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Theorem dprdss 15587
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 2436 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
2 eqid 2436 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
3 eqid 2436 . . 3  |-  (mrCls `  (SubGrp `  G ) )  =  (mrCls `  (SubGrp `  G ) )
4 dprdss.1 . . . 4  |-  ( ph  ->  G dom DProd  T )
5 dprdgrp 15563 . . . 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 15558 . . . . . 6  |-  Rel  dom DProd
98brrelex2i 4919 . . . . 5  |-  ( G dom DProd  T  ->  T  e. 
_V )
10 dmexg 5130 . . . . 5  |-  ( T  e.  _V  ->  dom  T  e.  _V )
114, 9, 103syl 19 . . . 4  |-  ( ph  ->  dom  T  e.  _V )
127, 11eqeltrrd 2511 . . 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 2789 . . . . . 6  |-  ( ph  ->  A. k  e.  I 
( S `  k
)  C_  ( T `  k ) )
16 fveq2 5728 . . . . . . . 8  |-  ( k  =  x  ->  ( S `  k )  =  ( S `  x ) )
17 fveq2 5728 . . . . . . . 8  |-  ( k  =  x  ->  ( T `  k )  =  ( T `  x ) )
1816, 17sseq12d 3377 . . . . . . 7  |-  ( k  =  x  ->  (
( S `  k
)  C_  ( T `  k )  <->  ( S `  x )  C_  ( T `  x )
) )
1918rspcv 3048 . . . . . 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 15566 . . . . 5  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  x
)  C_  ( (Cntz `  G ) `  ( T `  y )
) )
284, 7dprdf2 15565 . . . . . . . . 9  |-  ( ph  ->  T : I --> (SubGrp `  G ) )
2928adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  ->  T : I --> (SubGrp `  G ) )
3029, 25ffvelrnd 5871 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  y
)  e.  (SubGrp `  G ) )
31 eqid 2436 . . . . . . . 8  |-  ( Base `  G )  =  (
Base `  G )
3231subgss 14945 . . . . . . 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 5728 . . . . . . . . 9  |-  ( k  =  y  ->  ( S `  k )  =  ( S `  y ) )
36 fveq2 5728 . . . . . . . . 9  |-  ( k  =  y  ->  ( T `  k )  =  ( T `  y ) )
3735, 36sseq12d 3377 . . . . . . . 8  |-  ( k  =  y  ->  (
( S `  k
)  C_  ( T `  k )  <->  ( S `  y )  C_  ( T `  y )
) )
3837rspcv 3048 . . . . . . 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 15131 . . . . . 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 3358 . . . 4  |-  ( (
ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y ) )  -> 
( T `  x
)  C_  ( (Cntz `  G ) `  ( S `  y )
) )
4321, 42sstrd 3358 . . 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 14975 . . . . . . 7  |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS
`  ( Base `  G
) ) )
46 acsmre 13877 . . . . . . 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 3474 . . . . . . . . 9  |-  ( I 
\  { x }
)  C_  I
4915adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  I )  ->  A. k  e.  I  ( S `  k )  C_  ( T `  k )
)
50 ssralv 3407 . . . . . . . . 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 4108 . . . . . . . 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 5593 . . . . . . . 8  |-  ( S : I --> (SubGrp `  G )  ->  Fun  S )
56 funiunfv 5995 . . . . . . . 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 5593 . . . . . . . 8  |-  ( T : I --> (SubGrp `  G )  ->  Fun  T )
60 funiunfv 5995 . . . . . . . 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 3390 . . . . . 6  |-  ( (
ph  /\  x  e.  I )  ->  U. ( S " ( I  \  { x } ) )  C_  U. ( T " ( I  \  { x } ) ) )
63 imassrn 5216 . . . . . . . 8  |-  ( T
" ( I  \  { x } ) )  C_  ran  T
64 frn 5597 . . . . . . . . . 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 13817 . . . . . . . . . 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 3358 . . . . . . . 8  |-  ( (
ph  /\  x  e.  I )  ->  ran  T 
C_  ~P ( Base `  G
) )
6963, 68syl5ss 3359 . . . . . . 7  |-  ( (
ph  /\  x  e.  I )  ->  ( T " ( I  \  { x } ) )  C_  ~P ( Base `  G ) )
70 sspwuni 4176 . . . . . . 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 13849 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  (
(mrCls `  (SubGrp `  G
) ) `  U. ( S " ( I 
\  { x }
) ) )  C_  ( (mrCls `  (SubGrp `  G
) ) `  U. ( T " ( I 
\  { x }
) ) ) )
73 ss2in 3568 . . . . 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 15567 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( T `  x
)  i^i  ( (mrCls `  (SubGrp `  G )
) `  U. ( T
" ( I  \  { x } ) ) ) )  =  { ( 0g `  G ) } )
7974, 78sseqtrd 3384 . . 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 15560 . 2  |-  ( ph  ->  G dom DProd  S )
814a1d 23 . . . . 5  |-  ( ph  ->  ( G dom DProd  S  ->  G dom DProd  T ) )
82 ss2ixp 7075 . . . . . . 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 3426 . . . . . 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 3408 . . . . . 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 5595 . . . . 5  |-  ( S : I --> (SubGrp `  G )  ->  dom  S  =  I )
89 eqid 2436 . . . . . 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 15562 . . . . 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 2436 . . . . . 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 15562 . . . . 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 3354 . 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 2599   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2956    \ cdif 3317    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   {csn 3814   U.cuni 4015   U_ciun 4093   class class class wbr 4212   `'ccnv 4877   dom cdm 4878   ran crn 4879   "cima 4881   Fun wfun 5448   -->wf 5450   ` cfv 5454  (class class class)co 6081   X_cixp 7063   Fincfn 7109   Basecbs 13469   0gc0g 13723    gsumg cgsu 13724  Moorecmre 13807  mrClscmrc 13808  ACScacs 13810   Grpcgrp 14685  SubGrpcsubg 14938  Cntzccntz 15114   DProd cdprd 15554
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-subg 14941  df-cntz 15116  df-dprd 15556
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