Theorem frgpcyg 19922

Description: A free group is cyclic iff it has zero or one generator. (Contributed by Mario Carneiro, 21-Apr-2016.) (Proof shortened by AV, 18-Apr-2021.)

Hypothesis

Ref	Expression
frgpcyg.g	|- G = (freeGrp ` I )

Assertion

Ref	Expression
frgpcyg	|- ( I ~<_ 1o <-> G e. CycGrp )

Proof of Theorem frgpcyg

Dummy variables f g n x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdom2 7985	. . 3 |- ( I ~<_ 1o <-> ( I ~< 1o \/ I ~~ 1o ) )
2		sdom1 8160	. . . . 5 |- ( I ~< 1o <-> I = (/) )
3		frgpcyg.g	. . . . . . 7 |- G = (freeGrp ` I )
4		fveq2 6191	. . . . . . 7 |- ( I = (/) -> (freeGrp ` I ) = (freeGrp ` (/) ) )
5	3, 4	syl5eq 2668	. . . . . 6 |- ( I = (/) -> G = (freeGrp ` (/) ) )
6		0ex 4790	. . . . . . . 8 |- (/) e. _V
7		eqid 2622	. . . . . . . . 9 |- (freeGrp ` (/) ) = (freeGrp ` (/) )
8	7	frgpgrp 18175	. . . . . . . 8 |- ( (/) e. _V -> (freeGrp ` (/) ) e. Grp )
9	6, 8	ax-mp 5	. . . . . . 7 |- (freeGrp ` (/) ) e. Grp
10		eqid 2622	. . . . . . . 8 |- ( Base ` (freeGrp ` (/) ) ) = ( Base ` (freeGrp ` (/) ) )
11	7, 10	0frgp 18192	. . . . . . 7 |- ( Base ` (freeGrp ` (/) ) ) ~~ 1o
12	10	0cyg 18294	. . . . . . 7 |- ( ( (freeGrp ` (/) ) e. Grp /\ ( Base ` (freeGrp ` (/) ) ) ~~ 1o ) -> (freeGrp ` (/) ) e. CycGrp )
13	9, 11, 12	mp2an 708	. . . . . 6 |- (freeGrp ` (/) ) e. CycGrp
14	5, 13	syl6eqel 2709	. . . . 5 |- ( I = (/) -> G e. CycGrp )
15	2, 14	sylbi 207	. . . 4 |- ( I ~< 1o -> G e. CycGrp )
16		eqid 2622	. . . . 5 |- ( Base ` G ) = ( Base ` G )
17		eqid 2622	. . . . 5 |- (.g ` G ) = (.g ` G )
18		relen 7960	. . . . . . 7 |- Rel ~~
19	18	brrelexi 5158	. . . . . 6 |- ( I ~~ 1o -> I e. _V )
20	3	frgpgrp 18175	. . . . . 6 |- ( I e. _V -> G e. Grp )
21	19, 20	syl 17	. . . . 5 |- ( I ~~ 1o -> G e. Grp )
22		eqid 2622	. . . . . . . 8 |- ( ~FG ` I ) = ( ~FG ` I )
23		eqid 2622	. . . . . . . 8 |- (varFGrp ` I ) = (varFGrp ` I )
24	22, 23, 3, 16	vrgpf 18181	. . . . . . 7 |- ( I e. _V -> (varFGrp ` I ) : I --> ( Base ` G ) )
25	19, 24	syl 17	. . . . . 6 |- ( I ~~ 1o -> (varFGrp ` I ) : I --> ( Base ` G ) )
26		en1uniel 8028	. . . . . 6 |- ( I ~~ 1o -> U. I e. I )
27	25, 26	ffvelrnd 6360	. . . . 5 |- ( I ~~ 1o -> ( (varFGrp ` I ) ` U. I ) e. ( Base ` G ) )
28		zringgrp 19823	. . . . . . . . 9 |- ℤring e. Grp
29		uniexg 6955	. . . . . . . . . . . 12 |- ( I e. _V -> U. I e. _V )
30	19, 29	syl 17	. . . . . . . . . . 11 |- ( I ~~ 1o -> U. I e. _V )
31		1zzd 11408	. . . . . . . . . . 11 |- ( I ~~ 1o -> 1 e. ZZ )
32	30, 31	fsnd 6179	. . . . . . . . . 10 |- ( I ~~ 1o -> { <. U. I , 1 >. } : { U. I } --> ZZ )
33		en1b 8024	. . . . . . . . . . . 12 |- ( I ~~ 1o <-> I = { U. I } )
34	33	biimpi 206	. . . . . . . . . . 11 |- ( I ~~ 1o -> I = { U. I } )
35	34	feq2d 6031	. . . . . . . . . 10 |- ( I ~~ 1o -> ( { <. U. I , 1 >. } : I --> ZZ <-> { <. U. I , 1 >. } : { U. I } --> ZZ ) )
36	32, 35	mpbird 247	. . . . . . . . 9 |- ( I ~~ 1o -> { <. U. I , 1 >. } : I --> ZZ )
37		zringbas 19824	. . . . . . . . . 10 |- ZZ = ( Base ` ℤring )
38	3, 37, 23	frgpup3 18191	. . . . . . . . 9 |- ( (ℤring e. Grp /\ I e. _V /\ { <. U. I , 1 >. } : I --> ZZ ) -> E! f e. ( G GrpHom ℤring ) ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } )
39	28, 19, 36, 38	mp3an2i 1429	. . . . . . . 8 |- ( I ~~ 1o -> E! f e. ( G GrpHom ℤring ) ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } )
40	39	adantr 481	. . . . . . 7 |- ( ( I ~~ 1o /\ x e. ( Base ` G ) ) -> E! f e. ( G GrpHom ℤring ) ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } )
