ) ) = ( ( m + n ) .x. A ) )
36		oveq1 6657	. . . . . . . . . . . . . 14 |- ( x = m -> ( x .x. A ) = ( m .x. A ) )
37		ovex 6678	. . . . . . . . . . . . . 14 |- ( m .x. A ) e. _V
38	36, 6, 37	fvmpt 6282	. . . . . . . . . . . . 13 |- ( m e. ZZ -> ( F ` m ) = ( m .x. A ) )
39	38	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( ( G e. Grp /\ A e. X ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( F ` m ) = ( m .x. A ) )
40		oveq1 6657	. . . . . . . . . . . . . 14 |- ( x = n -> ( x .x. A ) = ( n .x. A ) )
41		ovex 6678	. . . . . . . . . . . . . 14 |- ( n .x. A ) e. _V
42	40, 6, 41	fvmpt 6282	. . . . . . . . . . . . 13 |- ( n e. ZZ -> ( F ` n ) = ( n .x. A ) )
43	42	ad2antll 765	. . . . . . . . . . . 12 |- ( ( ( G e. Grp /\ A e. X ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( F ` n ) = ( n .x. A ) )
44	39, 43	oveq12d 6668	. . . . . . . . . . 11 |- ( ( ( G e. Grp /\ A e. X ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( ( F ` m ) ( +g ` G ) ( F ` n ) ) = ( ( m .x. A ) ( +g ` G ) ( n .x. A ) ) )
45	29, 35, 44	3eqtr4d 2666	. . . . . . . . . 10 |- ( ( ( G e. Grp /\ A e. X ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( F ` ( m + n ) ) = ( ( F ` m ) ( +g ` G ) ( F ` n ) ) )
46		fnfvelrn 6356	. . . . . . . . . . 11 |- ( ( F Fn ZZ /\ ( m + n ) e. ZZ ) -> ( F ` ( m + n ) ) e. ran F )
47	19, 30, 46	syl2an 494	. . . . . . . . . 10 |- ( ( ( G e. Grp /\ A e. X ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( F ` ( m + n ) ) e. ran F )
48	45, 47	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( G e. Grp /\ A e. X ) /\ ( m e. ZZ /\ n e. ZZ ) ) -> ( ( F ` m ) ( +g ` G ) ( F ` n ) ) e. ran F )
49	48	anassrs 680	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) /\ n e. ZZ ) -> ( ( F ` m ) ( +g ` G ) ( F ` n ) ) e. ran F )
50	49	ralrimiva 2966	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> A. n e. ZZ ( ( F ` m ) ( +g ` G ) ( F ` n ) ) e. ran F )
51		oveq2 6658	. . . . . . . . . . 11 |- ( v = ( F ` n ) -> ( ( F ` m ) ( +g ` G ) v ) = ( ( F ` m ) ( +g ` G ) ( F ` n ) ) )
52	51	eleq1d 2686	. . . . . . . . . 10 |- ( v = ( F ` n ) -> ( ( ( F ` m ) ( +g ` G ) v ) e. ran F <-> ( ( F ` m ) ( +g ` G ) ( F ` n ) ) e. ran F ) )
53	52	ralrn 6362	. . . . . . . . 9 |- ( F Fn ZZ -> ( A. v e. ran F ( ( F ` m ) ( +g ` G ) v ) e. ran F <-> A. n e. ZZ ( ( F ` m ) ( +g ` G ) ( F ` n ) ) e. ran F ) )
54	19, 53	syl 17	. . . . . . . 8 |- ( ( G e. Grp /\ A e. X ) -> ( A. v e. ran F ( ( F ` m ) ( +g ` G ) v ) e. ran F <-> A. n e. ZZ ( ( F ` m ) ( +g ` G ) ( F ` n ) ) e. ran F ) )
55	54	adantr 481	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> ( A. v e. ran F ( ( F ` m ) ( +g ` G ) v ) e. ran F <-> A. n e. ZZ ( ( F ` m ) ( +g ` G ) ( F ` n ) ) e. ran F ) )
56	50, 55	mpbird 247	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> A. v e. ran F ( ( F ` m ) ( +g ` G ) v ) e. ran F )
57		eqid 2622	. . . . . . . . . . 11 |- ( invg ` G ) = ( invg ` G )
58	1, 2, 57	mulgneg 17560	. . . . . . . . . 10 |- ( ( G e. Grp /\ m e. ZZ /\ A e. X ) -> ( -u m .x. A ) = ( ( invg ` G ) ` ( m .x. A ) ) )
59	58	3expa 1265	. . . . . . . . 9 |- ( ( ( G e. Grp /\ m e. ZZ ) /\ A e. X ) -> ( -u m .x. A ) = ( ( invg ` G ) ` ( m .x. A ) ) )
60	59	an32s 846	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> ( -u m .x. A ) = ( ( invg ` G ) ` ( m .x. A ) ) )
61		znegcl 11412	. . . . . . . . . 10 |- ( m e. ZZ -> -u m e. ZZ )
62	61	adantl 482	. . . . . . . . 9 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> -u m e. ZZ )
63		oveq1 6657	. . . . . . . . . 10 |- ( x = -u m -> ( x .x. A ) = ( -u m .x. A ) )
64		ovex 6678	. . . . . . . . . 10 |- ( -u m .x. A ) e. _V
65	63, 6, 64	fvmpt 6282	. . . . . . . . 9 |- ( -u m e. ZZ -> ( F ` -u m ) = ( -u m .x. A ) )
66	62, 65	syl 17	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> ( F ` -u m ) = ( -u m .x. A ) )
