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Theorem frgpcyg 16770
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 7066 . . 3  |-  ( I  ~<_  1o  <->  ( I  ~<  1o  \/  I  ~~  1o ) )
2 sdom1 7237 . . . . 5  |-  ( I 
~<  1o  <->  I  =  (/) )
3 frgpcyg.g . . . . . . 7  |-  G  =  (freeGrp `  I )
4 fveq2 5661 . . . . . . 7  |-  ( I  =  (/)  ->  (freeGrp `  I
)  =  (freeGrp `  (/) ) )
53, 4syl5eq 2424 . . . . . 6  |-  ( I  =  (/)  ->  G  =  (freeGrp `  (/) ) )
6 0ex 4273 . . . . . . . 8  |-  (/)  e.  _V
7 eqid 2380 . . . . . . . . 9  |-  (freeGrp `  (/) )  =  (freeGrp `  (/) )
87frgpgrp 15314 . . . . . . . 8  |-  ( (/)  e.  _V  ->  (freeGrp `  (/) )  e. 
Grp )
96, 8ax-mp 8 . . . . . . 7  |-  (freeGrp `  (/) )  e. 
Grp
10 eqid 2380 . . . . . . . 8  |-  ( Base `  (freeGrp `  (/) ) )  =  ( Base `  (freeGrp `  (/) ) )
117, 100frgp 15331 . . . . . . 7  |-  ( Base `  (freeGrp `  (/) ) ) 
~~  1o
12100cyg 15422 . . . . . . 7  |-  ( ( (freeGrp `  (/) )  e. 
Grp  /\  ( Base `  (freeGrp `  (/) ) ) 
~~  1o )  -> 
(freeGrp `  (/) )  e. CycGrp )
139, 11, 12mp2an 654 . . . . . 6  |-  (freeGrp `  (/) )  e. CycGrp
145, 13syl6eqel 2468 . . . . 5  |-  ( I  =  (/)  ->  G  e. CycGrp
)
152, 14sylbi 188 . . . 4  |-  ( I 
~<  1o  ->  G  e. CycGrp )
16 eqid 2380 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
17 eqid 2380 . . . . 5  |-  (.g `  G
)  =  (.g `  G
)
18 relen 7043 . . . . . . 7  |-  Rel  ~~
1918brrelexi 4851 . . . . . 6  |-  ( I 
~~  1o  ->  I  e. 
_V )
203frgpgrp 15314 . . . . . 6  |-  ( I  e.  _V  ->  G  e.  Grp )
2119, 20syl 16 . . . . 5  |-  ( I 
~~  1o  ->  G  e. 
Grp )
22 eqid 2380 . . . . . . . 8  |-  ( ~FG  `  I
)  =  ( ~FG  `  I
)
23 eqid 2380 . . . . . . . 8  |-  (varFGrp `  I
)  =  (varFGrp `  I
)
2422, 23, 3, 16vrgpf 15320 . . . . . . 7  |-  ( I  e.  _V  ->  (varFGrp `  I
) : I --> ( Base `  G ) )
2519, 24syl 16 . . . . . 6  |-  ( I 
~~  1o  ->  (varFGrp `  I
) : I --> ( Base `  G ) )
26 en1b 7104 . . . . . . . 8  |-  ( I 
~~  1o  <->  I  =  { U. I } )
27 eqimss2 3337 . . . . . . . 8  |-  ( I  =  { U. I }  ->  { U. I }  C_  I )
2826, 27sylbi 188 . . . . . . 7  |-  ( I 
~~  1o  ->  { U. I }  C_  I )
29 uniexg 4639 . . . . . . . . 9  |-  ( I  e.  _V  ->  U. I  e.  _V )
3019, 29syl 16 . . . . . . . 8  |-  ( I 
~~  1o  ->  U. I  e.  _V )
31 snssg 3868 . . . . . . . 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 224 . . . . . 6  |-  ( I 
~~  1o  ->  U. I  e.  I )
3425, 33ffvelrnd 5803 . . . . 5  |-  ( I 
~~  1o  ->  ( (varFGrp `  I ) `  U. I )  e.  (
Base `  G )
)
35 zsubrg 16668 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
36 subrgsubg 15794 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
3735, 36ax-mp 8 . . . . . . . . . 10  |-  ZZ  e.  (SubGrp ` fld )
38 eqid 2380 . . . . . . . . . . 11  |-  (flds  ZZ )  =  (flds  ZZ )
3938subggrp 14867 . . . . . . . . . 10  |-  ( ZZ  e.  (SubGrp ` fld )  ->  (flds  ZZ )  e.  Grp )
4037, 39mp1i 12 . . . . . . . . 9  |-  ( I 
~~  1o  ->  (flds  ZZ )  e.  Grp )
41 1z 10236 . . . . . . . . . . . . 13  |-  1  e.  ZZ
42 f1osng 5649 . . . . . . . . . . . . 13  |-  ( ( U. I  e.  _V  /\  1  e.  ZZ )  ->  { <. U. I ,  1 >. } : { U. I } -1-1-onto-> { 1 } )
4330, 41, 42sylancl 644 . . . . . . . . . . . 12  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : { U. I }
-1-1-onto-> { 1 } )
44 f1of 5607 . . . . . . . . . . . 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 187 . . . . . . . . . . . 12  |-  ( I 
~~  1o  ->  I  =  { U. I }
)
4746feq2d 5514 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  ( {
<. U. I ,  1
>. } : I --> { 1 }  <->  { <. U. I ,  1
>. } : { U. I } --> { 1 } ) )
4845, 47mpbird 224 . . . . . . . . . 10  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : I --> { 1 } )
