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Theorem frgpcyg 16856
Description: A free group is cyclic iff it has zero or one generator. (Contributed by Mario Carneiro, 21-Apr-2016.)
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.
StepHypRef Expression
1 brdom2 7139 . . 3  |-  ( I  ~<_  1o  <->  ( I  ~<  1o  \/  I  ~~  1o ) )
2 sdom1 7310 . . . . 5  |-  ( I 
~<  1o  <->  I  =  (/) )
3 frgpcyg.g . . . . . . 7  |-  G  =  (freeGrp `  I )
4 fveq2 5730 . . . . . . 7  |-  ( I  =  (/)  ->  (freeGrp `  I
)  =  (freeGrp `  (/) ) )
53, 4syl5eq 2482 . . . . . 6  |-  ( I  =  (/)  ->  G  =  (freeGrp `  (/) ) )
6 0ex 4341 . . . . . . . 8  |-  (/)  e.  _V
7 eqid 2438 . . . . . . . . 9  |-  (freeGrp `  (/) )  =  (freeGrp `  (/) )
87frgpgrp 15396 . . . . . . . 8  |-  ( (/)  e.  _V  ->  (freeGrp `  (/) )  e. 
Grp )
96, 8ax-mp 8 . . . . . . 7  |-  (freeGrp `  (/) )  e. 
Grp
10 eqid 2438 . . . . . . . 8  |-  ( Base `  (freeGrp `  (/) ) )  =  ( Base `  (freeGrp `  (/) ) )
117, 100frgp 15413 . . . . . . 7  |-  ( Base `  (freeGrp `  (/) ) ) 
~~  1o
12100cyg 15504 . . . . . . 7  |-  ( ( (freeGrp `  (/) )  e. 
Grp  /\  ( Base `  (freeGrp `  (/) ) ) 
~~  1o )  -> 
(freeGrp `  (/) )  e. CycGrp )
139, 11, 12mp2an 655 . . . . . 6  |-  (freeGrp `  (/) )  e. CycGrp
145, 13syl6eqel 2526 . . . . 5  |-  ( I  =  (/)  ->  G  e. CycGrp
)
152, 14sylbi 189 . . . 4  |-  ( I 
~<  1o  ->  G  e. CycGrp )
16 eqid 2438 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
17 eqid 2438 . . . . 5  |-  (.g `  G
)  =  (.g `  G
)
18 relen 7116 . . . . . . 7  |-  Rel  ~~
1918brrelexi 4920 . . . . . 6  |-  ( I 
~~  1o  ->  I  e. 
_V )
203frgpgrp 15396 . . . . . 6  |-  ( I  e.  _V  ->  G  e.  Grp )
2119, 20syl 16 . . . . 5  |-  ( I 
~~  1o  ->  G  e. 
Grp )
22 eqid 2438 . . . . . . . 8  |-  ( ~FG  `  I
)  =  ( ~FG  `  I
)
23 eqid 2438 . . . . . . . 8  |-  (varFGrp `  I
)  =  (varFGrp `  I
)
2422, 23, 3, 16vrgpf 15402 . . . . . . 7  |-  ( I  e.  _V  ->  (varFGrp `  I
) : I --> ( Base `  G ) )
2519, 24syl 16 . . . . . 6  |-  ( I 
~~  1o  ->  (varFGrp `  I
) : I --> ( Base `  G ) )
26 en1b 7177 . . . . . . . 8  |-  ( I 
~~  1o  <->  I  =  { U. I } )
27 eqimss2 3403 . . . . . . . 8  |-  ( I  =  { U. I }  ->  { U. I }  C_  I )
2826, 27sylbi 189 . . . . . . 7  |-  ( I 
~~  1o  ->  { U. I }  C_  I )
29 uniexg 4708 . . . . . . . . 9  |-  ( I  e.  _V  ->  U. I  e.  _V )
3019, 29syl 16 . . . . . . . 8  |-  ( I 
~~  1o  ->  U. I  e.  _V )
31 snssg 3934 . . . . . . . 8  |-  ( U. I  e.  _V  ->  ( U. I  e.  I  <->  { U. I }  C_  I ) )
3230, 31syl 16 . . . . . . 7  |-  ( I 
~~  1o  ->  ( U. I  e.  I  <->  { U. I }  C_  I ) )
3328, 32mpbird 225 . . . . . 6  |-  ( I 
~~  1o  ->  U. I  e.  I )
3425, 33ffvelrnd 5873 . . . . 5  |-  ( I 
~~  1o  ->  ( (varFGrp `  I ) `  U. I )  e.  (
Base `  G )
)
35 zsubrg 16754 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
36 subrgsubg 15876 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
3735, 36ax-mp 8 . . . . . . . . . 10  |-  ZZ  e.  (SubGrp ` fld )
38 eqid 2438 . . . . . . . . . . 11  |-  (flds  ZZ )  =  (flds  ZZ )
3938subggrp 14949 . . . . . . . . . 10  |-  ( ZZ  e.  (SubGrp ` fld )  ->  (flds  ZZ )  e.  Grp )
4037, 39mp1i 12 . . . . . . . . 9  |-  ( I 
~~  1o  ->  (flds  ZZ )  e.  Grp )
41 1z 10313 . . . . . . . . . . . . 13  |-  1  e.  ZZ
42 f1osng 5718 . . . . . . . . . . . . 13  |-  ( ( U. I  e.  _V  /\  1  e.  ZZ )  ->  { <. U. I ,  1 >. } : { U. I } -1-1-onto-> { 1 } )
4330, 41, 42sylancl 645 . . . . . . . . . . . 12  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : { U. I }
-1-1-onto-> { 1 } )
44 f1of 5676 . . . . . . . . . . . 12  |-  ( {
<. U. I ,  1
