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Theorem dprdss 15579
Description: Create a direct product by finding subgroups inside each factor of another direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdss.1  |-  ( ph  ->  G dom DProd  T )
dprdss.2  |-  ( ph  ->  dom  T  =  I )
dprdss.3  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
dprdss.4  |-  ( (
ph  /\  k  e.  I )  ->  ( S `  k )  C_  ( T `  k
) )
Assertion
Ref Expression
dprdss  |-  ( ph  ->  ( G dom DProd  S  /\  ( G DProd  S )  C_  ( G DProd  T ) ) )
Distinct variable groups:    k, G    ph, k    S, k    T, k   
k, I

Proof of Theorem dprdss
Dummy variables  f 
a  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2435 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2435 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2435 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dprdss.1 . . . 4  |-  ( ph  ->  G dom DProd  T )
5 dprdgrp 15555 . . . 4  |-  ( G dom DProd  T  ->  G  e. 
Grp )
64, 5syl 16 . . 3  |-  ( ph  ->  G  e.  Grp )
7 dprdss.2 . . . 4  |-  ( ph  ->  dom  T  =  I )
8 reldmdprd 15550 . . . . . 6  |-  Rel  dom DProd
98brrelex2i 4911 . . . . 5  |-  ( G dom DProd  T  ->  T  e. 
_V )
10 dmexg 5122 . . . . 5  |-  ( T  e.  _V  ->  dom  T  e.  _V )
114, 9, 103syl 19 . . . 4  |-  ( ph  ->  dom  T  e.  _V )
127, 11eqeltrrd 2510 . . 3  |-  ( ph  ->  I  e.  _V )
13 dprdss.3 . . 3  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
14 dprdss.4 . . . . . . 7  |-  ( (
ph  /\  k  e.  I )  ->  ( S `  k )  C_  ( T `  k
) )
1514ralrimiva 2781 . . . . . 6  |-  ( ph  ->  A. k  e.  I 
( S `  k
)  C_  ( T `  k ) )
16 fveq2 5720 . . . . . . . 8  |-  ( k  =  x  ->  ( S `  k )  =  ( S `  x ) )
17 fveq2 5720 . . . . . . . 8  |-  ( k  =  x  ->  ( T `  k )  =  ( T `  x ) )
1816, 17sseq12d 3369 . . . . . . 7  |-  ( k  =  x  ->  (
( S `  k
)  C_  ( T `  k )  <->  ( S `  x )  C_  ( T `  x )
) )
1918rspcv 3040 . . . . . 6  |-  ( x  e.  I  ->  ( A. k  e.  I 
( S `  k
)  C_  ( T `  k )  ->  ( S `  x )  C_  ( T `  x
) ) )
2015, 19mpan9 456 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  C_  ( T `  x
) )
21203ad2antr1 1122 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( T `  x ) )
224adantr 452 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  G dom DProd  T )
237adantr 452 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  dom  T  =  I )
24 simpr1 963 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  x  e.  I )
25 simpr2 964 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
y  e.  I )
26 simpr3 965 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  x  =/=  y )
2722, 23, 24, 25, 26, 1dprdcntz 15558 . . . . 5  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  x
)  C_  ( (Cntz `  G ) `  ( T `  y )
) )
284, 7dprdf2 15557 . . . . . . . . 9  |-  ( ph  ->  T : I --> (SubGrp `  G ) )
2928adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  T : I --> (SubGrp `  G ) )
3029, 25ffvelrnd 5863 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  y
)  e.  (SubGrp `  G ) )
31 eqid 2435 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
3231subgss 14937 . . . . . . 7  |-  ( ( T `  y )  e.  (SubGrp `  G
)  ->  ( T `  y )  C_  ( Base `  G ) )
3330, 32syl 16 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  y
)  C_  ( Base `  G ) )
3415adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  A. k  e.  I 
( S `  k
)  C_  ( T `  k ) )
35 fveq2 5720 . . . . . . . . 9  |-  ( k  =  y  ->  ( S `  k )  =  ( S `  y ) )
36 fveq2 5720 . . . . . . . . 9  |-  ( k  =  y  ->  ( T `  k )  =  ( T `  y ) )
3735, 36sseq12d 3369 . . . . . . . 8  |-  ( k  =  y  ->  (
( S `  k
)  C_  ( T `  k )  <->  ( S `  y )  C_  ( T `  y )
) )
3837rspcv 3040 . . . . . . 7  |-  ( y  e.  I  ->  ( A. k  e.  I 
( S `  k
)  C_  ( T `  k )  ->  ( S `  y )  C_  ( T `  y
) ) )
3925, 34, 38sylc 58 . . . . . 6  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  y
)  C_  ( T `  y ) )
4031, 1cntz2ss 15123 . . . . . 6  |-  ( ( ( T `  y
)  C_  ( Base `  G )  /\  ( S `  y )  C_  ( T `  y
) )  ->  (
(Cntz `  G ) `  ( T `  y
) )  C_  (
(Cntz `  G ) `  ( S `  y
) ) )
4133, 39, 40syl2anc 643 . . . . 5  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( (Cntz `  G
) `  ( T `  y ) )  C_  ( (Cntz `  G ) `  ( S `  y