41		reurex 3160	. . . . . . 7 |- ( E! f e. ( G GrpHom ℤring ) ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } -> E. f e. ( G GrpHom ℤring ) ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } )
42	40, 41	syl 17	. . . . . 6 |- ( ( I ~~ 1o /\ x e. ( Base ` G ) ) -> E. f e. ( G GrpHom ℤring ) ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } )
43		fveq1 6190	. . . . . . . . . 10 |- ( ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } -> ( ( f o. (varFGrp ` I ) ) ` U. I ) = ( { <. U. I , 1 >. } ` U. I ) )
44		fvco3 6275	. . . . . . . . . . . 12 |- ( ( (varFGrp ` I ) : I --> ( Base ` G ) /\ U. I e. I ) -> ( ( f o. (varFGrp ` I ) ) ` U. I ) = ( f ` ( (varFGrp ` I ) ` U. I ) ) )
45	25, 26, 44	syl2anc 693	. . . . . . . . . . 11 |- ( I ~~ 1o -> ( ( f o. (varFGrp ` I ) ) ` U. I ) = ( f ` ( (varFGrp ` I ) ` U. I ) ) )
46		1z 11407	. . . . . . . . . . . 12 |- 1 e. ZZ
47		fvsng 6447	. . . . . . . . . . . 12 |- ( ( U. I e. _V /\ 1 e. ZZ ) -> ( { <. U. I , 1 >. } ` U. I ) = 1 )
48	30, 46, 47	sylancl 694	. . . . . . . . . . 11 |- ( I ~~ 1o -> ( { <. U. I , 1 >. } ` U. I ) = 1 )
49	45, 48	eqeq12d 2637	. . . . . . . . . 10 |- ( I ~~ 1o -> ( ( ( f o. (varFGrp ` I ) ) ` U. I ) = ( { <. U. I , 1 >. } ` U. I ) <-> ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) )
50	43, 49	syl5ib 234	. . . . . . . . 9 |- ( I ~~ 1o -> ( ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } -> ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) )
51	50	ad2antrr 762	. . . . . . . 8 |- ( ( ( I ~~ 1o /\ x e. ( Base ` G ) ) /\ f e. ( G GrpHom ℤring ) ) -> ( ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } -> ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) )
52	16, 37	ghmf 17664	. . . . . . . . . . . . 13 |- ( f e. ( G GrpHom ℤring ) -> f : ( Base ` G ) --> ZZ )
53	52	ad2antrl 764	. . . . . . . . . . . 12 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> f : ( Base ` G ) --> ZZ )
54	53	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) /\ x e. ( Base ` G ) ) -> ( f ` x ) e. ZZ )
55	54	an32s 846	. . . . . . . . . 10 |- ( ( ( I ~~ 1o /\ x e. ( Base ` G ) ) /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( f ` x ) e. ZZ )
56		mptresid 5456	. . . . . . . . . . . . . 14 |- ( x e. ( Base ` G ) |-> x ) = ( _I |` ( Base ` G ) )
57	3, 16, 23	frgpup3 18191	. . . . . . . . . . . . . . . . . 18 |- ( ( G e. Grp /\ I e. _V /\ (varFGrp ` I ) : I --> ( Base ` G ) ) -> E! g e. ( G GrpHom G ) ( g o. (varFGrp ` I ) ) = (varFGrp ` I ) )
58	21, 19, 25, 57	syl3anc 1326	. . . . . . . . . . . . . . . . 17 |- ( I ~~ 1o -> E! g e. ( G GrpHom G ) ( g o. (varFGrp ` I ) ) = (varFGrp ` I ) )
59		reurmo 3161	. . . . . . . . . . . . . . . . 17 |- ( E! g e. ( G GrpHom G ) ( g o. (varFGrp ` I ) ) = (varFGrp ` I ) -> E* g e. ( G GrpHom G ) ( g o. (varFGrp ` I ) ) = (varFGrp ` I ) )
60	58, 59	syl 17	. . . . . . . . . . . . . . . 16 |- ( I ~~ 1o -> E* g e. ( G GrpHom G ) ( g o. (varFGrp ` I ) ) = (varFGrp ` I ) )
61	60	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> E* g e. ( G GrpHom G ) ( g o. (varFGrp ` I ) ) = (varFGrp ` I ) )
62	21	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> G e. Grp )
63	16	idghm 17675	. . . . . . . . . . . . . . . 16 |- ( G e. Grp -> ( _I |` ( Base ` G ) ) e. ( G GrpHom G ) )
64	62, 63	syl 17	. . . . . . . . . . . . . . 15 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( _I |` ( Base ` G ) ) e. ( G GrpHom G ) )