67	38	adantl 482	. . . . . . . . 9 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> ( F ` m ) = ( m .x. A ) )
68	67	fveq2d 6195	. . . . . . . 8 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> ( ( invg ` G ) ` ( F ` m ) ) = ( ( invg ` G ) ` ( m .x. A ) ) )
69	60, 66, 68	3eqtr4d 2666	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> ( F ` -u m ) = ( ( invg ` G ) ` ( F ` m ) ) )
70		fnfvelrn 6356	. . . . . . . 8 |- ( ( F Fn ZZ /\ -u m e. ZZ ) -> ( F ` -u m ) e. ran F )
71	19, 61, 70	syl2an 494	. . . . . . 7 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> ( F ` -u m ) e. ran F )
72	69, 71	eqeltrrd 2702	. . . . . 6 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> ( ( invg ` G ) ` ( F ` m ) ) e. ran F )
73	56, 72	jca 554	. . . . 5 |- ( ( ( G e. Grp /\ A e. X ) /\ m e. ZZ ) -> ( A. v e. ran F ( ( F ` m ) ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` ( F ` m ) ) e. ran F ) )
74	73	ralrimiva 2966	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> A. m e. ZZ ( A. v e. ran F ( ( F ` m ) ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` ( F ` m ) ) e. ran F ) )
75		oveq1 6657	. . . . . . . . 9 |- ( u = ( F ` m ) -> ( u ( +g ` G ) v ) = ( ( F ` m ) ( +g ` G ) v ) )
76	75	eleq1d 2686	. . . . . . . 8 |- ( u = ( F ` m ) -> ( ( u ( +g ` G ) v ) e. ran F <-> ( ( F ` m ) ( +g ` G ) v ) e. ran F ) )
77	76	ralbidv 2986	. . . . . . 7 |- ( u = ( F ` m ) -> ( A. v e. ran F ( u ( +g ` G ) v ) e. ran F <-> A. v e. ran F ( ( F ` m ) ( +g ` G ) v ) e. ran F ) )
78		fveq2 6191	. . . . . . . 8 |- ( u = ( F ` m ) -> ( ( invg ` G ) ` u ) = ( ( invg ` G ) ` ( F ` m ) ) )
79	78	eleq1d 2686	. . . . . . 7 |- ( u = ( F ` m ) -> ( ( ( invg ` G ) ` u ) e. ran F <-> ( ( invg ` G ) ` ( F ` m ) ) e. ran F ) )
80	77, 79	anbi12d 747	. . . . . 6 |- ( u = ( F ` m ) -> ( ( A. v e. ran F ( u ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` u ) e. ran F ) <-> ( A. v e. ran F ( ( F ` m ) ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` ( F ` m ) ) e. ran F ) ) )
81	80	ralrn 6362	. . . . 5 |- ( F Fn ZZ -> ( A. u e. ran F ( A. v e. ran F ( u ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` u ) e. ran F ) <-> A. m e. ZZ ( A. v e. ran F ( ( F ` m ) ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` ( F ` m ) ) e. ran F ) ) )
82	19, 81	syl 17	. . . 4 |- ( ( G e. Grp /\ A e. X ) -> ( A. u e. ran F ( A. v e. ran F ( u ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` u ) e. ran F ) <-> A. m e. ZZ ( A. v e. ran F ( ( F ` m ) ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` ( F ` m ) ) e. ran F ) ) )
83	74, 82	mpbird 247	. . 3 |- ( ( G e. Grp /\ A e. X ) -> A. u e. ran F ( A. v e. ran F ( u ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` u ) e. ran F ) )
84	1, 26, 57	issubg2 17609	. . . 4 |- ( G e. Grp -> ( ran F e. (SubGrp ` G ) <-> ( ran F C_ X /\ ran F =/= (/) /\ A. u e. ran F ( A. v e. ran F ( u ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` u ) e. ran F ) ) ) )
85	84	adantr 481	. . 3 |- ( ( G e. Grp /\ A e. X ) -> ( ran F e. (SubGrp ` G ) <-> ( ran F C_ X /\ ran F =/= (/) /\ A. u e. ran F ( A. v e. ran F ( u ( +g ` G ) v ) e. ran F /\ ( ( invg ` G ) ` u ) e. ran F ) ) ) )
86	9, 24, 83, 85	mpbir3and 1245	. 2 |- ( ( G e. Grp /\ A e. X ) -> ran F e. (SubGrp ` G ) )
87	86, 22	jca 554	1 |- ( ( G e. Grp /\ A e. X ) -> ( ran F e. (SubGrp ` G ) /\ A e. ran F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   ran crn 5115   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   -ucneg 10267   ZZcz 11377   Basecbs 15857   +g cplusg 15941   Grpcgrp 17422   invgcminusg 17423  ._gcmg 17540  SubGrpcsubg 17588

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

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-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591

This theorem is referenced by:  cycsubg  17622  oddvds2  17983  cycsubgcyg  18302