Theorem prmcyg 18295

Description: A group with prime order is cyclic. (Contributed by Mario Carneiro, 27-Apr-2016.)

Hypothesis

Ref	Expression
cygctb.1	|- B = ( Base ` G )

Assertion

Ref	Expression
prmcyg	|- ( ( G e. Grp /\ ( # ` B ) e. Prime ) -> G e. CycGrp )

Proof of Theorem prmcyg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1nprm 15392	. . . 4 |- -. 1 e. Prime
2		simpr 477	. . . . . . . . 9 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ B C_ { ( 0g ` G ) } ) -> B C_ { ( 0g ` G ) } )
3		cygctb.1	. . . . . . . . . . . 12 |- B = ( Base ` G )
4		eqid 2622	. . . . . . . . . . . 12 |- ( 0g ` G ) = ( 0g ` G )
5	3, 4	grpidcl 17450	. . . . . . . . . . 11 |- ( G e. Grp -> ( 0g ` G ) e. B )
6	5	snssd 4340	. . . . . . . . . 10 |- ( G e. Grp -> { ( 0g ` G ) } C_ B )
7	6	ad2antrr 762	. . . . . . . . 9 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ B C_ { ( 0g ` G ) } ) -> { ( 0g ` G ) } C_ B )
8	2, 7	eqssd 3620	. . . . . . . 8 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ B C_ { ( 0g ` G ) } ) -> B = { ( 0g ` G ) } )
9	8	fveq2d 6195	. . . . . . 7 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ B C_ { ( 0g ` G ) } ) -> ( # ` B ) = ( # ` { ( 0g ` G ) } ) )
10		fvex 6201	. . . . . . . 8 |- ( 0g ` G ) e. _V
11		hashsng 13159	. . . . . . . 8 |- ( ( 0g ` G ) e. _V -> ( # ` { ( 0g ` G ) } ) = 1 )
12	10, 11	ax-mp 5	. . . . . . 7 |- ( # ` { ( 0g ` G ) } ) = 1
13	9, 12	syl6eq 2672	. . . . . 6 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ B C_ { ( 0g ` G ) } ) -> ( # ` B ) = 1 )
14		simplr 792	. . . . . 6 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ B C_ { ( 0g ` G ) } ) -> ( # ` B ) e. Prime )
15	13, 14	eqeltrrd 2702	. . . . 5 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ B C_ { ( 0g ` G ) } ) -> 1 e. Prime )
16	15	ex 450	. . . 4 |- ( ( G e. Grp /\ ( # ` B ) e. Prime ) -> ( B C_ { ( 0g ` G ) } -> 1 e. Prime ) )
17	1, 16	mtoi 190	. . 3 |- ( ( G e. Grp /\ ( # ` B ) e. Prime ) -> -. B C_ { ( 0g ` G ) } )
18		nss 3663	. . 3 |- ( -. B C_ { ( 0g ` G ) } <-> E. x ( x e. B /\ -. x e. { ( 0g ` G ) } ) )
19	17, 18	sylib 208	. 2 |- ( ( G e. Grp /\ ( # ` B ) e. Prime ) -> E. x ( x e. B /\ -. x e. { ( 0g ` G ) } ) )
20		eqid 2622	. . 3 |- ( od ` G ) = ( od ` G )
21		simpll 790	. . 3 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> G e. Grp )
22		simprl 794	. . 3 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> x e. B )
23		simprr 796	. . . . 5 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> -. x e. { ( 0g ` G ) } )
24	20, 4, 3	odeq1 17977	. . . . . . 7 |- ( ( G e. Grp /\ x e. B ) -> ( ( ( od ` G ) ` x ) = 1 <-> x = ( 0g ` G ) ) )
25	21, 22, 24	syl2anc 693	. . . . . 6 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( ( ( od ` G ) ` x ) = 1 <-> x = ( 0g ` G ) ) )
26		velsn 4193	. . . . . 6 |- ( x e. { ( 0g ` G ) } <-> x = ( 0g ` G ) )
27	25, 26	syl6bbr 278	. . . . 5 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( ( ( od ` G ) ` x ) = 1 <-> x e. { ( 0g ` G ) } ) )
28	23, 27	mtbird 315	. . . 4 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> -. ( ( od ` G ) ` x ) = 1 )