65	25	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> (varFGrp ` I ) : I --> ( Base ` G ) )
66		fcoi2 6079	. . . . . . . . . . . . . . . 16 |- ( (varFGrp ` I ) : I --> ( Base ` G ) -> ( ( _I |` ( Base ` G ) ) o. (varFGrp ` I ) ) = (varFGrp ` I ) )
67	65, 66	syl 17	. . . . . . . . . . . . . . 15 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( ( _I |` ( Base ` G ) ) o. (varFGrp ` I ) ) = (varFGrp ` I ) )
68	53	feqmptd 6249	. . . . . . . . . . . . . . . . 17 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> f = ( x e. ( Base ` G ) |-> ( f ` x ) ) )
69		eqidd 2623	. . . . . . . . . . . . . . . . 17 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( n e. ZZ |-> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) = ( n e. ZZ |-> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
70		oveq1 6657	. . . . . . . . . . . . . . . . 17 |- ( n = ( f ` x ) -> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
71	54, 68, 69, 70	fmptco 6396	. . . . . . . . . . . . . . . 16 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( ( n e. ZZ |-> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) o. f ) = ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
72	27	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( (varFGrp ` I ) ` U. I ) e. ( Base ` G ) )
73		eqid 2622	. . . . . . . . . . . . . . . . . . 19 |- ( n e. ZZ |-> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) = ( n e. ZZ |-> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
74	17, 73, 16	mulgghm2 19845	. . . . . . . . . . . . . . . . . 18 |- ( ( G e. Grp /\ ( (varFGrp ` I ) ` U. I ) e. ( Base ` G ) ) -> ( n e. ZZ |-> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) e. (ℤring GrpHom G ) )
75	62, 72, 74	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( n e. ZZ |-> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) e. (ℤring GrpHom G ) )
76		simprl 794	. . . . . . . . . . . . . . . . 17 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> f e. ( G GrpHom ℤring ) )
77		ghmco 17680	. . . . . . . . . . . . . . . . 17 |- ( ( ( n e. ZZ |-> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) e. (ℤring GrpHom G ) /\ f e. ( G GrpHom ℤring ) ) -> ( ( n e. ZZ |-> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) o. f ) e. ( G GrpHom G ) )
78	75, 76, 77	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( ( n e. ZZ |-> ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) o. f ) e. ( G GrpHom G ) )
79	71, 78	eqeltrrd 2702	. . . . . . . . . . . . . . 15 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) e. ( G GrpHom G ) )
80	34	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> I = { U. I } )
81	80	eleq2d 2687	. . . . . . . . . . . . . . . . . . 19 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( y e. I <-> y e. { U. I } ) )
82		simprr 796	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 )
83	82	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( ( f ` ( (varFGrp ` I ) ` U. I ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( 1 (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
84	16, 17	mulg1 17548	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( (varFGrp ` I ) ` U. I ) e. ( Base ` G ) -> ( 1 (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( (varFGrp ` I ) ` U. I ) )
85	72, 84	syl 17	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( 1 (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( (varFGrp ` I ) ` U. I ) )