>. } : { U. I } -1-1-onto-> { 1 }  ->  {
<. U. I ,  1
>. } : { U. I } --> { 1 } )
4543, 44syl 16 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : { U. I }
--> { 1 } )
4626biimpi 188 . . . . . . . . . . . 12  |-  ( I 
~~  1o  ->  I  =  { U. I }
)
4746feq2d 5583 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  ( {
<. U. I ,  1
>. } : I --> { 1 }  <->  { <. U. I ,  1
>. } : { U. I } --> { 1 } ) )
4845, 47mpbird 225 . . . . . . . . . 10  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : I --> { 1 } )
49 snssi 3944 . . . . . . . . . . 11  |-  ( 1  e.  ZZ  ->  { 1 }  C_  ZZ )
5041, 49ax-mp 8 . . . . . . . . . 10  |-  { 1 }  C_  ZZ
51 fss 5601 . . . . . . . . . 10  |-  ( ( { <. U. I ,  1
>. } : I --> { 1 }  /\  { 1 }  C_  ZZ )  ->  { <. U. I ,  1
>. } : I --> ZZ )
5248, 50, 51sylancl 645 . . . . . . . . 9  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : I --> ZZ )
5338subrgbas 15879 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
5435, 53ax-mp 8 . . . . . . . . . 10  |-  ZZ  =  ( Base `  (flds  ZZ ) )
553, 54, 23frgpup3 15412 . . . . . . . . 9  |-  ( ( (flds  ZZ )  e.  Grp  /\  I  e.  _V  /\  { <. U. I ,  1
>. } : I --> ZZ )  ->  E! f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I
) )  =  { <. U. I ,  1
>. } )
5640, 19, 52, 55syl3anc 1185 . . . . . . . 8  |-  ( I 
~~  1o  ->  E! f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. } )
5756adantr 453 . . . . . . 7  |-  ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  ->  E! f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. } )
58 reurex 2924 . . . . . . 7  |-  ( E! f  e.  ( G 
GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. }  ->  E. f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I
) )  =  { <. U. I ,  1
>. } )
5957, 58syl 16 . . . . . 6  |-  ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  ->  E. f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. } )
60 fveq1 5729 . . . . . . . . . 10  |-  ( ( f  o.  (varFGrp `  I
) )  =  { <. U. I ,  1
>. }  ->  ( (
f  o.  (varFGrp `  I
) ) `  U. I )  =  ( { <. U. I ,  1
>. } `  U. I
) )
61 fvco3 5802 . . . . . . . . . . . 12  |-  ( ( (varFGrp `  I ) : I --> ( Base `  G
)  /\  U. I  e.  I )  ->  (
( f  o.  (varFGrp `  I
) ) `  U. I )  =  ( f `  ( (varFGrp `  I ) `  U. I ) ) )
6225, 33, 61syl2anc 644 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  ( ( f  o.  (varFGrp `  I
) ) `  U. I )  =  ( f `  ( (varFGrp `  I ) `  U. I ) ) )
63 fvsng 5929 . . . . . . . . . . . 12  |-  ( ( U. I  e.  _V  /\  1  e.  ZZ )  ->  ( { <. U. I ,  1 >. } `  U. I )  =  1 )
6430, 41, 63sylancl 645 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  ( {
<. U. I ,  1
>. } `  U. I
)  =  1 )
6562, 64eqeq12d 2452 . . . . . . . . . 10  |-  ( I 
~~  1o  ->  ( ( ( f  o.  (varFGrp `  I
) ) `  U. I )  =  ( { <. U. I ,  1
>. } `  U. I
)  <->  ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1 ) )
6660, 65syl5ib 212 . . . . . . . . 9  |-  ( I 
~~  1o  ->  ( ( f  o.  (varFGrp `  I
) )  =  { <. U. I ,  1
>. }  ->  ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1 ) )
6766ad2antrr 708 . . . . . . . 8  |-  ( ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  /\  f  e.  ( G  GrpHom  (flds  ZZ ) ) )  -> 
( ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. }  ->  ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1 ) )
6816, 54ghmf 15012 . . . . . . . . . . . . 13  |-  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  ->  f :
( Base `  G ) --> ZZ )
6968ad2antrl 710 . . . . . . . . . . . 12  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  f :
( Base `  G ) --> ZZ )
