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Theorem subgmulg 14950
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 2435 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
31, 2subg0 14942 . . . . 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 3745 . . 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 2435 . . . . . . . . . . 11  |-  ( +g  `  G )  =  ( +g  `  G )
71, 6ressplusg 13563 . . . . . . . . . 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 11322 . . . . . . . 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 5722 . . . . . 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 3745 . . . . 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 10367 . . . . . . . . . . . 12  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  N  e.  RR )
15 0re 9083 . . . . . . . . . . . 12  |-  0  e.  RR
16 axlttri 9139 . . . . . . . . . . . 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 10369 . . . . . . . . . . . . 13  |-  ( ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  /\  N  <  0 )  ->  -u N  e.  ZZ )
2414lt0neg1d 9588 . . . . . . . . . . . . . 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 10284 . . . . . . . . . . . . 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 2435 . . . . . . . . . . . . . . . 16  |-  ( Base `  G )  =  (
Base `  G )
2928subgss 14937 . . . . . . . . . . . . . . 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 3341 . . . . . . . . . . . . 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 2435 . . . . . . . . . . . . 13  |-  seq  1
( ( +g  `  G
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  G ) ,  ( NN  X.  { X } ) )
3628, 6, 34, 35mulgnn 14888 . . . . . . . . . . . 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 14949 . . . . . . . . . . . 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 2510 . . . . . . . . . 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 2435 . . . . . . . . . . 11  |-  ( inv g `  G )  =  ( inv g `  G )
43 eqid 2435 . . . . . . . . . . 11  |-  ( inv g `  H )  =  ( inv g `  H )
441, 42, 43subginv 14943 . . . . . . . . . 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 5722 . . . . . . . . 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 5724 . . . . . . . 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 2467 . . . . . . 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 3757 . . . . 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 2467 . . . 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 3757 . . 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 2467 . 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 14884 . . 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 14940 . . . . 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 2511 . . 3  |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  X  e.  ( Base `  H
) )
61 eqid 2435 . . . 4  |-  ( Base `  H )  =  (
Base `  H )
62 eqid 2435 . . . 4  |-  ( +g  `  H )  =  ( +g  `  H )
63 eqid 2435 . . . 4  |-  ( 0g
`  H )  =  ( 0g `  H
)
64 subgmulg.t . . . 4  |-  .xb  =  (.g
`  H )
65 eqid 2435 . . . 4  |-  seq  1
( ( +g  `  H
) ,  ( NN 
X.  { X }
) )  =  seq  1 ( ( +g  `  H ) ,  ( NN  X.  { X } ) )
6661, 62, 63, 43, 64, 65mulgval 14884 . . 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 2477 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 1652    e. wcel 1725    C_ wss 3312   ifcif 3731   {csn 3806   class class class wbr 4204    X. cxp 4868   ` cfv 5446  (class class class)co 6073   RRcr 8981   0cc0 8982   1c1 8983    < clt 9112   -ucneg 9284   NNcn 9992   ZZcz 10274    seq cseq 11315   Basecbs 13461   ↾s cress 13462   +g cplusg 13521   0gc0g 13715   inv gcminusg 14678  .gcmg 14681  SubGrpcsubg 14930
This theorem is referenced by:  cycsubgcyg  15502  ablfac2  15639  zcyg  16764  mulgrhm2  16780  zzsmulg  24257  remulg  24262  rezh  24347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-seq 11316  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-mulg 14807  df-subg 14933
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