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Theorem frgpcyg 16744
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 7034 . . 3  |-  ( I  ~<_  1o  <->  ( I  ~<  1o  \/  I  ~~  1o ) )
2 sdom1 7205 . . . . 5  |-  ( I 
~<  1o  <->  I  =  (/) )
3 frgpcyg.g . . . . . . 7  |-  G  =  (freeGrp `  I )
4 fveq2 5632 . . . . . . 7  |-  ( I  =  (/)  ->  (freeGrp `  I
)  =  (freeGrp `  (/) ) )
53, 4syl5eq 2410 . . . . . 6  |-  ( I  =  (/)  ->  G  =  (freeGrp `  (/) ) )
6 0ex 4252 . . . . . . . 8  |-  (/)  e.  _V
7 eqid 2366 . . . . . . . . 9  |-  (freeGrp `  (/) )  =  (freeGrp `  (/) )
87frgpgrp 15281 . . . . . . . 8  |-  ( (/)  e.  _V  ->  (freeGrp `  (/) )  e. 
Grp )
96, 8ax-mp 8 . . . . . . 7  |-  (freeGrp `  (/) )  e. 
Grp
10 eqid 2366 . . . . . . . 8  |-  ( Base `  (freeGrp `  (/) ) )  =  ( Base `  (freeGrp `  (/) ) )
117, 100frgp 15298 . . . . . . 7  |-  ( Base `  (freeGrp `  (/) ) ) 
~~  1o
12100cyg 15389 . . . . . . 7  |-  ( ( (freeGrp `  (/) )  e. 
Grp  /\  ( Base `  (freeGrp `  (/) ) ) 
~~  1o )  -> 
(freeGrp `  (/) )  e. CycGrp )
139, 11, 12mp2an 653 . . . . . 6  |-  (freeGrp `  (/) )  e. CycGrp
145, 13syl6eqel 2454 . . . . 5  |-  ( I  =  (/)  ->  G  e. CycGrp
)
152, 14sylbi 187 . . . 4  |-  ( I 
~<  1o  ->  G  e. CycGrp )
16 eqid 2366 . . . . 5  |-  ( Base `  G )  =  (
Base `  G )
17 eqid 2366 . . . . 5  |-  (.g `  G
)  =  (.g `  G
)
18 relen 7011 . . . . . . 7  |-  Rel  ~~
1918brrelexi 4832 . . . . . 6  |-  ( I 
~~  1o  ->  I  e. 
_V )
203frgpgrp 15281 . . . . . 6  |-  ( I  e.  _V  ->  G  e.  Grp )
2119, 20syl 15 . . . . 5  |-  ( I 
~~  1o  ->  G  e. 
Grp )
22 eqid 2366 . . . . . . . 8  |-  ( ~FG  `  I
)  =  ( ~FG  `  I
)
23 eqid 2366 . . . . . . . 8  |-  (varFGrp `  I
)  =  (varFGrp `  I
)
2422, 23, 3, 16vrgpf 15287 . . . . . . 7  |-  ( I  e.  _V  ->  (varFGrp `  I
) : I --> ( Base `  G ) )
2519, 24syl 15 . . . . . 6  |-  ( I 
~~  1o  ->  (varFGrp `  I
) : I --> ( Base `  G ) )
26 en1b 7072 . . . . . . . 8  |-  ( I 
~~  1o  <->  I  =  { U. I } )
27 eqimss2 3317 . . . . . . . 8  |-  ( I  =  { U. I }  ->  { U. I }  C_  I )
2826, 27sylbi 187 . . . . . . 7  |-  ( I 
~~  1o  ->  { U. I }  C_  I )
29 uniexg 4620 . . . . . . . . 9  |-  ( I  e.  _V  ->  U. I  e.  _V )
3019, 29syl 15 . . . . . . . 8  |-  ( I 
~~  1o  ->  U. I  e.  _V )
31 snssg 3847 . . . . . . . 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 5770 . . . . . 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 16642 . . . . . . . . . . 11  |-  ZZ  e.  (SubRing ` fld )
37 subrgsubg 15761 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
3836, 37ax-mp 8 . . . . . . . . . 10  |-  ZZ  e.  (SubGrp ` fld )
39 eqid 2366 . . . . . . . . . . 11  |-  (flds  ZZ )  =  (flds  ZZ )
4039subggrp 14834 . . . . . . . . . 10  |-  ( ZZ  e.  (SubGrp ` fld )  ->  (flds  ZZ )  e.  Grp )
4138, 40mp1i 11 . . . . . . . . 9  |-  ( I 
~~  1o  ->  (flds  ZZ )  e.  Grp )
42 1z 10204 . . . . . . . . . . . . 13  |-  1  e.  ZZ
43 f1osng 5620 . . . . . . . . . . . . 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 5578 . . . . . . . . . . . 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 5485 . . . . . . . . . . 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 3857 . . . . . . . . . . 11  |-  ( 1  e.  ZZ  ->  { 1 }  C_  ZZ )
5142, 50ax-mp 8 . . . . . . . . . 10  |-  { 1 }  C_  ZZ
52 fss 5503 . . . . . . . . . 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 15764 . . . . . . . . . . 11  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
5536, 54ax-mp 8 . . . . . . . . . 10  |-  ZZ  =  ( Base `  (flds  ZZ ) )
563, 55, 23frgpup3 15297 . . . . . . . . 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 1183 . . . . . . . 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 2839 . . . . . . 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 5631 . . . . . . . . . 10  |-  ( ( f  o.  (varFGrp `  I
) )  =  { <. U. I ,  1
>. }  ->  ( (
f  o.  (varFGrp `  I
) ) `  U. I )  =  ( { <. U. I ,  1
>. } `  U. I
) )
