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Theorem dpjidclOLD 16984
Description: The key property of projections: the sum of all the projections of  A is  A. (Contributed by Mario Carneiro, 26-Apr-2016.) Obsolete version of dpjidcl 16977 as of 14-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
dpjidclOLD.1  |-  ( ph  ->  G dom DProd  S )
dpjidclOLD.2  |-  ( ph  ->  dom  S  =  I )
dpjidclOLD.p  |-  P  =  ( GdProj S )
dpjidclOLD.3  |-  ( ph  ->  A  e.  ( G DProd 
S ) )
dpjidclOLD.0  |-  .0.  =  ( 0g `  G )
dpjidclOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
Assertion
Ref Expression
dpjidclOLD  |-  ( ph  ->  ( ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e.  W  /\  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A
) ) ) ) )
Distinct variable groups:    x, h,  .0.    h, i, G, x    P, h, x    ph, i, x    h, I, i, x   
x, W    A, h, x    S, h, i, x
Allowed substitution hints:    ph( h)    A( i)    P( i)    W( h, i)    .0. ( i)

Proof of Theorem dpjidclOLD
Dummy variables  k 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dpjidclOLD.3 . . . 4  |-  ( ph  ->  A  e.  ( G DProd 
S ) )
2 dpjidclOLD.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
3 dpjidclOLD.0 . . . . . 6  |-  .0.  =  ( 0g `  G )
4 dpjidclOLD.w . . . . . 6  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
53, 4eldprdOLD 16908 . . . . 5  |-  ( dom 
S  =  I  -> 
( A  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) ) )
62, 5syl 16 . . . 4  |-  ( ph  ->  ( A  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) ) )
71, 6mpbid 210 . . 3  |-  ( ph  ->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
87simprd 463 . 2  |-  ( ph  ->  E. f  e.  W  A  =  ( G  gsumg  f ) )
9 dpjidclOLD.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
109adantr 465 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  G dom DProd  S )
112adantr 465 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  dom  S  =  I )
129ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G dom DProd  S )
132ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  dom  S  =  I )
14 dpjidclOLD.p . . . . . 6  |-  P  =  ( GdProj S )
15 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  x  e.  I )
1612, 13, 14, 15dpjf 16976 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( P `  x
) : ( G DProd 
S ) --> ( S `
 x ) )
171ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  e.  ( G DProd 
S ) )
1816, 17ffvelrnd 6033 . . . 4  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  e.  ( S `
 x ) )
19 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  e.  W
)
204, 10, 11, 19dprdffiOLD 16924 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' f
" ( _V  \  {  .0.  } ) )  e.  Fin )
21 eldifi 3631 . . . . . . . 8  |-  ( x  e.  ( I  \ 
( `' f "
( _V  \  {  .0.  } ) ) )  ->  x  e.  I
)
22 eqid 2467 . . . . . . . . . 10  |-  ( proj1 `  G )  =  ( proj1 `  G )
2312, 13, 14, 22, 15dpjval 16975 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( P `  x
)  =  ( ( S `  x ) ( proj1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) )
2423fveq1d 5874 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  =  ( ( ( S `  x
) ( proj1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A ) )
2521, 24sylan2 474 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
( P `  x
) `  A )  =  ( ( ( S `  x ) ( proj1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A ) )
26 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  =  ( G  gsumg  f ) )
27 eqid 2467 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
28 eqid 2467 . . . . . . . . . . 11  |-  (Cntz `  G )  =  (Cntz `  G )
29 dprdgrp 16909 . . . . . . . . . . . . 13  |-  ( G dom DProd  S  ->  G  e. 
Grp )
30 grpmnd 15933 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  G  e.  Mnd )
3110, 29, 303syl 20 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  G  e.  Mnd )
3231adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  G  e.  Mnd )
33 reldmdprd 16899 . . . . . . . . . . . . . . 15  |-  Rel  dom DProd
3433brrelex2i 5047 . . . . . . . . . . . . . 14  |-  ( G dom DProd  S  ->  S  e. 
