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Theorem frgpcyg 16529
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 6893 . . 3  |-  ( I  ~<_  1o  <->  ( I  ~<  1o  \/  I  ~~  1o ) )
2 sdom1 7064 . . . . 5  |-  ( I 
~<  1o  <->  I  =  (/) )
3 frgpcyg.g . . . . . . 7  |-  G  =  (freeGrp `  I )
4 fveq2 5527 . . . . . . 7  |-  ( I  =  (/)  ->  (freeGrp `  I
)  =  (freeGrp `  (/) ) )
53, 4syl5eq 2329 . . . . . 6  |-  ( I  =  (/)  ->  G  =  (freeGrp `  (/) ) )
6 0ex 4152 . . . . . . . 8  |-  (/)  e.  _V
7 eqid 2285 . . . . . . . . 9  |-  (freeGrp `  (/) )  =  (freeGrp `  (/) )
87frgpgrp 15073 . . . . . . . 8  |-  ( (/)  e.  _V  ->  (freeGrp `  (/) )  e. 
Grp )
96, 8ax-mp 8 . . . . . . 7  |-  (freeGrp `  (/) )  e. 
Grp
10 eqid 2285 . . . . . . . 8  |-  ( Base `  (freeGrp `  (/) ) )  =  ( Base `  (freeGrp `  (/) ) )
117, 100frgp 15090 . . . . . . 7  |-  ( Base `  (freeGrp `  (/) ) ) 
~~  1o
12100cyg 15181 . . . . . . 7  |-  ( ( (freeGrp `  (/) )  e. 
Grp  /\  ( Base `  (freeGrp `  (/) ) ) 
~~  1o )  -> 
(freeGrp `  (/) )  e. CycGrp )
139, 11, 12mp2an 653 . . . . . 6  |-  (freeGrp `  (/) )  e. CycGrp
145, 13syl6eqel 2373 . . . . 5  |-  ( I  =  (/)  ->  G  e. CycGrp
)
152, 14sylbi 187 . . . 4  |-  ( I 
~<  1o  ->  G  e. CycGrp )
16 eqid 2285 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
17 eqid 2285 . . . . 5  |-  (.g `  G
)  =  (.g `  G
)
18 relen 6870 . . . . . . 7  |-  Rel  ~~
1918brrelexi 4731 . . . . . 6  |-  ( I 
~~  1o  ->  I  e. 
_V )
203frgpgrp 15073 . . . . . 6  |-  ( I  e.  _V  ->  G  e.  Grp )
2119, 20syl 15 . . . . 5  |-  ( I 
~~  1o  ->  G  e. 
Grp )
22 eqid 2285 . . . . . . . 8  |-  ( ~FG  `  I
)  =  ( ~FG  `  I
)
23 eqid 2285 . . . . . . . 8  |-  (varFGrp `  I
)  =  (varFGrp `  I
)
2422, 23, 3, 16vrgpf 15079 . . . . . . 7  |-  ( I  e.  _V  ->  (varFGrp `  I
) : I --> ( Base `  G ) )
2519, 24syl 15 . . . . . 6  |-  ( I 
~~  1o  ->  (varFGrp `  I
) : I --> ( Base `  G ) )
26 en1b 6931 . . . . . . . 8  |-  ( I 
~~  1o  <->  I  =  { U. I } )
27 eqimss2 3233 . . . . . . . 8  |-  ( I  =  { U. I }  ->  { U. I }  C_  I )
2826, 27sylbi 187 . . . . . . 7  |-  ( I 
~~  1o  ->  { U. I }  C_  I )
29 uniexg 4519 . . . . . . . . 9  |-  ( I  e.  _V  ->  U. I  e.  _V )
3019, 29syl 15 . . . . . . . 8  |-  ( I 
~~  1o  ->  U. I  e.  _V )
31 snssg 3756 . . . . . . . 8  |-  ( U. I  e.  _V  ->  ( U. I  e.  I  <->  { U. I }  C_  I ) )
3230, 31syl 15 . . . . . . 7  |-  ( I 
~~  1o  ->  ( U. I  e.  I  <->  { U. I }  C_  I ) )
3328, 32mpbird 223 . . . . . 6  |-  ( I 
~~  1o  ->  U. I  e.  I )
34 ffvelrn 5665 . . . . . 6  |-  ( ( (varFGrp `  I ) : I --> ( Base `  G
)  /\  U. I  e.  I )  ->  (
(varFGrp `  I ) `  U. I )  e.  (
Base `  G )
)
3525, 33, 34syl2anc 642 . . . . 5  |-  ( I 
~~  1o  ->  ( (varFGrp `  I ) `  U. I )  e.  (
Base `  G )
)
36 zsubrg 16427 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
37 subrgsubg 15553 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
3836, 37ax-mp 8 . . . . . . . . . 10  |-  ZZ  e.  (SubGrp ` fld )
39 eqid 2285 . . . . . . . . . . 11  |-  (flds  ZZ )  =  (flds  ZZ )
4039subggrp 14626 . . . . . . . . . 10  |-  ( ZZ  e.  (SubGrp ` fld )  ->  (flds  ZZ )  e.  Grp )
4138, 40mp1i 11 . . . . . . . . 9  |-  ( I 
~~  1o  ->  (flds  ZZ )  e.  Grp )
42 1z 10055 . . . . . . . . . . . . 13  |-  1  e.  ZZ
43 f1osng 5516 . . . . . . . . . . . . 13  |-  ( ( U. I  e.  _V  /\  1  e.  ZZ )  ->  { <. U. I ,  1 >. } : { U. I } -1-1-onto-> { 1 } )
4430, 42, 43sylancl 643 . . . . . . . . . . . 12  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : { U. I }
-1-1-onto-> { 1 } )
45 f1of 5474 . . . . . . . . . . . 12  |-  ( {
<. U. I ,  1
>. } : { U. I } -1-1-onto-> { 1 }  ->  {
<. U. I ,  1
>. } : { U. I } --> { 1 } )
4644, 45syl 15 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : { U. I }
--> { 1 } )
4726biimpi 186 . . . . . . . . . . . 12  |-  ( I 