86	83, 85	eqtrd 2656	. . . . . . . . . . . . . . . . . . . 20 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( ( f ` ( (varFGrp ` I ) ` U. I ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( (varFGrp ` I ) ` U. I ) )
87		elsni 4194	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( y e. { U. I } -> y = U. I )
88	87	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( y e. { U. I } -> ( (varFGrp ` I ) ` y ) = ( (varFGrp ` I ) ` U. I ) )
89	88	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 |- ( y e. { U. I } -> ( f ` ( (varFGrp ` I ) ` y ) ) = ( f ` ( (varFGrp ` I ) ` U. I ) ) )
90	89	oveq1d 6665	. . . . . . . . . . . . . . . . . . . . 21 |- ( y e. { U. I } -> ( ( f ` ( (varFGrp ` I ) ` y ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( ( f ` ( (varFGrp ` I ) ` U. I ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
91	90, 88	eqeq12d 2637	. . . . . . . . . . . . . . . . . . . 20 |- ( y e. { U. I } -> ( ( ( f ` ( (varFGrp ` I ) ` y ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( (varFGrp ` I ) ` y ) <-> ( ( f ` ( (varFGrp ` I ) ` U. I ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( (varFGrp ` I ) ` U. I ) ) )
92	86, 91	syl5ibrcom 237	. . . . . . . . . . . . . . . . . . 19 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( y e. { U. I } -> ( ( f ` ( (varFGrp ` I ) ` y ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( (varFGrp ` I ) ` y ) ) )
93	81, 92	sylbid 230	. . . . . . . . . . . . . . . . . 18 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( y e. I -> ( ( f ` ( (varFGrp ` I ) ` y ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( (varFGrp ` I ) ` y ) ) )
94	93	imp 445	. . . . . . . . . . . . . . . . 17 |- ( ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) /\ y e. I ) -> ( ( f ` ( (varFGrp ` I ) ` y ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( (varFGrp ` I ) ` y ) )
95	94	mpteq2dva 4744	. . . . . . . . . . . . . . . 16 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( y e. I |-> ( ( f ` ( (varFGrp ` I ) ` y ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) = ( y e. I |-> ( (varFGrp ` I ) ` y ) ) )
96	65	ffvelrnda 6359	. . . . . . . . . . . . . . . . 17 |- ( ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) /\ y e. I ) -> ( (varFGrp ` I ) ` y ) e. ( Base ` G ) )
97	65	feqmptd 6249	. . . . . . . . . . . . . . . . 17 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> (varFGrp ` I ) = ( y e. I |-> ( (varFGrp ` I ) ` y ) ) )
98		eqidd 2623	. . . . . . . . . . . . . . . . 17 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) = ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
99		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( x = ( (varFGrp ` I ) ` y ) -> ( f ` x ) = ( f ` ( (varFGrp ` I ) ` y ) ) )
100	99	oveq1d 6665	. . . . . . . . . . . . . . . . 17 |- ( x = ( (varFGrp ` I ) ` y ) -> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) = ( ( f ` ( (varFGrp ` I ) ` y ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
101	96, 97, 98, 100	fmptco 6396	. . . . . . . . . . . . . . . 16 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) o. (varFGrp ` I ) ) = ( y e. I |-> ( ( f ` ( (varFGrp ` I ) ` y ) ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
102	95, 101, 97	3eqtr4d 2666	. . . . . . . . . . . . . . 15 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) o. (varFGrp ` I ) ) = (varFGrp ` I ) )