62 fvco3 5703 . . . . . . . . . . . 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 5827 . . . . . . . . . . . 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 2380 . . . . . . . . . 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 14897 . . . . . . . . . . . . 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 5770 . . . . . . . . . . . 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 5107 . . . . . . . . . . . . . 14  |-  ( x  e.  ( Base `  G
)  |->  x )  =  (  _I  |`  ( Base `  G ) )
753, 16, 23frgpup3 15297 . . . . . . . . . . . . . . . . . 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 1183 . . . . . . . . . . . . . . . . 17  |-  ( I 
~~  1o  ->  E! g  e.  ( G  GrpHom  G ) ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I ) )
77 reu5 2838 . . . . . . . . . . . . . . . . . 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 14908 . . . . . . . . . . . . . . . 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 5522 . . . . . . . . . . . . . . . 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 5682 . . . . . . . . . . . . . . . . 17  |-  ( ( I  ~~  1o  /\  ( f  e.  ( G  GrpHom  (flds  ZZ ) )  /\  (
f `  ( (varFGrp `  I
) `  U. I ) )  =  1 ) )  ->  f  =  ( x  e.  ( Base `  G )  |->  ( f `  x ) ) )
88 eqidd 2367 . . . . . . . . . . . . . . . . 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 5988 . . . . . . . . . . . . . . . . 17  |-  ( n  =  ( f `  x )  ->  (
n (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  =  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) )
9072, 87, 88, 89fmptco 5802 . . . . . . . . . . . . . . . 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 2366 . . . . . . . . . . . . . . . . . . 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 16676 . . . . . . . . . . . . . . . . . 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 14912 . . . . . . . . . . . . . . . . 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 2441 . . . . . . . . . . . . . . 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 2433 . . . . . . . . . . . . . . . . . . 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 5996 . . . . . . . . . . . . . . . . . . . . 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 14784 . . . . . . . . . . . . . . . . . . . . . 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 2398 . . . . . . . . . . . . . . . . . . . 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 3753 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  { U. I }  ->  y  =  U. I )
107106fveq2d 5636 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  { U. I }  ->  ( (varFGrp `  I
) `  y )  =  ( (varFGrp `  I
) `  U. I ) )
108107fveq2d 5636 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( y  e.  { U. I }  ->  ( f `  ( (varFGrp `  I ) `  y
) )  =  ( f `  ( (varFGrp `  I ) `  U. I ) ) )
109108oveq1d 5996 . . . . . . . . . . . . . . . . . . . . 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 2380 . . . . . . . . . . . . . . . . . . . 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 4208 . . . . . . . . . . . . . . . 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 5770 . . . . . . . . . . . . . . . . . 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 5682 . . . . . . . . . . . . . . . . 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 2367 . . . . . . . . . . . . . . . . 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 5632 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( (varFGrp `  I
) `  y )  ->  ( f `  x
)  =  ( f `
 ( (varFGrp `  I
) `  y )
) )
120119oveq1d 5996 . . . . . . . . . . . . . . . . 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 5802 . . . . . . . . . . . . . . . 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 2408 . . . . . . . . . . . . . . 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 4944 . . . . . . . . . . . . . . . . 17  |-  ( g  =  (  _I  |`  ( Base `  G ) )  ->  ( g  o.  (varFGrp `  I ) )  =  ( (  _I  |`  ( Base `  G ) )  o.  (varFGrp `  I ) ) )
124123eqeq1d 2374 . . . . . . . . . . . . . . . 16  |-  ( g  =  (  _I  |`  ( Base `  G ) )  ->  ( ( g  o.  (varFGrp `  I ) )  =  (varFGrp `  I )  <->  ( (  _I  |`  ( Base `  G