7069ffvelrnda 5872 . . . . . . . . . . 11  |-  ( ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  /\  x  e.  ( Base `  G
) )  ->  (
f `  x )  e.  ZZ )
7170an32s 781 . . . . . . . . . 10  |-  ( ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( f `  x )  e.  ZZ )
72 mptresid 5197 . . . . . . . . . . . . . 14  |-  ( x  e.  ( Base `  G
)  |->  x )  =  (  _I  |`  ( Base `  G ) )
733, 16, 23frgpup3 15412 . . . . . . . . . . . . . . . . . 18  |-  ( ( G  e.  Grp  /\  I  e.  _V  /\  (varFGrp `  I
) : I --> ( Base `  G ) )  ->  E! g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) )
7421, 19, 25, 73syl3anc 1185 . . . . . . . . . . . . . . . . 17  |-  ( I 
~~  1o  ->  E! g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I ) )
75 reurmo 2925 . . . . . . . . . . . . . . . . 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 ) )
7674, 75syl 16 . . . . . . . . . . . . . . . 16  |-  ( I 
~~  1o  ->  E* g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) )
7776adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  E* g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) )
7821adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  G  e.  Grp )
7916idghm 15023 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Grp  ->  (  _I  |`  ( Base `  G
) )  e.  ( G  GrpHom  G ) )
8078, 79syl 16 . . . . . . . . . . . . . . 15  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  (  _I  |`  ( Base `  G
) )  e.  ( G  GrpHom  G ) )
8125adantr 453 . . . . . . . . . . . . . . . 16  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  (varFGrp `  I ) : I --> ( Base `  G
) )
82 fcoi2 5620 . . . . . . . . . . . . . . . 16  |-  ( (varFGrp `  I ) : I --> ( Base `  G
)  ->  ( (  _I  |`  ( Base `  G
) )  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) )
8381, 82syl 16 . . . . . . . . . . . . . . 15  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( (  _I  |`  ( Base `  G
) )  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) )
8469feqmptd 5781 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  f  =  ( x  e.  ( Base `  G )  |->  ( f `  x ) ) )
85 eqidd 2439 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
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 ) ) ) )
86 oveq1 6090 . . . . . . . . . . . . . . . . 17  |-  ( n  =  ( f `  x )  ->  (
n (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  =  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) )
8770, 84, 85, 86fmptco 5903 . . . . . . . . . . . . . . . 16  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
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 ) ) ) )
8834adantr 453 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( (varFGrp `  I
) `  U. I )  e.  ( Base `  G
) )
89 eqid 2438 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  ZZ  |->  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )  =  ( n  e.  ZZ  |->  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )
9038, 17, 89, 16mulgghm2 16788 . . . . . . . . . . . . . . . . . 18  |-  ( ( G  e.  Grp  /\  ( (varFGrp `  I ) `  U. I )  e.  (
Base `  G )
)  ->  ( n  e.  ZZ  |->  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )  e.  ( (flds  ZZ )  GrpHom  G ) )
9178, 88, 90syl2anc 644 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( n  e.  ZZ  |->  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )  e.  ( (flds  ZZ )  GrpHom  G ) )
92 simprl 734 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  f  e.  ( G  GrpHom  (flds  ZZ ) ) )
93 ghmco 15027 . . . . . . . . . . . . . . . . 17  |-  ( ( ( n  e.  ZZ  |->  ( n (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) )  e.  ( (flds  ZZ )  GrpHom  G )  /\  f  e.  ( G  GrpHom  (flds  ZZ ) ) )  -> 