_V )
35 dmexg 6726 . . . . . . . . . . . . . 14  |-  ( S  e.  _V  ->  dom  S  e.  _V )
3610, 34, 353syl 20 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  dom  S  e.  _V )
3711, 36eqeltrrd 2556 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  I  e.  _V )
3837adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  I  e.  _V )
394, 10, 11, 19, 27dprdffOLD 16922 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f : I --> ( Base `  G
) )
4039adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  f : I --> ( Base `  G ) )
4119adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f  e.  W )
424, 12, 13, 41, 28dprdfcntzOLD 16925 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ran  f  C_  (
(Cntz `  G ) `  ran  f ) )
4321, 42sylan2 474 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ran  f  C_  ( (Cntz `  G ) `  ran  f ) )
44 snssi 4177 . . . . . . . . . . . . 13  |-  ( x  e.  ( I  \ 
( `' f "
( _V  \  {  .0.  } ) ) )  ->  { x }  C_  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) ) )
4544adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  { x }  C_  ( I  \ 
( `' f "
( _V  \  {  .0.  } ) ) ) )
4645difss2d 3639 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  { x }  C_  I )
47 cnvimass 5363 . . . . . . . . . . . . . . 15  |-  ( `' f " ( _V 
\  {  .0.  }
) )  C_  dom  f
48 fdm 5741 . . . . . . . . . . . . . . . 16  |-  ( f : I --> ( Base `  G )  ->  dom  f  =  I )
4939, 48syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  dom  f  =  I )
5047, 49syl5sseq 3557 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' f
" ( _V  \  {  .0.  } ) ) 
C_  I )
5150adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( `' f " ( _V  \  {  .0.  }
) )  C_  I
)
52 ssconb 3642 . . . . . . . . . . . . 13  |-  ( ( { x }  C_  I  /\  ( `' f
" ( _V  \  {  .0.  } ) ) 
C_  I )  -> 
( { x }  C_  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) )  <->  ( `' f " ( _V  \  {  .0.  } ) ) 
C_  ( I  \  { x } ) ) )
5346, 51, 52syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( { x }  C_  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) )  <->  ( `' f " ( _V  \  {  .0.  } ) ) 
C_  ( I  \  { x } ) ) )
5445, 53mpbid 210 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( `' f " ( _V  \  {  .0.  }
) )  C_  (
I  \  { x } ) )
5520adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( `' f " ( _V  \  {  .0.  }
) )  e.  Fin )
5627, 3, 28, 32, 38, 40, 43, 54, 55gsumzresOLD 16789 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( G  gsumg  ( f  |`  (
I  \  { x } ) ) )  =  ( G  gsumg  f ) )
5726, 56eqtr4d 2511 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  =  ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) )
58 eqid 2467 . . . . . . . . . . 11  |-  { h  e.  X_ i  e.  ( I  \  { x } ) ( ( S  |`  ( I  \  { x } ) ) `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }  =  {
h  e.  X_ i  e.  ( I  \  {
x } ) ( ( S  |`  (
I  \  { x } ) ) `  i )  |  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin }
59 difss 3636 . . . . . . . . . . . . . 14  |-  ( I 
\  { x }
)  C_  I
6059a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( I  \  {
x } )  C_  I )
6112, 13, 60dprdres 16945 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G dom DProd  ( S  |`  ( I  \  {
x } ) )  /\  ( G DProd  ( S  |`  ( I  \  { x } ) ) )  C_  ( G DProd  S ) ) )
6261simpld 459 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G dom DProd  ( S  |`  ( I  \  {
x } ) ) )
6312, 13dprdf2 16911 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
64 fssres 5757 . . . . . . . . . . . . 13  |-  ( ( S : I --> (SubGrp `  G )  /\  (
I  \  { x } )  C_  I
)  ->  ( S  |`  ( I  \  {
x } ) ) : ( I  \  { x } ) --> (SubGrp `  G )
)
6563, 59, 64sylancl 662 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S  |`  (
I  \  { x } ) ) : ( I  \  {
x } ) --> (SubGrp `  G ) )