) )  o.  (varFGrp `  I
) )  =  (varFGrp `  I ) ) )
125 coeq1 4944 . . . . . . . . . . . . . . . . 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 2374 . . . . . . . . . . . . . . . 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 3166 . . . . . . . . . . . . . . 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 1192 . . . . . . . . . . . . . 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 2410 . . . . . . . . . . . . 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 5721 . . . . . . . . . . . . . 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 2697 . . . . . . . . . . . . 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 2726 . . . . . . . . . . 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 2377 . . . . . . . . . . 11  |-  ( n  =  ( f `  x )  ->  (
x  =  ( n (.g `  G ) ( (varFGrp `  I ) `  U. I ) )  <->  x  =  ( ( f `  x ) (.g `  G
) ( (varFGrp `  I
) `  U. I ) ) ) )
137136rspcev 2969 . . . . . . . . . 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 2752 . . . . . 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 15384 . . . 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 15387 . . 3  |-  ( G  e. CycGrp  ->  G  e.  Abel )
1473frgpnabl 15373 . . . . 5  |-  ( 1o 
~<  I  ->  -.  G  e.  Abel )
148147con2i 112 . . . 4  |-  ( G  e.  Abel  ->  -.  1o  ~<  I )
149 ablgrp 15304 . . . . . 6  |-  ( G  e.  Abel  ->  G  e. 
Grp )
150 eqid 2366 . . . . . . 7  |-  ( 0g
`  G )  =  ( 0g `  G
)
15116, 150grpidcl 14720 . . . . . 6  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  ( Base `  G
) )
1523, 16elbasfv 13399 . . . . . 6  |-  ( ( 0g `  G )  e.  ( Base `  G
)  ->  I  e.  _V )
153149, 151, 1523syl 18 . . . . 5  |-  ( G  e.  Abel  ->  I  e. 
_V )
154 1onn 6779 . . . . . 6  |-  1o  e.  om
155 nnfi 7196 . . . . . 6  |-  ( 1o  e.  om  ->  1o  e.  Fin )
156154, 155ax-mp 8 . . . . 5  |-  1o  e.  Fin
157 fidomtri2 7774 . . . . 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 1647    e. wcel 1715   A.wral 2628   E.wrex 2629   E!wreu 2630   E*wrmo 2631   _Vcvv 2873    C_ wss 3238   (/)c0 3543   {csn 3729   <.cop 3732   U.cuni 3929   class class class wbr 4125    e. cmpt 4179    _I cid 4407   omcom 4759    |` cres 4794    o. ccom 4796   -->wf 5354   -1-1-onto->wf1o 5357   ` cfv 5358  (class class class)co 5981   1oc1o 6614    ~~ cen 7003    ~<_ cdom 7004    ~< csdm 7005   Fincfn 7006   1c1 8885   ZZcz 10175   Basecbs 13356   ↾s cress 13357   0gc0g 13610   Grpcgrp 14572  .gcmg 14576  SubGrpcsubg 14825    GrpHom cghm 14890   ~FG cefg 15225  freeGrpcfrgp 15226  varFGrpcvrgp 15227   Abelcabel 15300  CycGrpccyg 15374  SubRingcsubrg 15751  ℂfldccnfld 16593
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-inf2 7489  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961  ax-addf 8963  ax-mulf 8964
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-ot 3739  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-2o 6622  df-oadd 6625  df-er 6802  df-ec 6804  df-qs 6808  df-map 6917  df-pm 6918  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-sup 7341  df-card 7719  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-10 9959  df-n0 10115  df-z 10176  df-dec 10276  df-uz 10382  df-rp 10506  df-fz 10936  df-fzo 11026  df-seq 11211  df-hash 11506  df-word 11610  df-concat 11611  df-s1 11612  df-substr 11613  df-splice 11614  df-reverse 11615  df-s2 11699  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-starv 13431  df-sca 13432  df-vsca 13433  df-tset 13435  df-ple 13436  df-ds 13438  df-unif 13439  df-0g 13614  df-gsum 13615  df-imas 13621  df-divs 13622  df-mnd 14577  df-mhm 14625  df-submnd 14626  df-frmd 14681  df-vrmd 14682  df-grp 14699  df-minusg 14700  df-mulg 14702  df-subg 14828  df-ghm 14891  df-efg 15228  df-frgp 15229  df-vrgp 15230  df-cmn 15301  df-abl 15302  df-cyg 15375  df-mgp 15536  df-rng 15550  df-cring 15551  df-ur 15552  df-subrg 15753  df-cnfld 16594
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