( ( n  e.  ZZ  |->  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )  o.  f )  e.  ( G  GrpHom  G ) )
9491, 92, 93syl2anc 644 . . . . . . . . . . . . . . . 16  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( (
n  e.  ZZ  |->  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )  o.  f )  e.  ( G  GrpHom  G ) )
9587, 94eqeltrrd 2513 . . . . . . . . . . . . . . 15  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( x  e.  ( Base `  G
)  |->  ( ( f `
 x ) (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )  e.  ( G  GrpHom  G ) )
9646adantr 453 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  I  =  { U. I } )
9796eleq2d 2505 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( y  e.  I  <->  y  e.  { U. I } ) )
98 simprr 735 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1 )
9998oveq1d 6098 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( (
f `  ( (varFGrp `  I
) `  U. I ) ) (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  =  ( 1 (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) )
10016, 17mulg1 14899 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( (varFGrp `  I ) `  U. I )  e.  (
Base `  G )  ->  ( 1 (.g `  G
) ( (varFGrp `  I
) `  U. I ) )  =  ( (varFGrp `  I ) `  U. I ) )
10188, 100syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( 1 (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  =  ( (varFGrp `  I ) `  U. I ) )
10299, 101eqtrd 2470 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( (
f `  ( (varFGrp `  I
) `  U. I ) ) (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  =  ( (varFGrp `  I ) `  U. I ) )
103 elsni 3840 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  { U. I }  ->  y  =  U. I )
104103fveq2d 5734 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  { U. I }  ->  ( (varFGrp `  I
) `  y )  =  ( (varFGrp `  I
) `  U. I ) )
105104fveq2d 5734 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  { U. I }  ->  ( f `  ( (varFGrp `  I ) `  y
) )  =  ( f `  ( (varFGrp `  I ) `  U. I ) ) )
106105oveq1d 6098 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
107106, 104eqeq12d 2452 . . . . . . . . . . . . . . . . . . . 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 ) ) )
108102, 107syl5ibrcom 215 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( y  e.  { U. I }  ->  ( ( f `  ( (varFGrp `  I ) `  y
) ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) )  =  ( (varFGrp `  I ) `  y
) ) )
10997, 108sylbid 208 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( y  e.  I  ->  ( ( f `  ( (varFGrp `  I ) `  y
) ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) )  =  ( (varFGrp `  I ) `  y
) ) )
110109imp 420 . . . . . . . . . . . . . . . . 17  |-  ( ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  /\  y  e.  I )  ->  (
( f `  (
(varFGrp `  I ) `  y
) ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) )  =  ( (varFGrp `  I ) `  y
) )
111110mpteq2dva 4297 . . . . . . . . . . . . . . . 16  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
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
) ) )
11281ffvelrnda 5872 . . . . . . . . . . . . . . . . 17  |-  ( ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  /\  y  e.  I )  ->  (
(varFGrp `  I ) `  y
)  e.  ( Base `  G ) )
11381feqmptd 5781 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  (varFGrp `  I )  =  ( y  e.  I  |->  ( (varFGrp `  I ) `  y
) ) )
114 eqidd 2439 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
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 ) ) ) )
115 fveq2 5730 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( (varFGrp `  I