49 snssi 3878 . . . . . . . . . . 11  |-  ( 1  e.  ZZ  ->  { 1 }  C_  ZZ )
5041, 49ax-mp 8 . . . . . . . . . 10  |-  { 1 }  C_  ZZ
51 fss 5532 . . . . . . . . . 10  |-  ( ( { <. U. I ,  1
>. } : I --> { 1 }  /\  { 1 }  C_  ZZ )  ->  { <. U. I ,  1
>. } : I --> ZZ )
5248, 50, 51sylancl 644 . . . . . . . . 9  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : I --> ZZ )
5338subrgbas 15797 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
5435, 53ax-mp 8 . . . . . . . . . 10  |-  ZZ  =  ( Base `  (flds  ZZ ) )
553, 54, 23frgpup3 15330 . . . . . . . . 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 1184 . . . . . . . 8  |-  ( I 
~~  1o  ->  E! f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. } )
5756adantr 452 . . . . . . 7  |-  ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  ->  E! f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. } )
58 reurex 2858 . . . . . . 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 5660 . . . . . . . . . 10  |-  ( ( f  o.  (varFGrp `  I
) )  =  { <. U. I ,  1
>. }  ->  ( (
f  o.  (varFGrp `  I
) ) `  U. I )  =  ( { <. U. I ,  1
>. } `  U. I
) )
61 fvco3 5732 . . . . . . . . . . . 12  |-  ( ( (varFGrp `  I ) : I --> ( Base `  G
)  /\  U. I  e.  I )  ->  (
( f  o.  (varFGrp `  I
) ) `  U. I )  =  ( f `  ( (varFGrp `  I ) `  U. I ) ) )
6225, 33, 61syl2anc 643 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  ( ( f  o.  (varFGrp `  I
) ) `  U. I )  =  ( f `  ( (varFGrp `  I ) `  U. I ) ) )
63 fvsng 5859 . . . . . . . . . . . 12  |-  ( ( U. I  e.  _V  /\  1  e.  ZZ )  ->  ( { <. U. I ,  1 >. } `  U. I )  =  1 )
6430, 41, 63sylancl 644 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  ( {
<. U. I ,  1
>. } `  U. I
)  =  1 )
6562, 64eqeq12d 2394 . . . . . . . . . 10  |-  ( I 
~~  1o  ->  ( ( ( f  o.  (varFGrp `  I
) ) `  U. I )  =  ( { <. U. I ,  1
>. } `  U. I
)  <->  ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1 ) )
6660, 65syl5ib 211 . . . . . . . . 9  |-  ( I 
~~  1o  ->  ( ( f  o.  (varFGrp `  I
) )  =  { <. U. I ,  1
>. }  ->  ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1 ) )
6766ad2antrr 707 . . . . . . . 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 14930 . . . . . . . . . . . . 13  |-  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  ->  f :
( Base `  G ) --> ZZ )
6968ad2antrl 709 . . . . . . . . . . . 12  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  f :
( Base `  G ) --> ZZ )
7069ffvelrnda 5802 . . . . . . . . . . 11  |-  ( ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  /\  x  e.  ( Base `  G
) )  ->  (
f `  x )  e.  ZZ )
7170an32s 780 . . . . . . . . . 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 5128 . . . . . . . . . . . . . 14  |-  ( x  e.  ( Base `  G
)  |->  x )  =  (  _I  |`  ( Base `  G ) )
733, 16, 23frgpup3 15330 . . . . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . . . . 17  |-  ( I 
~~  1o  ->  E! g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I ) )
75 reurmo 2859 . . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . 16  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  G  e.  Grp )
7916idghm 14941 . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . 16  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  (varFGrp `  I ) : I --> ( Base `  G
) )
82 fcoi2 5551 . . . . . . . . . . . . . . . 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 5711 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  f  =  ( x  e.  ( Base `  G )  |->  ( f `  x ) ) )
85 eqidd 2381 . . . . . . . . . . . . . . . . 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 6020 . . . . . . . . . . . . . . . . 17  |-  ( n  =  ( f `  x )  ->  (
n (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  =  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) )