~~  1o  ->  I  =  { U. I }
)
4847feq2d 5382 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  ( {
<. U. I ,  1
>. } : I --> { 1 }  <->  { <. U. I ,  1
>. } : { U. I } --> { 1 } ) )
4946, 48mpbird 223 . . . . . . . . . 10  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : I --> { 1 } )
50 snssi 3761 . . . . . . . . . . 11  |-  ( 1  e.  ZZ  ->  { 1 }  C_  ZZ )
5142, 50ax-mp 8 . . . . . . . . . 10  |-  { 1 }  C_  ZZ
52 fss 5399 . . . . . . . . . 10  |-  ( ( { <. U. I ,  1
>. } : I --> { 1 }  /\  { 1 }  C_  ZZ )  ->  { <. U. I ,  1
>. } : I --> ZZ )
5349, 51, 52sylancl 643 . . . . . . . . 9  |-  ( I 
~~  1o  ->  { <. U. I ,  1 >. } : I --> ZZ )
5439subrgbas 15556 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
5536, 54ax-mp 8 . . . . . . . . . 10  |-  ZZ  =  ( Base `  (flds  ZZ ) )
563, 55, 23frgpup3 15089 . . . . . . . . 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
>. } )
5741, 19, 53, 56syl3anc 1182 . . . . . . . 8  |-  ( I 
~~  1o  ->  E! f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. } )
5857adantr 451 . . . . . . 7  |-  ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  ->  E! f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. } )
59 reurex 2756 . . . . . . 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
>. } )
6058, 59syl 15 . . . . . 6  |-  ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  ->  E. f  e.  ( G  GrpHom  (flds  ZZ ) ) ( f  o.  (varFGrp `  I ) )  =  { <. U. I ,  1
>. } )
61 fveq1 5526 . . . . . . . . . 10  |-  ( ( f  o.  (varFGrp `  I
) )  =  { <. U. I ,  1
>. }  ->  ( (
f  o.  (varFGrp `  I
) ) `  U. I )  =  ( { <. U. I ,  1
>. } `  U. I
) )
62 fvco3 5598 . . . . . . . . . . . 12  |-  ( ( (varFGrp `  I ) : I --> ( Base `  G
)  /\  U. I  e.  I )  ->  (
( f  o.  (varFGrp `  I
) ) `  U. I )  =  ( f `  ( (varFGrp `  I ) `  U. I ) ) )
6325, 33, 62syl2anc 642 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  ( ( f  o.  (varFGrp `  I
) ) `  U. I )  =  ( f `  ( (varFGrp `  I ) `  U. I ) ) )
64 fvsng 5716 . . . . . . . . . . . 12  |-  ( ( U. I  e.  _V  /\  1  e.  ZZ )  ->  ( { <. U. I ,  1 >. } `  U. I )  =  1 )
6530, 42, 64sylancl 643 . . . . . . . . . . 11  |-  ( I 
~~  1o  ->  ( {
<. U. I ,  1
>. } `  U. I
)  =  1 )
6663, 65eqeq12d 2299 . . . . . . . . . 10  |-  ( I 
~~  1o  ->  ( ( ( f  o.  (varFGrp `  I
) ) `  U. I )  =  ( { <. U. I ,  1
>. } `  U. I
)  <->  ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1 ) )
6761, 66syl5ib 210 . . . . . . . . 9  |-  ( I 
~~  1o  ->  ( ( f  o.  (varFGrp `  I
) )  =  { <. U. I ,  1
>. }  ->  ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1 ) )
6867ad2antrr 706 . . . . . . . 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 ) )
6916, 55ghmf 14689 . . . . . . . . . . . . 13  |-  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  ->  f :
( Base `  G ) --> ZZ )
7069ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  f :
( Base `  G ) --> ZZ )
71 ffvelrn 5665 . . . . . . . . . . . 12  |-  ( ( f : ( Base `  G ) --> ZZ  /\  x  e.  ( Base `  G ) )  -> 
( f `  x
)  e.  ZZ )
7270, 71sylan 457 . . . . . . . . . . 11  |-  ( ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  /\  x  e.  ( Base `  G
) )  ->  (
f `  x )  e.  ZZ )
7372an32s 779 . . . . . . . . . 10  |-  ( ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( f `  x )  e.  ZZ )
74 mptresid 5006 . . . . . . . . . . . . . 14  |-  ( x  e.  ( Base `  G
)  |->  x )  =  (  _I  |`  ( Base `  G ) )
753, 16, 23frgpup3 15089 . . . . . . . . . . . . . . . . . 18  |-  ( ( G  e.  Grp  /\  I  e.  _V  /\  (varFGrp `  I
) : I --> ( Base `  G ) )  ->  E! g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) )
7621, 19, 25, 75syl3anc 1182 . . . . . . . . . . . . . . . . 17  |-  ( I 
~~  1o  ->  E! g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I ) )
77 reu5 2755 . . . . . . . . . . . . . . . . . 18  |-  ( E! g  e.  ( G 