) `  y )  ->  ( f `  x
)  =  ( f `
 ( (varFGrp `  I
) `  y )
) )
116115oveq1d 6098 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( (varFGrp `  I
) `  y )  ->  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) )  =  ( ( f `  ( (varFGrp `  I ) `  y
) ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) )
117112, 113, 114, 116fmptco 5903 . . . . . . . . . . . . . . . 16  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
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 ) ) ) )
118111, 117, 1133eqtr4d 2480 . . . . . . . . . . . . . . 15  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( (
x  e.  ( Base `  G )  |->  ( ( f `  x ) (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )  o.  (varFGrp `  I ) )  =  (varFGrp `  I ) )
119 coeq1 5032 . . . . . . . . . . . . . . . . 17  |-  ( g  =  (  _I  |`  ( Base `  G ) )  ->  ( g  o.  (varFGrp `  I ) )  =  ( (  _I  |`  ( Base `  G ) )  o.  (varFGrp `  I ) ) )
120119eqeq1d 2446 . . . . . . . . . . . . . . . 16  |-  ( g  =  (  _I  |`  ( Base `  G ) )  ->  ( ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I )  <->  ( (  _I  |`  ( Base `  G
) )  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) ) )
121 coeq1 5032 . . . . . . . . . . . . . . . . 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 ) ) )
122121eqeq1d 2446 . . . . . . . . . . . . . . . 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 ) ) )
123120, 122rmoi 3252 . . . . . . . . . . . . . . 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 ) ) ) )
12477, 80, 83, 95, 118, 123syl122anc 1194 . . . . . . . . . . . . . 14  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  (  _I  |`  ( Base `  G
) )  =  ( x  e.  ( Base `  G )  |->  ( ( f `  x ) (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) ) )
12572, 124syl5eq 2482 . . . . . . . . . . . . 13  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( x  e.  ( Base `  G
)  |->  x )  =  ( x  e.  (
Base `  G )  |->  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) ) )
126 mpteqb 5821 . . . . . . . . . . . . . 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 ) ) ) )
127 id 21 . . . . . . . . . . . . . 14  |-  ( x  e.  ( Base `  G
)  ->  x  e.  ( Base `  G )
)
128126, 127mprg 2777 . . . . . . . . . . . . 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 ) ) )
129125, 128sylib 190 . . . . . . . . . . . 12  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  A. x  e.  ( Base `  G
) x  =  ( ( f `  x
) (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )
130129r19.21bi 2806 . . . . . . . . . . 11  |-  ( ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  /\  x  e.  ( Base `  G
) )  ->  x  =  ( ( f `
 x ) (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )
131130an32s 781 . . . . . . . . . 10  |-  ( ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  x  =  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) )
13286eqeq2d 2449 . . . . . . . . . . 11  |-  ( n  =  ( f `  x )  ->  (
x  =  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  <->  x  =  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) ) )
133132rspcev 3054 . . . . . . . . . 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 ) ) )
13471, 131, 133syl2anc 644 . . . . . . . . 9  |-  ( ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  E. n  e.  ZZ  x  =  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )
135134expr 600 . . . . . . . 8  |-  ( ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  /\  f  e.  ( G  GrpHom  (flds  ZZ ) ) )  -> 
( ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1  ->  E. n  e.  ZZ  x  =  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) ) )