8770, 84, 85, 86fmptco 5833 . . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( (varFGrp `  I
) `  U. I )  e.  ( Base `  G
) )
89 eqid 2380 . . . . . . . . . . . . . . . . . . 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 16702 . . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  f  e.  ( G  GrpHom  (flds  ZZ ) ) )
93 ghmco 14945 . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . 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 2455 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  I  =  { U. I } )
9796eleq2d 2447 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( y  e.  I  <->  y  e.  { U. I } ) )
98 simprr 734 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1 )
9998oveq1d 6028 . . . . . . . . . . . . . . . . . . . . 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 14817 . . . . . . . . . . . . . . . . . . . . . 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 2412 . . . . . . . . . . . . . . . . . . . 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 3774 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  { U. I }  ->  y  =  U. I )
104103fveq2d 5665 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  { U. I }  ->  ( (varFGrp `  I
) `  y )  =  ( (varFGrp `  I
) `  U. I ) )
105104fveq2d 5665 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  { U. I }  ->  ( f `  ( (varFGrp `  I ) `  y
) )  =  ( f `  ( (varFGrp `  I ) `  U. I ) ) )
106105oveq1d 6028 . . . . . . . . . . . . . . . . . . . . 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 2394 . . . . . . . . . . . . . . . . . . . 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 214 . . . . . . . . . . . . . . . . . . 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 207 . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . 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 4229 . . . . . . . . . . . . . . . 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 5802 . . . . . . . . . . . . . . . . 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 5711 . . . . . . . . . . . . . . . . 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 2381 . . . . . . . . . . . . . . . . 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 5661 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( (varFGrp `  I
) `  y )  ->  ( f `  x
)  =  ( f `
 ( (varFGrp `  I
) `  y )
) )
116115oveq1d 6028 . . . . . . . . . . . . . . . . 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 5833 . . . . . . . . . . . . . . . 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 2422 . . . . . . . . . . . . . . 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 4963 . . . . . . . . . . . . . . . . 17  |-  ( g  =  (  _I  |`  ( Base `  G ) )  ->  ( g  o.  (varFGrp `  I ) )  =  ( (  _I  |`  ( Base `  G ) )  o.  (varFGrp `  I ) ) )
120119eqeq1d 2388 . . . . . . . . . . . . . . . 16  |-  ( g  =  (  _I  |`  ( Base `  G ) )  ->  ( ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I )  <->  ( (  _I  |`  ( Base `  G
) )  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) ) )
121 coeq1 4963 . . . . . . . . . . . . . . . . 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 2388 . . . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . 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 2424 . . . . . . . . . . . . 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 5751 . . . . . . . . . . . . . 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 20 . . . . . . . . . . . . . 14  |-  ( x  e.  ( Base `  G
)  ->  x  e.  ( Base `  G )
)
128126, 127mprg 2711 . . . . . . . . . . . . 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 189 . . . . . . . . . . . 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 2740 . . . . . . . . . . 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 780 . . . . . . . . . 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 2391 . . . . . . . . . . 11  |-  ( n  =  ( f `  x )  ->  (
