Theorem cyggenod 18286

Description: An element is the generator of a finite group iff the order of the generator equals the order of the group. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
iscyg.1	|- B = ( Base ` G )
iscyg.2	|- .x. = (.g ` G )
iscyg3.e	|- E = { x e. B | ran ( n e. ZZ |-> ( n .x. x ) ) = B }
cyggenod.o	|- O = ( od ` G )

Assertion

Ref	Expression
cyggenod	|- ( ( G e. Grp /\ B e. Fin ) -> ( X e. E <-> ( X e. B /\ ( O ` X ) = ( # ` B ) ) ) )

Distinct variable groups:   x, n, B   n, O   n, X, x   n, G, x   .x. , n, x

Allowed substitution hints:   E( x, n)   O( x)

Proof of Theorem cyggenod

Step	Hyp	Ref	Expression
1		iscyg.1	. . 3 |- B = ( Base ` G )
2		iscyg.2	. . 3 |- .x. = (.g ` G )
3		iscyg3.e	. . 3 |- E = { x e. B | ran ( n e. ZZ |-> ( n .x. x ) ) = B }
4	1, 2, 3	iscyggen 18282	. 2 |- ( X e. E <-> ( X e. B /\ ran ( n e. ZZ |-> ( n .x. X ) ) = B ) )
5		simplr 792	. . . . . 6 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> B e. Fin )
6		simplll 798	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) /\ n e. ZZ ) -> G e. Grp )
7		simpr 477	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) /\ n e. ZZ ) -> n e. ZZ )
8		simplr 792	. . . . . . . . 9 |- ( ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) /\ n e. ZZ ) -> X e. B )
9	1, 2	mulgcl 17559	. . . . . . . . 9 |- ( ( G e. Grp /\ n e. ZZ /\ X e. B ) -> ( n .x. X ) e. B )
10	6, 7, 8, 9	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) /\ n e. ZZ ) -> ( n .x. X ) e. B )
11		eqid 2622	. . . . . . . 8 |- ( n e. ZZ |-> ( n .x. X ) ) = ( n e. ZZ |-> ( n .x. X ) )
12	10, 11	fmptd 6385	. . . . . . 7 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> ( n e. ZZ |-> ( n .x. X ) ) : ZZ --> B )
13		frn 6053	. . . . . . 7 |- ( ( n e. ZZ |-> ( n .x. X ) ) : ZZ --> B -> ran ( n e. ZZ |-> ( n .x. X ) ) C_ B )
14	12, 13	syl 17	. . . . . 6 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> ran ( n e. ZZ |-> ( n .x. X ) ) C_ B )
15		ssfi 8180	. . . . . 6 |- ( ( B e. Fin /\ ran ( n e. ZZ |-> ( n .x. X ) ) C_ B ) -> ran ( n e. ZZ |-> ( n .x. X ) ) e. Fin )
16	5, 14, 15	syl2anc 693	. . . . 5 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> ran ( n e. ZZ |-> ( n .x. X ) ) e. Fin )
17		hashen 13135	. . . . 5 |- ( ( ran ( n e. ZZ |-> ( n .x. X ) ) e. Fin /\ B e. Fin ) -> ( ( # ` ran ( n e. ZZ |-> ( n .x. X ) ) ) = ( # ` B ) <-> ran ( n e. ZZ |-> ( n .x. X ) ) ~~ B ) )
18	16, 5, 17	syl2anc 693	. . . 4 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> ( ( # ` ran ( n e. ZZ |-> ( n .x. X ) ) ) = ( # ` B ) <-> ran ( n e. ZZ |-> ( n .x. X ) ) ~~ B ) )
19		cyggenod.o	. . . . . . . 8 |- O = ( od ` G )
20	1, 19, 2, 11	dfod2 17981	. . . . . . 7 |- ( ( G e. Grp /\ X e. B ) -> ( O ` X ) = if ( ran ( n e. ZZ |-> ( n .x. X ) ) e. Fin , ( # ` ran ( n e. ZZ |-> ( n .x. X ) ) ) , 0 ) )