29		prmnn 15388	. . . . . . . . . 10 |- ( ( # ` B ) e. Prime -> ( # ` B ) e. NN )
30	29	ad2antlr 763	. . . . . . . . 9 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( # ` B ) e. NN )
31	30	nnnn0d 11351	. . . . . . . 8 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( # ` B ) e. NN0 )
32		fvex 6201	. . . . . . . . . 10 |- ( Base ` G ) e. _V
33	3, 32	eqeltri 2697	. . . . . . . . 9 |- B e. _V
34		hashclb 13149	. . . . . . . . 9 |- ( B e. _V -> ( B e. Fin <-> ( # ` B ) e. NN0 ) )
35	33, 34	ax-mp 5	. . . . . . . 8 |- ( B e. Fin <-> ( # ` B ) e. NN0 )
36	31, 35	sylibr 224	. . . . . . 7 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> B e. Fin )
37	3, 20	oddvds2 17983	. . . . . . 7 |- ( ( G e. Grp /\ B e. Fin /\ x e. B ) -> ( ( od ` G ) ` x ) || ( # ` B ) )
38	21, 36, 22, 37	syl3anc 1326	. . . . . 6 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( ( od ` G ) ` x ) || ( # ` B ) )
39		simplr 792	. . . . . . 7 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( # ` B ) e. Prime )
40	3, 20	odcl2 17982	. . . . . . . 8 |- ( ( G e. Grp /\ B e. Fin /\ x e. B ) -> ( ( od ` G ) ` x ) e. NN )
41	21, 36, 22, 40	syl3anc 1326	. . . . . . 7 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( ( od ` G ) ` x ) e. NN )
42		dvdsprime 15400	. . . . . . 7 |- ( ( ( # ` B ) e. Prime /\ ( ( od ` G ) ` x ) e. NN ) -> ( ( ( od ` G ) ` x ) || ( # ` B ) <-> ( ( ( od ` G ) ` x ) = ( # ` B ) \/ ( ( od ` G ) ` x ) = 1 ) ) )
43	39, 41, 42	syl2anc 693	. . . . . 6 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( ( ( od ` G ) ` x ) || ( # ` B ) <-> ( ( ( od ` G ) ` x ) = ( # ` B ) \/ ( ( od ` G ) ` x ) = 1 ) ) )
44	38, 43	mpbid 222	. . . . 5 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( ( ( od ` G ) ` x ) = ( # ` B ) \/ ( ( od ` G ) ` x ) = 1 ) )
45	44	ord 392	. . . 4 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( -. ( ( od ` G ) ` x ) = ( # ` B ) -> ( ( od ` G ) ` x ) = 1 ) )
46	28, 45	mt3d 140	. . 3 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> ( ( od ` G ) ` x ) = ( # ` B ) )
47	3, 20, 21, 22, 46	iscygodd 18290	. 2 |- ( ( ( G e. Grp /\ ( # ` B ) e. Prime ) /\ ( x e. B /\ -. x e. { ( 0g ` G ) } ) ) -> G e. CycGrp )
48	19, 47	exlimddv 1863	1 |- ( ( G e. Grp /\ ( # ` B ) e. Prime ) -> G e. CycGrp )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   class class class wbr 4653   ` cfv 5888   Fincfn 7955   1c1 9937   NNcn 11020   NN0cn0 11292   #chash 13117   || cdvds 14983   Primecprime 15385   Basecbs 15857   0gc0g 16100   Grpcgrp 17422   odcod 17944  CycGrpccyg 18279

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-fal 1489  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-disj 4621  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-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  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-fzo 12466  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-clim 14219  df-sum 14417  df-dvds 14984  df-prm 15386  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-sbg 17427  df-mulg 17541  df-subg 17591  df-eqg 17593  df-od 17948  df-cyg 18280

This theorem is referenced by:  lt6abl  18296