x  =  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  <->  x  =  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) ) )
133132rspcev 2988 . . . . . . . . . 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 643 . . . . . . . . 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 599 . . . . . . . 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 42 . . . . . . 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 2766 . . . . . 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 15417 . . . 4  |-  ( I 
~~  1o  ->  G  e. CycGrp
)
14015, 139jaoi 369 . . 3  |-  ( ( I  ~<  1o  \/  I  ~~  1o )  ->  G  e. CycGrp )
1411, 140sylbi 188 . 2  |-  ( I  ~<_  1o  ->  G  e. CycGrp )
142 cygabl 15420 . . 3  |-  ( G  e. CycGrp  ->  G  e.  Abel )
1433frgpnabl 15406 . . . . 5  |-  ( 1o 
~<  I  ->  -.  G  e.  Abel )
144143con2i 114 . . . 4  |-  ( G  e.  Abel  ->  -.  1o  ~<  I )
145 ablgrp 15337 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
146 eqid 2380 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
14716, 146grpidcl 14753 . . . . . 6  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( Base `  G
) )
1483, 16elbasfv 13432 . . . . . 6  |-  ( ( 0g `  G )  e.  ( Base `  G
)  ->  I  e.  _V )
149145, 147, 1483syl 19 . . . . 5  |-  ( G  e.  Abel  ->  I  e. 
_V )
150 1onn 6811 . . . . . 6  |-  1o  e.  om
151 nnfi 7228 . . . . . 6  |-  ( 1o  e.  om  ->  1o  e.  Fin )
152150, 151ax-mp 8 . . . . 5  |-  1o  e.  Fin
153 fidomtri2 7807 . . . . 5  |-  ( ( I  e.  _V  /\  1o  e.  Fin )  -> 
( I  ~<_  1o  <->  -.  1o  ~<  I ) )
154149, 152, 153sylancl 644 . . . 4  |-  ( G  e.  Abel  ->  ( I  ~<_  1o  <->  -.  1o  ~<  I ) )
155144, 154mpbird 224 . . 3  |-  ( G  e.  Abel  ->  I  ~<_  1o )
156142, 155syl 16 . 2  |-  ( G  e. CycGrp  ->  I  ~<_  1o )
157141, 156impbii 181 1  |-  ( I  ~<_  1o  <->  G  e. CycGrp )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   E.wrex 2643   E!wreu 2644   E*wrmo 2645   _Vcvv 2892    C_ wss 3256   (/)c0 3564   {csn 3750   <.cop 3753   U.cuni 3950   class class class wbr 4146    e. cmpt 4200    _I cid 4427   omcom 4778    |` cres 4813    o. ccom 4815   -->wf 5383   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013   1oc1o 6646    ~~ cen 7035    ~<_ cdom 7036    ~< csdm 7037   Fincfn 7038   1c1 8917   ZZcz 10207   Basecbs 13389   ↾s cress 13390   0gc0g 13643   Grpcgrp 14605  .gcmg 14609  SubGrpcsubg 14858    GrpHom cghm 14923   ~FG cefg 15258  freeGrpcfrgp 15259  varFGrpcvrgp 15260   Abelcabel 15333  CycGrpccyg 15407  SubRingcsubrg 15784  ℂfldccnfld 16619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-addf 8995  ax-mulf 8996
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-ot 3760  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-ec 6836  df-qs 6840  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-uz 10414  df-rp 10538  df-fz 10969  df-fzo 11059  df-seq 11244  df-hash 11539  df-word 11643  df-concat 11644  df-s1 11645  df-substr 11646  df-splice 11647  df-reverse 11648  df-s2 11732  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-starv 13464  df-sca 13465  df-vsca 13466  df-tset 13468  df-ple 13469  df-ds 13471  df-unif 13472  df-0g 13647  df-gsum 13648  df-imas 13654  df-divs 13655  df-mnd 14610  df-mhm 14658  df-submnd 14659  df-frmd 14714  df-vrmd 14715  df-grp 14732  df-minusg 14733  df-mulg 14735  df-subg 14861  df-ghm 14924  df-efg 15261  df-frgp 15262  df-vrgp 15263  df-cmn 15334  df-abl 15335  df-cyg 15408  df-mgp 15569  df-rng 15583  df-cring 15584  df-ur 15585  df-subrg 15786  df-cnfld 16620
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