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Theorem subgmulg 14635
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 2283 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2subg0 14627 . . . . 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 3579 . . 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 2283 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
71, 6ressplusg 13250 . . . . . . . . . 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 11053 . . . . . . . 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 5527 . . . . . 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 3579 . . . . 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 10117 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  N  e.  RR )
15 0re 8838 . . . . . . . . . . . 12  |-  0  e.  RR
16 axlttri 8894 . . . . . . . . . . . 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 10119 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  -u N  e.  ZZ )
2414lt0neg1d 9342 . . . . . . . . . . . . . 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 10034 . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . 16  |-  ( Base `  G )  =  (
Base `  G )
2928subgss 14622 . . . . . . . . . . . . . . 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 3181 . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . 13  |-  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
3628, 6, 34, 35mulgnn 14573 . . . . . . . . . . . 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 14634 . . . . . . . . . . . 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 2358 . . . . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( inv g `  G )  =  ( inv g `  G )
43 eqid 2283 . . . . . . . . . . 11  |-  ( inv g `  H )  =  ( inv g `  H )
441, 42, 43subginv 14628 . . . . . . . . . 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 5527 . . . . . . . . 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 5529 . . . . . . . 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 2315 . . . . . . 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 3591 . . . . 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 2315 . . . 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 3591 . . 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 2315 . 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 14569 . . 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 14625 . . . . 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 2359 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
61 eqid 2283 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
62 eqid 2283 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
63 eqid 2283 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
64 subgmulg.t . . . 4  |-  .xb  =  (.g
`  H )
65 eqid 2283 . . . 4  |-  seq  1
( ( +g  `  H
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) )
6661, 62, 63, 43, 64, 65mulgval 14569 . . 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 2325 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 1623    e. wcel 1684    C_ wss 3152   ifcif 3565   {csn 3640   class class class wbr 4023    X. cxp 4687   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    < clt 8867   -ucneg 9038   NNcn 9746   ZZcz 10024    seq cseq 11046   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   0gc0g 13400   inv gcminusg 14363  .gcmg 14366  SubGrpcsubg 14615
This theorem is referenced by:  cycsubgcyg  15187  ablfac2  15324  zcyg  16445  mulgrhm2  16461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mulg 14492  df-subg 14618
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