103		coeq1 5279	. . . . . . . . . . . . . . . . 17 |- ( g = ( _I |` ( Base ` G ) ) -> ( g o. (varFGrp ` I ) ) = ( ( _I |` ( Base ` G ) ) o. (varFGrp ` I ) ) )
104	103	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( g = ( _I |` ( Base ` G ) ) -> ( ( g o. (varFGrp ` I ) ) = (varFGrp ` I ) <-> ( ( _I |` ( Base ` G ) ) o. (varFGrp ` I ) ) = (varFGrp ` I ) ) )
105		coeq1 5279	. . . . . . . . . . . . . . . . 17 |- ( g = ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) -> ( g o. (varFGrp ` I ) ) = ( ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) o. (varFGrp ` I ) ) )
106	105	eqeq1d 2624	. . . . . . . . . . . . . . . 16 |- ( g = ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) -> ( ( g o. (varFGrp ` I ) ) = (varFGrp ` I ) <-> ( ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) o. (varFGrp ` I ) ) = (varFGrp ` I ) ) )
107	104, 106	rmoi 3530	. . . . . . . . . . . . . . 15 |- ( ( E* g e. ( G GrpHom G ) ( g o. (varFGrp ` I ) ) = (varFGrp ` I ) /\ ( ( _I |` ( Base ` G ) ) e. ( G GrpHom G ) /\ ( ( _I |` ( Base ` G ) ) o. (varFGrp ` I ) ) = (varFGrp ` I ) ) /\ ( ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) e. ( G GrpHom G ) /\ ( ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) o. (varFGrp ` I ) ) = (varFGrp ` I ) ) ) -> ( _I |` ( Base ` G ) ) = ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
108	61, 64, 67, 79, 102, 107	syl122anc 1335	. . . . . . . . . . . . . 14 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( _I |` ( Base ` G ) ) = ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
109	56, 108	syl5eq 2668	. . . . . . . . . . . . 13 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> ( x e. ( Base ` G ) |-> x ) = ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
110		mpteqb 6299	. . . . . . . . . . . . . 14 |- ( A. x e. ( Base ` G ) x e. ( Base ` G ) -> ( ( x e. ( Base ` G ) |-> x ) = ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) <-> A. x e. ( Base ` G ) x = ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
111		id 22	. . . . . . . . . . . . . 14 |- ( x e. ( Base ` G ) -> x e. ( Base ` G ) )
112	110, 111	mprg 2926	. . . . . . . . . . . . 13 |- ( ( x e. ( Base ` G ) |-> x ) = ( x e. ( Base ` G ) |-> ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) <-> A. x e. ( Base ` G ) x = ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
113	109, 112	sylib 208	. . . . . . . . . . . 12 |- ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> A. x e. ( Base ` G ) x = ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
114	113	r19.21bi 2932	. . . . . . . . . . 11 |- ( ( ( I ~~ 1o /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) /\ x e. ( Base ` G ) ) -> x = ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
115	114	an32s 846	. . . . . . . . . 10 |- ( ( ( I ~~ 1o /\ x e. ( Base ` G ) ) /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> x = ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
116	70	eqeq2d 2632	. . . . . . . . . . 11 |- ( n = ( f ` x ) -> ( x = ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) <-> x = ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
117	116	rspcev 3309	. . . . . . . . . 10 |- ( ( ( f ` x ) e. ZZ /\ x = ( ( f ` x ) (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) -> E. n e. ZZ x = ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
118	55, 115, 117	syl2anc 693	. . . . . . . . 9 |- ( ( ( I ~~ 1o /\ x e. ( Base ` G ) ) /\ ( f e. ( G GrpHom ℤring ) /\ ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 ) ) -> E. n e. ZZ x = ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