) ) )
4227, 41sstrd 3350 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
) )
4321, 42sstrd 3350 . . 3  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( S `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
) )
446adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  G  e.  Grp )
4531subgacs 14967 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
46 acsmre 13869 . . . . . . 7  |-  ( (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
4744, 45, 463syl 19 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  e.  (Moore `  ( Base `  G
) ) )
48 difss 3466 . . . . . . . . 9  |-  ( I 
\  { x }
)  C_  I
4915adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  A. k  e.  I  ( S `  k )  C_  ( T `  k )
)
50 ssralv 3399 . . . . . . . . 9  |-  ( ( I  \  { x } )  C_  I  ->  ( A. k  e.  I  ( S `  k )  C_  ( T `  k )  ->  A. k  e.  ( I  \  { x } ) ( S `
 k )  C_  ( T `  k ) ) )
5148, 49, 50mpsyl 61 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  A. k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  ( T `  k ) )
52 ss2iun 4100 . . . . . . . 8  |-  ( A. k  e.  ( I  \  { x } ) ( S `  k
)  C_  ( T `  k )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  U_ k  e.  ( I  \  { x } ) ( T `
 k ) )
5351, 52syl 16 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k ) 
C_  U_ k  e.  ( I  \  { x } ) ( T `
 k ) )
5413adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
55 ffun 5585 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  Fun  S )
56 funiunfv 5987 . . . . . . . 8  |-  ( Fun 
S  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k )  =  U. ( S
" ( I  \  { x } ) ) )
5754, 55, 563syl 19 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( S `  k )  =  U. ( S
" ( I  \  { x } ) ) )
5828adantr 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  T : I --> (SubGrp `  G ) )
59 ffun 5585 . . . . . . . 8  |-  ( T : I --> (SubGrp `  G )  ->  Fun  T )
60 funiunfv 5987 . . . . . . . 8  |-  ( Fun 
T  ->  U_ k  e.  ( I  \  {
x } ) ( T `  k )  =  U. ( T
" ( I  \  { x } ) ) )
6158, 59, 603syl 19 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  U_ k  e.  ( I  \  {
x } ) ( T `  k )  =  U. ( T
" ( I  \  { x } ) ) )
6253, 57, 613sstr3d 3382 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  U. ( T " ( I  \  { x } ) ) )
63 imassrn 5208 . . . . . . . 8  |-  ( T
" ( I  \  { x } ) )  C_  ran  T
64 frn 5589 . . . . . . . . . 10  |-  ( T : I --> (SubGrp `  G )  ->  ran  T 
C_  (SubGrp `  G )
)
6558, 64syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  ran  T 
C_  (SubGrp `  G )
)
66 mresspw 13809 . . . . . . . . . 10  |-  ( (SubGrp `  G )  e.  (Moore `  ( Base `  G
) )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6747, 66syl 16 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  (SubGrp `  G )  C_  ~P ( Base `  G )
)
6865, 67sstrd 3350 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ran  T 
C_  ~P ( Base `  G
) )
6963, 68syl5ss 3351 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( T " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
70 sspwuni 4168 . . . . . . 7  |-  ( ( T " ( I 
\  { x }
) )  C_  ~P ( Base `  G )  <->  U. ( T " (
I  \  { x } ) )  C_  ( Base `  G )
)
7169, 70sylib 189 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  U. ( T " ( I  \  { x } ) )  C_  ( Base `  G ) )
7247, 3, 62, 71mrcssd 13841 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) )
73 ss2in 3560 . . . . 5  |-  ( ( ( S `  x
)  C_  ( T `  x )  /\  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) )  ->  ( ( S `
 x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( S " (
I  \  { x } ) ) ) )  C_  ( ( T `  x )  i^i  ( (mrCls `  (SubGrp `  G ) ) `  U. ( T " (
I  \  { x } ) ) ) ) )
7420, 72, 73syl2anc 643 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  ( ( T `  x )  i^i  (
(mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) ) )
754adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  G dom DProd  T )
767adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  dom  T  =  I )
77 simpr 448 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  x  e.  I )
7875, 76, 77, 2, 3dprddisj 15559 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( T `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( T
