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Theorem subgmulg 14651
Description: A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
Hypotheses
Ref Expression
subgmulgcl.t  |-  .x.  =  (.g
`  G )
subgmulg.h  |-  H  =  ( Gs  S )
subgmulg.t  |-  .xb  =  (.g
`  H )
Assertion
Ref Expression
subgmulg  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  =  ( N  .xb  X
) )

Proof of Theorem subgmulg
StepHypRef Expression
1 subgmulg.h . . . . . 6  |-  H  =  ( Gs  S )
2 eqid 2296 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2subg0 14643 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
433ad2ant1 976 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( 0g `  G )  =  ( 0g `  H
) )
54ifeq1d 3592 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  if ( N  =  0 ,  ( 0g `  G ) ,  if ( 0  <  N ,  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )  =  if ( N  =  0 ,  ( 0g `  H
) ,  if ( 0  <  N , 
(  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
6 eqid 2296 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
71, 6ressplusg 13266 . . . . . . . . . 10  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
873ad2ant1 976 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( +g  `  G )  =  ( +g  `  H
) )
98seqeq2d 11069 . . . . . . . 8  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) )
109adantr 451 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  ->  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) )
1110fveq1d 5543 . . . . . 6  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  ->  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
)  =  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  N ) )
1211ifeq1d 3592 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  ->  if ( 0  <  N ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )  =  if ( 0  <  N ,  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )
13 simp2 956 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  N  e.  ZZ )
1413zred 10133 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  N  e.  RR )
15 0re 8854 . . . . . . . . . . . 12  |-  0  e.  RR
16 axlttri 8910 . . . . . . . . . . . 12  |-  ( ( N  e.  RR  /\  0  e.  RR )  ->  ( N  <  0  <->  -.  ( N  =  0  \/  0  <  N
) ) )
1714, 15, 16sylancl 643 . . . . . . . . . . 11  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  <  0  <->  -.  ( N  =  0  \/  0  <  N ) ) )
18 ioran 476 . . . . . . . . . . 11  |-  ( -.  ( N  =  0  \/  0  <  N
)  <->  ( -.  N  =  0  /\  -.  0  <  N ) )
1917, 18syl6bb 252 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  <  0  <->  ( -.  N  =  0  /\  -.  0  <  N ) ) )
2019biimpar 471 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  N  <  0 )
21 simpl1 958 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  S  e.  (SubGrp `  G )
)
2213adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  N  e.  ZZ )
2322znegcld 10135 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  -u N  e.  ZZ )
2414lt0neg1d 9358 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  <  0  <->  0  <  -u N ) )
2524biimpa 470 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  0  <  -u N )
26 elnnz 10050 . . . . . . . . . . . . 13  |-  ( -u N  e.  NN  <->  ( -u N  e.  ZZ  /\  0  <  -u N ) )
2723, 25, 26sylanbrc 645 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  -u N  e.  NN )
28 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ( Base `  G )  =  (
Base `  G )
2928subgss 14638 . . . . . . . . . . . . . . 15  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
30293ad2ant1 976 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  S  C_  ( Base `  G
) )
31 simp3 957 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  S )
3230, 31sseldd 3194 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
3332adantr 451 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  X  e.  ( Base `  G
) )
34 subgmulgcl.t . . . . . . . . . . . . 13  |-  .x.  =  (.g
`  G )
35 eqid 2296 . . . . . . . . . . . . 13  |-  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
3628, 6, 34, 35mulgnn 14589 . . . . . . . . . . . 12  |-  ( (
-u N  e.  NN  /\  X  e.  ( Base `  G ) )  -> 
( -u N  .x.  X
)  =  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )
3727, 33, 36syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  ( -u N  .x.  X )  =  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u N
) )
3831adantr 451 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  X  e.  S )
3934subgmulgcl 14650 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubGrp `  G )  /\  -u N  e.  ZZ  /\  X  e.  S )  ->  ( -u N  .x.  X )  e.  S )
4021, 23, 38, 39syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  ( -u N  .x.  X )  e.  S )
4137, 40eqeltrrd 2371 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N )  e.  S
)
42 eqid 2296 . . . . . . . . . . 11  |-  ( inv g `  G )  =  ( inv g `  G )
43 eqid 2296 . . . . . . . . . . 11  |-  ( inv g `  H )  =  ( inv g `  H )
441, 42, 43subginv 14644 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N )  e.  S
)  ->  ( ( inv g `  G ) `