GrpHom  G ) ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I )  <->  ( E. g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I )  /\  E* g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I ) ) )
7877simprbi 450 . . . . . . . . . . . . . . . . 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 ) )
7976, 78syl 15 . . . . . . . . . . . . . . . 16  |-  ( I 
~~  1o  ->  E* g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) )
8079adantr 451 . . . . . . . . . . . . . . 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 ) )
8121adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  G  e.  Grp )
8216idghm 14700 . . . . . . . . . . . . . . . 16  |-  ( G  e.  Grp  ->  (  _I  |`  ( Base `  G
) )  e.  ( G  GrpHom  G ) )
8381, 82syl 15 . . . . . . . . . . . . . . 15  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  (  _I  |`  ( Base `  G
) )  e.  ( G  GrpHom  G ) )
8425adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  (varFGrp `  I ) : I --> ( Base `  G
) )
85 fcoi2 5418 . . . . . . . . . . . . . . . 16  |-  ( (varFGrp `  I ) : I --> ( Base `  G
)  ->  ( (  _I  |`  ( Base `  G
) )  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) )
8684, 85syl 15 . . . . . . . . . . . . . . 15  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( (  _I  |`  ( Base `  G
) )  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) )
8770feqmptd 5577 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  f  =  ( x  e.  ( Base `  G )  |->  ( f `  x ) ) )
88 eqidd 2286 . . . . . . . . . . . . . . . . 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 ) ) ) )
89 oveq1 5867 . . . . . . . . . . . . . . . . 17  |-  ( n  =  ( f `  x )  ->  (
n (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  =  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) )
9072, 87, 88, 89fmptco 5693 . . . . . . . . . . . . . . . 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 ) ) ) )
9135adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( (varFGrp `  I
) `  U. I )  e.  ( Base `  G
) )
92 eqid 2285 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  ZZ  |->  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )  =  ( n  e.  ZZ  |->  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )
9339, 17, 92, 16mulgghm2 16461 . . . . . . . . . . . . . . . . . 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 ) )
9481, 91, 93syl2anc 642 . . . . . . . . . . . . . . . . 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 ) )
95 simprl 732 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  f  e.  ( G  GrpHom  (flds  ZZ ) ) )
96 ghmco 14704 . . . . . . . . . . . . . . . . 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 ) )
9794, 95, 96syl2anc 642 . . . . . . . . . . . . . . . 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 ) )
9890, 97eqeltrrd 2360 . . . . . . . . . . . . . . 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 ) )
9947adantr 451 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  I  =  { U. I } )
10099eleq2d 2352 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( y  e.  I  <->  y  e.  { U. I } ) )
101 simprr 733 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  ( f `  ( (varFGrp `  I ) `  U. I ) )  =  1 )
102101oveq1d 5875 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
10316, 17mulg1 14576 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( (varFGrp `  I ) `  U. I )  e.  (
Base `  G )  ->  ( 1 (.g `  G
) ( (varFGrp `  I
) `  U. I ) )  =  ( (varFGrp `  I ) `  U. I ) )
10491, 103syl 15 . . . . . . . . . . . . . . . . . . . . 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 ) )
105102, 104eqtrd 2317 . . . . . . . . . . . . . . . . . . . 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 ) )
106 elsni 3666 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  { U. I }  ->  y  =  U. I )