66 fdm 5741 . . . . . . . . . . . 12  |-  ( ( S  |`  ( I  \  { x } ) ) : ( I 
\  { x }
) --> (SubGrp `  G )  ->  dom  ( S  |`  ( I  \  { x } ) )  =  ( I  \  {
x } ) )
6765, 66syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  dom  ( S  |`  ( I  \  { x } ) )  =  ( I  \  {
x } ) )
6839adantr 465 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f : I --> ( Base `  G ) )
6968feqmptd 5927 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f  =  ( k  e.  I  |->  ( f `
 k ) ) )
7069reseq1d 5278 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  =  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) ) )
71 resmpt 5329 . . . . . . . . . . . . . 14  |-  ( ( I  \  { x } )  C_  I  ->  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) )  =  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) )
7259, 71ax-mp 5 . . . . . . . . . . . . 13  |-  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) )  =  ( k  e.  ( I  \  { x } ) 
|->  ( f `  k
) )
7370, 72syl6eq 2524 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  =  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) )
74 eldifi 3631 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( I  \  { x } )  ->  k  e.  I
)
754, 12, 13, 41dprdfclOLD 16923 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  I )  ->  (
f `  k )  e.  ( S `  k
) )
7674, 75sylan2 474 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( f `  k
)  e.  ( S `
 k ) )
77 fvres 5886 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( I  \  { x } )  ->  ( ( S  |`  ( I  \  {
x } ) ) `
 k )  =  ( S `  k
) )
7877adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( ( S  |`  ( I  \  { x } ) ) `  k )  =  ( S `  k ) )
7976, 78eleqtrrd 2558 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( f `  k
)  e.  ( ( S  |`  ( I  \  { x } ) ) `  k ) )
8020adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin )
81 ssdif 3644 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  \  { x } )  C_  I  ->  ( ( I  \  { x } ) 
\  ( `' f
" ( _V  \  {  .0.  } ) ) )  C_  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )
8259, 81ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( ( I  \  { x } )  \  ( `' f " ( _V  \  {  .0.  }
) ) )  C_  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) )
8382sseli 3505 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( ( I 
\  { x }
)  \  ( `' f " ( _V  \  {  .0.  } ) ) )  ->  k  e.  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) ) )
84 ssid 3528 . . . . . . . . . . . . . . . . . 18  |-  ( `' f " ( _V 
\  {  .0.  }
) )  C_  ( `' f " ( _V  \  {  .0.  }
) )
8584a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' f "
( _V  \  {  .0.  } ) )  C_  ( `' f " ( _V  \  {  .0.  }
) ) )
8668, 85suppssrOLD 6022 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) ) )  ->  ( f `  k )  =  .0.  )
8783, 86sylan2 474 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( ( I  \  { x } ) 
\  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
f `  k )  =  .0.  )
8887suppss2OLD 6525 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' ( k  e.  ( I  \  { x } ) 
|->  ( f `  k
) ) " ( _V  \  {  .0.  }
) )  C_  ( `' f " ( _V  \  {  .0.  }
) ) )
89 ssfi 7752 . . . . . . . . . . . . . 14  |-  ( ( ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' f
" ( _V  \  {  .0.  } ) ) )  ->  ( `' ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
9080, 88, 89syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' ( k  e.  ( I  \  { x } ) 
|->  ( f `  k
) ) " ( _V  \  {  .0.  }
) )  e.  Fin )
9158, 62, 67, 79, 90dprdwdOLD 16921 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) )  e.  { h  e.  X_ i  e.  (
I  \  { x } ) ( ( S  |`  ( I  \  { x } ) ) `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin } )
9273, 91eqeltrd 2555 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  e. 