119	118	expr 643	. . . . . . . 8 |- ( ( ( I ~~ 1o /\ x e. ( Base ` G ) ) /\ f e. ( G GrpHom ℤring ) ) -> ( ( f ` ( (varFGrp ` I ) ` U. I ) ) = 1 -> E. n e. ZZ x = ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
120	51, 119	syld 47	. . . . . . 7 |- ( ( ( I ~~ 1o /\ x e. ( Base ` G ) ) /\ f e. ( G GrpHom ℤring ) ) -> ( ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } -> E. n e. ZZ x = ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
121	120	rexlimdva 3031	. . . . . 6 |- ( ( I ~~ 1o /\ x e. ( Base ` G ) ) -> ( E. f e. ( G GrpHom ℤring ) ( f o. (varFGrp ` I ) ) = { <. U. I , 1 >. } -> E. n e. ZZ x = ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) ) )
122	42, 121	mpd 15	. . . . 5 |- ( ( I ~~ 1o /\ x e. ( Base ` G ) ) -> E. n e. ZZ x = ( n (.g ` G ) ( (varFGrp ` I ) ` U. I ) ) )
123	16, 17, 21, 27, 122	iscygd 18289	. . . 4 |- ( I ~~ 1o -> G e. CycGrp )
124	15, 123	jaoi 394	. . 3 |- ( ( I ~< 1o \/ I ~~ 1o ) -> G e. CycGrp )
125	1, 124	sylbi 207	. 2 |- ( I ~<_ 1o -> G e. CycGrp )
126		cygabl 18292	. . 3 |- ( G e. CycGrp -> G e. Abel )
127	3	frgpnabl 18278	. . . . 5 |- ( 1o ~< I -> -. G e. Abel )
128	127	con2i 134	. . . 4 |- ( G e. Abel -> -. 1o ~< I )
129		ablgrp 18198	. . . . . 6 |- ( G e. Abel -> G e. Grp )
130		eqid 2622	. . . . . . 7 |- ( 0g ` G ) = ( 0g ` G )
131	16, 130	grpidcl 17450	. . . . . 6 |- ( G e. Grp -> ( 0g ` G ) e. ( Base ` G ) )
132	3, 16	elbasfv 15920	. . . . . 6 |- ( ( 0g ` G ) e. ( Base ` G ) -> I e. _V )
133	129, 131, 132	3syl 18	. . . . 5 |- ( G e. Abel -> I e. _V )
134		1onn 7719	. . . . . 6 |- 1o e. om
135		nnfi 8153	. . . . . 6 |- ( 1o e. om -> 1o e. Fin )
136	134, 135	ax-mp 5	. . . . 5 |- 1o e. Fin
137		fidomtri2 8820	. . . . 5 |- ( ( I e. _V /\ 1o e. Fin ) -> ( I ~<_ 1o <-> -. 1o ~< I ) )
138	133, 136, 137	sylancl 694	. . . 4 |- ( G e. Abel -> ( I ~<_ 1o <-> -. 1o ~< I ) )
139	128, 138	mpbird 247	. . 3 |- ( G e. Abel -> I ~<_ 1o )
140	126, 139	syl 17	. 2 |- ( G e. CycGrp -> I ~<_ 1o )
141	125, 140	impbii 199	1 |- ( I ~<_ 1o <-> G e. CycGrp )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   E*wrmo 2915   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   |` cres 5116   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   1oc1o 7553   ~~ cen 7952   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   1c1 9937   ZZcz 11377   Basecbs 15857   0gc0g 16100   Grpcgrp 17422  ._gcmg 17540   GrpHom cghm 17657   ~_FG cefg 18119  freeGrpcfrgp 18120  var_FGrpcvrgp 18121   Abelcabl 18194  CycGrpccyg 18279  ℤ_ringzring 19818

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-addf 10015  ax-mulf 10016

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-ot 4186  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-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  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-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-reverse 13305  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-gsum 16103  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-frmd 17386  df-vrmd 17387  df-grp 17425  df-minusg 17426  df-mulg 17541  df-subg 17591  df-ghm 17658  df-efg 18122  df-frgp 18123  df-vrgp 18124  df-cmn 18195  df-abl 18196  df-cyg 18280  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-cnfld 19747  df-zring 19819

This theorem is referenced by: (None)