107106fveq2d 5531 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  { U. I }  ->  ( (varFGrp `  I
) `  y )  =  ( (varFGrp `  I
) `  U. I ) )
108107fveq2d 5531 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  { U. I }  ->  ( f `  ( (varFGrp `  I ) `  y
) )  =  ( f `  ( (varFGrp `  I ) `  U. I ) ) )
109108oveq1d 5875 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
110109, 107eqeq12d 2299 . . . . . . . . . . . . . . . . . . . 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 ) ) )
111105, 110syl5ibrcom 213 . . . . . . . . . . . . . . . . . . 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
) ) )
112100, 111sylbid 206 . . . . . . . . . . . . . . . . . 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
) ) )
113112imp 418 . . . . . . . . . . . . . . . . 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
) )
114113mpteq2dva 4108 . . . . . . . . . . . . . . . 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
) ) )
115 ffvelrn 5665 . . . . . . . . . . . . . . . . . 18  |-  ( ( (varFGrp `  I ) : I --> ( Base `  G
)  /\  y  e.  I )  ->  (
(varFGrp `  I ) `  y
)  e.  ( Base `  G ) )
11684, 115sylan 457 . . . . . . . . . . . . . . . . 17  |-  ( ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  /\  y  e.  I )  ->  (
(varFGrp `  I ) `  y
)  e.  ( Base `  G ) )
11784feqmptd 5577 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  (varFGrp `  I )  =  ( y  e.  I  |->  ( (varFGrp `  I ) `  y
) ) )
118 eqidd 2286 . . . . . . . . . . . . . . . . 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 ) ) ) )
119 fveq2 5527 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( (varFGrp `  I
) `  y )  ->  ( f `  x
)  =  ( f `
 ( (varFGrp `  I
) `  y )
) )
120119oveq1d 5875 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( (varFGrp `  I
) `  y )  ->  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) )  =  ( ( f `  ( (varFGrp `  I ) `  y
) ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) )
121116, 117, 118, 120fmptco 5693 . . . . . . . . . . . . . . . 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 ) ) ) )
122114, 121, 1173eqtr4d 2327 . . . . . . . . . . . . . . 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 ) )
123 coeq1 4843 . . . . . . . . . . . . . . . . 17  |-  ( g  =  (  _I  |`  ( Base `  G ) )  ->  ( g  o.  (varFGrp `  I ) )  =  ( (  _I  |`  ( Base `  G ) )  o.  (varFGrp `  I ) ) )
124123eqeq1d 2293 . . . . . . . . . . . . . . . 16  |-  ( g  =  (  _I  |`  ( Base `  G ) )  ->  ( ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I )  <->  ( (  _I  |`  ( Base `  G
) )  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) ) )
125 coeq1 4843 . . . . . . . . . . . . . . . . 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 ) ) )
126125eqeq1d 2293 . . . . . . . . . . . . . . . 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 ) ) )
127124, 126rmoi 3082 . . . . . . . . . . . . . . 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 ) ) ) )
12880, 83, 86, 98, 122, 127syl122anc 1191 . . . . . . . . . . . . . 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 ) ) ) )
12974, 128syl5eq 2329 . . . . . . . . . . . . 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 ) ) ) )
130 mpteqb 5616 . . . . . . . . . . . . . 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 ) ) ) )
131 id 19 . . . . . . . . . . . . . 14  |-  ( x  e.  ( Base `  G
)  ->  x  e.  ( Base `  G )
)
132130, 131mprg 2614 . . . . . . . . . . . . 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 ) ) )
133129, 132sylib 188 . . . . . . . . . . . 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 ) ) )
134133r19.21bi 2643 . . . . . . . . . . 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 ) ) )
135134an32s 779 . . . . . . . . . 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 ) ) )
13689eqeq2d 2296 . . . . . . . . . . 11  |-  ( n  =  ( f `  x )  ->  (