 (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  -u N
) )  =  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
4521, 41, 44syl2anc 642 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  (
( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )  =  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
4620, 45syldan 456 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  (
( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )  =  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
479adantr 451 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) )
4847fveq1d 5543 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N )  =  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 -u N ) )
4948fveq2d 5545 . . . . . . . 8  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  (
( inv g `  H ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )  =  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
5046, 49eqtrd 2328 . . . . . . 7  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  (
( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )  =  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
5150anassrs 629 . . . . . 6  |-  ( ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  /\  -.  0  <  N )  -> 
( ( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) )  =  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )
5251ifeq2da 3604 . . . . 5  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  ->  if ( 0  <  N ,  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )  =  if ( 0  <  N ,  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )
5312, 52eqtrd 2328 . . . 4  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  -.  N  =  0 )  ->  if ( 0  <  N ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) )  =  if ( 0  <  N ,  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `
 N ) ,  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )
5453ifeq2da 3604 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  if ( N  =  0 ,  ( 0g `  H ) ,  if ( 0  <  N ,  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )  =  if ( N  =  0 ,  ( 0g `  H
) ,  if ( 0  <  N , 
(  seq  1 ( ( +g  `  H
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( inv g `  H
) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
555, 54eqtrd 2328 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  if ( N  =  0 ,  ( 0g `  G ) ,  if ( 0  <  N ,  (  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) )  =  if ( N  =  0 ,  ( 0g `  H
) ,  if ( 0  <  N , 
(  seq  1 ( ( +g  `  H
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( inv g `  H
) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
5628, 6, 2, 42, 34, 35mulgval 14585 . . 3  |-  ( ( N  e.  ZZ  /\  X  e.  ( Base `  G ) )  -> 
( N  .x.  X
)  =  if ( N  =  0 ,  ( 0g `  G
) ,  if ( 0  <  N , 
(  seq  1 ( ( +g  `  G
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( inv g `  G
) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
5713, 32, 56syl2anc 642 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  =  if ( N  =  0 ,  ( 0g
`  G ) ,  if ( 0  < 
N ,  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  N ) ,  ( ( inv g `  G ) `  (  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
581subgbas 14641 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
59583ad2ant1 976 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  S  =  ( Base `  H
) )
6031, 59eleqtrd 2372 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
61 eqid 2296 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
62 eqid 2296 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
63 eqid 2296 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
64 subgmulg.t . . . 4  |-  .xb  =  (.g
`  H )
65 eqid 2296 . . . 4  |-  seq  1
( ( +g  `  H
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) )
6661, 62, 63, 43, 64, 65mulgval 14585 . . 3  |-  ( ( N  e.  ZZ  /\  X  e.  ( Base `  H ) )  -> 
( N  .xb  X
)  =  if ( N  =  0 ,  ( 0g `  H
) ,  if ( 0  <  N , 
(  seq  1 ( ( +g  `  H
) ,  ( NN 
X.  { X }
) ) `  N
) ,  ( ( inv g `  H
) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
6713, 60, 66syl2anc 642 . 2  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .xb  X )  =  if ( N  =  0 ,  ( 0g
`  H ) ,  if ( 0  < 
N ,  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  N ) ,  ( ( inv g `  H ) `  (  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) ) `  -u N ) ) ) ) )
6855, 57, 673eqtr4d 2338 1  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  =  ( N  .xb  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   ifcif 3578   {csn 3653   class class class wbr 4039    X. cxp 4703   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    < clt 8883   -ucneg 9054   NNcn 9762   ZZcz 10040    seq cseq 11062   Basecbs 13164   ↾s cress 13165   +g cplusg 13224   0gc0g 13416   inv gcminusg 14379  .gcmg 14382  SubGrpcsubg 14631
This theorem is referenced by:  cycsubgcyg  15203  ablfac2  15340  zcyg  16461  mulgrhm2  16477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mulg 14508  df-subg 14634
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