{ h  e.  X_ i  e.  ( I  \  { x } ) ( ( S  |`  ( I  \  { x } ) ) `  i )  |  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin } )
933, 58, 62, 67, 92eldprdiOLD 16935 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  (
I  \  { x } ) ) )  e.  ( G DProd  ( S  |`  ( I  \  { x } ) ) ) )
9421, 93sylan2 474 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( G  gsumg  ( f  |`  (
I  \  { x } ) ) )  e.  ( G DProd  ( S  |`  ( I  \  { x } ) ) ) )
9557, 94eqeltrd 2555 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  e.  ( G DProd  ( S  |`  ( I  \  {
x } ) ) ) )
96 eqid 2467 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
97 eqid 2467 . . . . . . . . . 10  |-  ( LSSum `  G )  =  (
LSSum `  G )
9863, 15ffvelrnd 6033 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S `  x
)  e.  (SubGrp `  G ) )
99 dprdsubg 16941 . . . . . . . . . . 11  |-  ( G dom DProd  ( S  |`  ( I  \  { x } ) )  -> 
( G DProd  ( S  |`  ( I  \  {
x } ) ) )  e.  (SubGrp `  G ) )
10062, 99syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G DProd  ( S  |`  ( I  \  {
x } ) ) )  e.  (SubGrp `  G ) )
10112, 13, 15, 3dpjdisj 16972 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( S `  x )  i^i  ( G DProd  ( S  |`  (
I  \  { x } ) ) ) )  =  {  .0.  } )
10212, 13, 15, 28dpjcntz 16971 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S `  x
)  C_  ( (Cntz `  G ) `  ( G DProd  ( S  |`  (
I  \  { x } ) ) ) ) )
10396, 97, 3, 28, 98, 100, 101, 102, 22pj1rid 16591 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  A  e.  ( G DProd  ( S  |`  ( I  \  {
x } ) ) ) )  ->  (
( ( S `  x ) ( proj1 `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) `  A
)  =  .0.  )
10421, 103sylanl2 651 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f " ( _V 
\  {  .0.  }
) ) ) )  /\  A  e.  ( G DProd  ( S  |`  ( I  \  { x } ) ) ) )  ->  ( (
( S `  x
) ( proj1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A )  =  .0.  )
10595, 104mpdan 668 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
( ( S `  x ) ( proj1 `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) `  A
)  =  .0.  )
10625, 105eqtrd 2508 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
( P `  x
) `  A )  =  .0.  )
107106suppss2OLD 6525 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' ( x  e.  I  |->  ( ( P `  x
) `  A )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' f " ( _V  \  {  .0.  }
) ) )
108 ssfi 7752 . . . . 5  |-  ( ( ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( x  e.  I  |->  ( ( P `  x ) `  A
) ) " ( _V  \  {  .0.  }
) )  C_  ( `' f " ( _V  \  {  .0.  }
) ) )  -> 
( `' ( x  e.  I  |->  ( ( P `  x ) `
 A ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
10920, 107, 108syl2anc 661 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' ( x  e.  I  |->  ( ( P `  x
) `  A )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
1104, 10, 11, 18, 109dprdwdOLD 16921 . . 3  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e.  W )
111 simprr 756 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  A  =  ( G  gsumg  f ) )
11239feqmptd 5927 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  =  ( x  e.  I  |->  ( f `  x ) ) )
113 simplrr 760 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  =  ( G 
gsumg  f ) )
11412, 29, 303syl 20 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G  e.  Mnd )
11512, 34, 353syl 20 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  dom  S  e.  _V )
11613, 115eqeltrrd 2556 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  I  e.  _V )
1174, 12, 13, 41dprdffiOLD 16924 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin )
118 disjdif 3905 . . . . . . . . . . . . 13  |-  ( { x }  i^i  (
I  \  { x } ) )  =  (/)
119118a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( { x }  i^i  ( I  \  {
x } ) )  =  (/) )
120 undif2 3909 . . . . . . . . . . . . 13  |-  ( { x }  u.  (
I  \  { x } ) )  =  ( { x }  u.  I )
12115snssd 4178 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  { x }  C_  I )
122 ssequn1 3679 . . . . . . . . . . . . . 14  |-  ( { x }  C_  I  <->  ( { x }  u.  I )  =  I )