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
7974, 78sseqtrd 3376 . . 3  |-  ( (
ph  /\  x  e.  I )  ->  (
( S `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( S
" ( I  \  { x } ) ) ) )  C_  { ( 0g `  G
) } )
801, 2, 3, 6, 12, 13, 43, 79dmdprdd 15552 . 2  |-  ( ph  ->  G dom DProd  S )
814a1d 23 . . . . 5  |-  ( ph  ->  ( G dom DProd  S  ->  G dom DProd  T ) )
82 ss2ixp 7067 . . . . . . 7  |-  ( A. k  e.  I  ( S `  k )  C_  ( T `  k
)  ->  X_ k  e.  I  ( S `  k )  C_  X_ k  e.  I  ( T `  k ) )
8315, 82syl 16 . . . . . 6  |-  ( ph  -> 
X_ k  e.  I 
( S `  k
)  C_  X_ k  e.  I  ( T `  k ) )
84 rabss2 3418 . . . . . 6  |-  ( X_ k  e.  I  ( S `  k )  C_  X_ k  e.  I 
( T `  k
)  ->  { h  e.  X_ k  e.  I 
( S `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
C_  { h  e.  X_ k  e.  I 
( T `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } )
85 ssrexv 3400 . . . . . 6  |-  ( { h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } 
C_  { h  e.  X_ k  e.  I 
( T `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }  ->  ( E. f  e.  { h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f )  ->  E. f  e.  {
h  e.  X_ k  e.  I  ( T `  k )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } a  =  ( G 
gsumg  f ) ) )
8683, 84, 853syl 19 . . . . 5  |-  ( ph  ->  ( E. f  e. 
{ h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f )  ->  E. f  e.  {
h  e.  X_ k  e.  I  ( T `  k )  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin } a  =  ( G 
gsumg  f ) ) )
8781, 86anim12d 547 . . . 4  |-  ( ph  ->  ( ( G dom DProd  S  /\  E. f  e. 
{ h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) )  ->  ( G dom DProd  T  /\  E. f  e.  { h  e.  X_ k  e.  I  ( T `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) ) ) )
88 fdm 5587 . . . . 5  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
89 eqid 2435 . . . . . 6  |-  { h  e.  X_ k  e.  I 
( S `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }  =  { h  e.  X_ k  e.  I 
( S `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }
902, 89eldprd 15554 . . . . 5  |-  ( dom 
S  =  I  -> 
( a  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  { h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) ) ) )
9113, 88, 903syl 19 . . . 4  |-  ( ph  ->  ( a  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  { h  e.  X_ k  e.  I  ( S `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) ) ) )
92 eqid 2435 . . . . . 6  |-  { h  e.  X_ k  e.  I 
( T `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }  =  { h  e.  X_ k  e.  I 
( T `  k
)  |  ( `' h " ( _V 
\  { ( 0g
`  G ) } ) )  e.  Fin }
932, 92eldprd 15554 . . . . 5  |-  ( dom 
T  =  I  -> 
( a  e.  ( G DProd  T )  <->  ( G dom DProd  T  /\  E. f  e.  { h  e.  X_ k  e.  I  ( T `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) ) ) )
947, 93syl 16 . . . 4  |-  ( ph  ->  ( a  e.  ( G DProd  T )  <->  ( G dom DProd  T  /\  E. f  e.  { h  e.  X_ k  e.  I  ( T `  k )  |  ( `' h " ( _V  \  {
( 0g `  G
) } ) )  e.  Fin } a  =  ( G  gsumg  f ) ) ) )
9587, 91, 943imtr4d 260 . . 3  |-  ( ph  ->  ( a  e.  ( G DProd  S )  -> 
a  e.  ( G DProd 
T ) ) )
9695ssrdv 3346 . 2  |-  ( ph  ->  ( G DProd  S ) 
C_  ( G DProd  T
) )
9780, 96jca 519 1  |-  ( ph  ->  ( G dom DProd  S  /\  ( G DProd  S )  C_  ( G DProd  T ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    \ cdif 3309    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   {csn 3806   U.cuni 4007   U_ciun 4085   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873   Fun wfun 5440   -->wf 5442   ` cfv 5446  (class class class)co 6073   X_cixp 7055   Fincfn 7101   Basecbs 13461   0gc0g 13715    gsumg cgsu 13716  Moorecmre 13799  mrClscmrc 13800  ACScacs 13802   Grpcgrp 14677  SubGrpcsubg 14930  Cntzccntz 15106   DProd cdprd 15546
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-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-int 4043  df-iun 4087  df-iin 4088  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-1o 6716  df-oadd 6720  df-er 6897  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  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-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-subg 14933  df-cntz 15108  df-dprd 15548
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