x  =  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  <->  x  =  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) ) )
137136rspcev 2886 . . . . . . . . . 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 ) ) )
13873, 135, 137syl2anc 642 . . . . . . . . 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 ) ) )
139138expr 598 . . . . . . . 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 ) ) ) )
14068, 139syld 40 . . . . . . 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 ) ) ) )
141140rexlimdva 2669 . . . . . 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 ) ) ) )
14260, 141mpd 14 . . . . 5  |-  ( ( I  ~~  1o  /\  x  e.  ( Base `  G ) )  ->  E. n  e.  ZZ  x  =  ( n
(.g `  G ) ( (varFGrp `  I ) `  U. I ) ) )
14316, 17, 21, 35, 142iscygd 15176 . . . 4  |-  ( I 
~~  1o  ->  G  e. CycGrp
)
14415, 143jaoi 368 . . 3  |-  ( ( I  ~<  1o  \/  I  ~~  1o )  ->  G  e. CycGrp )
1451, 144sylbi 187 . 2  |-  ( I  ~<_  1o  ->  G  e. CycGrp )
146 cygabl 15179 . . 3  |-  ( G  e. CycGrp  ->  G  e.  Abel )
1473frgpnabl 15165 . . . . 5  |-  ( 1o 
~<  I  ->  -.  G  e.  Abel )
148147con2i 112 . . . 4  |-  ( G  e.  Abel  ->  -.  1o  ~<  I )
149 ablgrp 15096 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
150 eqid 2285 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
15116, 150grpidcl 14512 . . . . . 6  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( Base `  G
) )
1523, 16elbasfv 13193 . . . . . 6  |-  ( ( 0g `  G )  e.  ( Base `  G
)  ->  I  e.  _V )
153149, 151, 1523syl 18 . . . . 5  |-  ( G  e.  Abel  ->  I  e. 
_V )
154 1onn 6639 . . . . . 6  |-  1o  e.  om
155 nnfi 7055 . . . . . 6  |-  ( 1o  e.  om  ->  1o  e.  Fin )
156154, 155ax-mp 8 . . . . 5  |-  1o  e.  Fin
157 fidomtri2 7629 . . . . 5  |-  ( ( I  e.  _V  /\  1o  e.  Fin )  -> 
( I  ~<_  1o  <->  -.  1o  ~<  I ) )
158153, 156, 157sylancl 643 . . . 4  |-  ( G  e.  Abel  ->  ( I  ~<_  1o  <->  -.  1o  ~<  I ) )
159148, 158mpbird 223 . . 3  |-  ( G  e.  Abel  ->  I  ~<_  1o )
160146, 159syl 15 . 2  |-  ( G  e. CycGrp  ->  I  ~<_  1o )
161145, 160impbii 180 1  |-  ( I  ~<_  1o  <->  G  e. CycGrp )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   E!wreu 2547   E*wrmo 2548   _Vcvv 2790    C_ wss 3154   (/)c0 3457   {csn 3642   <.cop 3645   U.cuni 3829   class class class wbr 4025    e. cmpt 4079    _I cid 4306   omcom 4658    |` cres 4693    o. ccom 4695   -->wf 5253   -1-1-onto->wf1o 5256   ` cfv 5257  (class class class)co 5860   1oc1o 6474    ~~ cen 6862    ~<_ cdom 6863    ~< csdm 6864   Fincfn 6865   1c1 8740   ZZcz 10026   Basecbs 13150   ↾s cress 13151   0gc0g 13402   Grpcgrp 14364  .gcmg 14368  SubGrpcsubg 14617    GrpHom cghm 14682   ~FG cefg 15017  freeGrpcfrgp 15018  varFGrpcvrgp 15019   Abelcabel 15092  CycGrpccyg 15166  SubRingcsubrg 15543  ℂfldccnfld 16379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-addf 8818  ax-mulf 8819
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-ot 3652  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-ec 6664  df-qs 6668  df-map 6776  df-pm 6777  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-card 7574  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-rp 10357  df-fz 10785  df-fzo 10873  df-seq 11049  df-hash 11340  df-word 11411  df-concat 11412  df-s1 11413  df-substr 11414  df-splice 11415  df-reverse 11416  df-s2 11500  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-0g 13406  df-gsum 13407  df-imas 13413  df-divs 13414  df-mnd 14369  df-mhm 14417  df-submnd 14418  df-frmd 14473  df-vrmd 14474  df-grp 14491  df-minusg 14492  df-mulg 14494  df-subg 14620  df-ghm 14683  df-efg 15020  df-frgp 15021  df-vrgp 15022  df-cmn 15093  df-abl 15094  df-cyg 15167  df-mgp 15328  df-rng 15342  df-cring 15343  df-ur 15344  df-subrg 15545  df-cnfld 16380
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