13667, 135syld 43 . . . . . . 7  |-  ( ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  /\  f  e.  ( G  GrpHom  (flds  ZZ ) ) )  -> 
( ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. }  ->  E. n  e.  ZZ  x  =  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) ) )
137136rexlimdva 2832 . . . . . 6  |-  ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  -> 
( E. f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I
) )  =  { <. U. I ,  1
>. }  ->  E. n  e.  ZZ  x  =  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) ) )
13859, 137mpd 15 . . . . 5  |-  ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  ->  E. n  e.  ZZ  x  =  ( n
(.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )
13916, 17, 21, 34, 138iscygd 15499 . . . 4  |-  ( I 
~~  1o  ->  G  e. CycGrp
)
14015, 139jaoi 370 . . 3  |-  ( ( I  ~<  1o  \/  I  ~~  1o )  ->  G  e. CycGrp )
1411, 140sylbi 189 . 2  |-  ( I  ~<_  1o  ->  G  e. CycGrp )
142 cygabl 15502 . . 3  |-  ( G  e. CycGrp  ->  G  e.  Abel )
1433frgpnabl 15488 . . . . 5  |-  ( 1o 
~<  I  ->  -.  G  e.  Abel )
144143con2i 115 . . . 4  |-  ( G  e.  Abel  ->  -.  1o  ~<  I )
145 ablgrp 15419 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
146 eqid 2438 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
14716, 146grpidcl 14835 . . . . . 6  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( Base `  G
) )
1483, 16elbasfv 13514 . . . . . 6  |-  ( ( 0g `  G )  e.  ( Base `  G
)  ->  I  e.  _V )
149145, 147, 1483syl 19 . . . . 5  |-  ( G  e.  Abel  ->  I  e. 
_V )
150 1onn 6884 . . . . . 6  |-  1o  e.  om
151 nnfi 7301 . . . . . 6  |-  ( 1o  e.  om  ->  1o  e.  Fin )
152150, 151ax-mp 8 . . . . 5  |-  1o  e.  Fin
153 fidomtri2 7883 . . . . 5  |-  ( ( I  e.  _V  /\  1o  e.  Fin )  -> 
( I  ~<_  1o  <->  -.  1o  ~<  I ) )
154149, 152, 153sylancl 645 . . . 4  |-  ( G  e.  Abel  ->  ( I  ~<_  1o  <->  -.  1o  ~<  I ) )
155144, 154mpbird 225 . . 3  |-  ( G  e.  Abel  ->  I  ~<_  1o )
156142, 155syl 16 . 2  |-  ( G  e. CycGrp  ->  I  ~<_  1o )
157141, 156impbii 182 1  |-  ( I  ~<_  1o  <->  G  e. CycGrp )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708   E!wreu 2709   E*wrmo 2710   _Vcvv 2958    C_ wss 3322   (/)c0 3630   {csn 3816   <.cop 3819   U.cuni 4017   class class class wbr 4214    e. cmpt 4268    _I cid 4495   omcom 4847    |` cres 4882    o. ccom 4884   -->wf 5452   -1-1-onto->wf1o 5455   ` cfv 5456  (class class class)co 6083   1oc1o 6719    ~~ cen 7108    ~<_ cdom 7109    ~< csdm 7110   Fincfn 7111   1c1 8993   ZZcz 10284   Basecbs 13471   ↾s cress 13472   0gc0g 13725   Grpcgrp 14687  .gcmg 14691  SubGrpcsubg 14940    GrpHom cghm 15005   ~FG cefg 15340  freeGrpcfrgp 15341  varFGrpcvrgp 15342   Abelcabel 15415  CycGrpccyg 15489  SubRingcsubrg 15866  ℂfldccnfld 16705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-addf 9071  ax-mulf 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-seq 11326  df-hash 11621  df-word 11725  df-concat 11726  df-s1 11727  df-substr 11728  df-splice 11729  df-reverse 11730  df-s2 11814  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-starv 13546  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-unif 13554  df-0g 13729  df-gsum 13730  df-imas 13736  df-divs 13737  df-mnd 14692  df-mhm 14740  df-submnd 14741  df-frmd 14796  df-vrmd 14797  df-grp 14814  df-minusg 14815  df-mulg 14817  df-subg 14943  df-ghm 15006  df-efg 15343  df-frgp 15344  df-vrgp 15345  df-cmn 15416  df-abl 15417  df-cyg 15490  df-mgp 15651  df-rng 15665  df-cring 15666  df-ur 15667  df-subrg 15868  df-cnfld 16706
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