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Theorem subgmulg 14886
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 2388 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2subg0 14878 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  H ) )
433ad2ant1 978 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( 0g `  G )  =  ( 0g `  H
) )
54ifeq1d 3697 . . 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 2388 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
71, 6ressplusg 13499 . . . . . . . . . 10  |-  ( S  e.  (SubGrp `  G
)  ->  ( +g  `  G )  =  ( +g  `  H ) )
873ad2ant1 978 . . . . . . . . 9  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( +g  `  G )  =  ( +g  `  H
) )
98seqeq2d 11258 . . . . . . . 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 452 . . . . . . 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 5671 . . . . . 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 3697 . . . . 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 958 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  N  e.  ZZ )
1413zred 10308 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  N  e.  RR )
15 0re 9025 . . . . . . . . . . . 12  |-  0  e.  RR
16 axlttri 9081 . . . . . . . . . . . 12  |-  ( ( N  e.  RR  /\  0  e.  RR )  ->  ( N  <  0  <->  -.  ( N  =  0  \/  0  <  N
) ) )
1714, 15, 16sylancl 644 . . . . . . . . . . 11  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  <  0  <->  -.  ( N  =  0  \/  0  <  N ) ) )
18 ioran 477 . . . . . . . . . . 11  |-  ( -.  ( N  =  0  \/  0  <  N
)  <->  ( -.  N  =  0  /\  -.  0  <  N ) )
1917, 18syl6bb 253 . . . . . . . . . 10  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  <  0  <->  ( -.  N  =  0  /\  -.  0  <  N ) ) )
2019biimpar 472 . . . . . . . . 9  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  ( -.  N  =  0  /\  -.  0  <  N
) )  ->  N  <  0 )
21 simpl1 960 . . . . . . . . . 10  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  S  e.  (SubGrp `  G )
)
2213adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  N  e.  ZZ )
2322znegcld 10310 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  -u N  e.  ZZ )
2414lt0neg1d 9529 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  <  0  <->  0  <  -u N ) )
2524biimpa 471 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  0  <  -u N )
26 elnnz 10225 . . . . . . . . . . . . 13  |-  ( -u N  e.  NN  <->  ( -u N  e.  ZZ  /\  0  <  -u N ) )
2723, 25, 26sylanbrc 646 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  -u N  e.  NN )
28 eqid 2388 . . . . . . . . . . . . . . . 16  |-  ( Base `  G )  =  (
Base `  G )
2928subgss 14873 . . . . . . . . . . . . . . 15  |-  ( S  e.  (SubGrp `  G
)  ->  S  C_  ( Base `  G ) )
30293ad2ant1 978 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  S  C_  ( Base `  G
) )
31 simp3 959 . . . . . . . . . . . . . 14  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  S )
3230, 31sseldd 3293 . . . . . . . . . . . . 13  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  ( Base `  G
) )
3332adantr 452 . . . . . . . . . . . 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 2388 . . . . . . . . . . . . 13  |-  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
3628, 6, 34, 35mulgnn 14824 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 452 . . . . . . . . . . . 12  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  X  e.  S )
3934subgmulgcl 14885 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubGrp `  G )  /\  -u N  e.  ZZ  /\  X  e.  S )  ->  ( -u N  .x.  X )  e.  S )
4021, 23, 38, 39syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  ( -u N  .x.  X )  e.  S )
4137, 40eqeltrrd 2463 . . . . . . . . . 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 2388 . . . . . . . . . . 11  |-  ( inv g `  G )  =  ( inv g `  G )
43 eqid 2388 . . . . . . . . . . 11  |-  ( inv g `  H )  =  ( inv g `  H )
441, 42, 43subginv 14879 . . . . . . . . . 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 643 . . . . . . . . 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 457 . . . . . . . 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 452 . . . . . . . . . 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 5671 . . . . . . . . 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 5673 . . . . . . . 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 2420 . . . . . . 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 630 . . . . . 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 3709 . . . . 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 2420 . . . 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 3709 . . 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 2420 . 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 14820 . . 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 643 . 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 14876 . . . . 5  |-  ( S  e.  (SubGrp `  G
)  ->  S  =  ( Base `  H )
)
59583ad2ant1 978 . . . 4  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  S  =  ( Base `  H
) )
6031, 59eleqtrd 2464 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
61 eqid 2388 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
62 eqid 2388 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
63 eqid 2388 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
64 subgmulg.t . . . 4  |-  .xb  =  (.g
`  H )
65 eqid 2388 . . . 4  |-  seq  1
( ( +g  `  H
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) )
6661, 62, 63, 43, 64, 65mulgval 14820 . . 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 643 . 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 2430 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    C_ wss 3264   ifcif 3683   {csn 3758   class class class wbr 4154    X. cxp 4817   ` cfv 5395  (class class class)co 6021   RRcr 8923   0cc0 8924   1c1 8925    < clt 9054   -ucneg 9225   NNcn 9933   ZZcz 10215    seq cseq 11251   Basecbs 13397   ↾s cress 13398   +g cplusg 13457   0gc0g 13651   inv gcminusg 14614  .gcmg 14617  SubGrpcsubg 14866
This theorem is referenced by:  cycsubgcyg  15438  ablfac2  15575  zcyg  16696  mulgrhm2  16712  zzsmulg  24082  remulg  24087  rrhre  24184
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-2 9991  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-seq 11252  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-0g 13655  df-mnd 14618  df-grp 14740  df-minusg 14741  df-mulg 14743  df-subg 14869
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