21	20	adantlr 751	. . . . . 6 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> ( O ` X ) = if ( ran ( n e. ZZ |-> ( n .x. X ) ) e. Fin , ( # ` ran ( n e. ZZ |-> ( n .x. X ) ) ) , 0 ) )
22	16	iftrued 4094	. . . . . 6 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> if ( ran ( n e. ZZ |-> ( n .x. X ) ) e. Fin , ( # ` ran ( n e. ZZ |-> ( n .x. X ) ) ) , 0 ) = ( # ` ran ( n e. ZZ |-> ( n .x. X ) ) ) )
23	21, 22	eqtr2d 2657	. . . . 5 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> ( # ` ran ( n e. ZZ |-> ( n .x. X ) ) ) = ( O ` X ) )
24	23	eqeq1d 2624	. . . 4 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> ( ( # ` ran ( n e. ZZ |-> ( n .x. X ) ) ) = ( # ` B ) <-> ( O ` X ) = ( # ` B ) ) )
25		fisseneq 8171	. . . . . . 7 |- ( ( B e. Fin /\ ran ( n e. ZZ |-> ( n .x. X ) ) C_ B /\ ran ( n e. ZZ |-> ( n .x. X ) ) ~~ B ) -> ran ( n e. ZZ |-> ( n .x. X ) ) = B )
26	25	3expia 1267	. . . . . 6 |- ( ( B e. Fin /\ ran ( n e. ZZ |-> ( n .x. X ) ) C_ B ) -> ( ran ( n e. ZZ |-> ( n .x. X ) ) ~~ B -> ran ( n e. ZZ |-> ( n .x. X ) ) = B ) )
27		enrefg 7987	. . . . . . . 8 |- ( B e. Fin -> B ~~ B )
28	27	adantr 481	. . . . . . 7 |- ( ( B e. Fin /\ ran ( n e. ZZ |-> ( n .x. X ) ) C_ B ) -> B ~~ B )
29		breq1 4656	. . . . . . 7 |- ( ran ( n e. ZZ |-> ( n .x. X ) ) = B -> ( ran ( n e. ZZ |-> ( n .x. X ) ) ~~ B <-> B ~~ B ) )
30	28, 29	syl5ibrcom 237	. . . . . 6 |- ( ( B e. Fin /\ ran ( n e. ZZ |-> ( n .x. X ) ) C_ B ) -> ( ran ( n e. ZZ |-> ( n .x. X ) ) = B -> ran ( n e. ZZ |-> ( n .x. X ) ) ~~ B ) )
31	26, 30	impbid 202	. . . . 5 |- ( ( B e. Fin /\ ran ( n e. ZZ |-> ( n .x. X ) ) C_ B ) -> ( ran ( n e. ZZ |-> ( n .x. X ) ) ~~ B <-> ran ( n e. ZZ |-> ( n .x. X ) ) = B ) )
32	5, 14, 31	syl2anc 693	. . . 4 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> ( ran ( n e. ZZ |-> ( n .x. X ) ) ~~ B <-> ran ( n e. ZZ |-> ( n .x. X ) ) = B ) )
33	18, 24, 32	3bitr3rd 299	. . 3 |- ( ( ( G e. Grp /\ B e. Fin ) /\ X e. B ) -> ( ran ( n e. ZZ |-> ( n .x. X ) ) = B <-> ( O ` X ) = ( # ` B ) ) )
34	33	pm5.32da 673	. 2 |- ( ( G e. Grp /\ B e. Fin ) -> ( ( X e. B /\ ran ( n e. ZZ |-> ( n .x. X ) ) = B ) <-> ( X e. B /\ ( O ` X ) = ( # ` B ) ) ) )
35	4, 34	syl5bb 272	1 |- ( ( G e. Grp /\ B e. Fin ) -> ( X e. E <-> ( X e. B /\ ( O ` X ) = ( # ` B ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   Fincfn 7955   0cc0 9936   ZZcz 11377   #chash 13117   Basecbs 15857   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-int 4476  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-se 5074  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-isom 5897  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-omul 7565  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  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-hash 13118  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:  iscygodd  18290  cyggexb  18300