123121, 122sylib 196 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( { x }  u.  I )  =  I )
124120, 123syl5req 2521 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  I  =  ( { x }  u.  (
I  \  { x } ) ) )
12527, 3, 96, 28, 114, 116, 68, 42, 117, 119, 124gsumzsplitOLD 16816 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  f )  =  ( ( G  gsumg  ( f  |`  { x } ) ) ( +g  `  G ) ( G  gsumg  ( f  |`  (
I  \  { x } ) ) ) ) )
12668, 121feqresmpt 5928 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  { x } )  =  ( k  e.  { x }  |->  ( f `  k ) ) )
127126oveq2d 6311 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  { x } ) )  =  ( G  gsumg  ( k  e.  {
x }  |->  ( f `
 k ) ) ) )
12868, 15ffvelrnd 6033 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f `  x
)  e.  ( Base `  G ) )
129 fveq2 5872 . . . . . . . . . . . . . . 15  |-  ( k  =  x  ->  (
f `  k )  =  ( f `  x ) )
13027, 129gsumsn 16852 . . . . . . . . . . . . . 14  |-  ( ( G  e.  Mnd  /\  x  e.  I  /\  ( f `  x
)  e.  ( Base `  G ) )  -> 
( G  gsumg  ( k  e.  {
x }  |->  ( f `
 k ) ) )  =  ( f `
 x ) )
131114, 15, 128, 130syl3anc 1228 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( k  e.  {
x }  |->  ( f `
 k ) ) )  =  ( f `
 x ) )
132127, 131eqtrd 2508 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  { x } ) )  =  ( f `  x
) )
133132oveq1d 6310 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( G  gsumg  ( f  |`  { x } ) ) ( +g  `  G
) ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) )  =  ( ( f `  x
) ( +g  `  G
) ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) ) )
134113, 125, 1333eqtrd 2512 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  =  ( ( f `  x ) ( +g  `  G
) ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) ) )
13512, 13, 15, 97dpjlsm 16973 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G DProd  S )  =  ( ( S `
 x ) (
LSSum `  G ) ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) )
13617, 135eleqtrd 2557 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  e.  ( ( S `  x ) ( LSSum `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) )
1374, 10, 11, 19dprdfclOLD 16923 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f `  x
)  e.  ( S `
 x ) )
13896, 97, 3, 28, 98, 100, 101, 102, 22, 136, 137, 93pj1eq 16589 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( A  =  ( ( f `  x
) ( +g  `  G
) ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) )  <->  ( (
( ( S `  x ) ( proj1 `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) `  A
)  =  ( f `
 x )  /\  ( ( ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ( proj1 `  G ) ( S `
 x ) ) `
 A )  =  ( G  gsumg  ( f  |`  (
I  \  { x } ) ) ) ) ) )
139134, 138mpbid 210 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( ( ( S `  x ) ( proj1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A )  =  ( f `  x )  /\  (
( ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ( proj1 `  G )
( S `  x
) ) `  A
)  =  ( G 
gsumg  ( f  |`  (
I  \  { x } ) ) ) ) )
140139simpld 459 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( ( S `
 x ) (
proj1 `  G
) ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) `  A )  =  ( f `  x ) )
14124, 140eqtrd 2508 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  =  ( f `
 x ) )
142141mpteq2dva 4539 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  =  ( x  e.  I  |->  ( f `  x
) ) )
143112, 142eqtr4d 2511 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  =  ( x  e.  I  |->  ( ( P `  x
) `  A )
) )
144143oveq2d 6311 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( G  gsumg  f )  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) ) )
145111, 144eqtrd 2508 . . 3  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A
) ) ) )
146110, 145jca 532 . 2  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( ( x  e.  I  |->  ( ( P `  x ) `
 A ) )  e.  W  /\  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) ) ) )
1478, 146rexlimddv 2963 1  |-  ( ph  ->  ( ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e.  W  /\  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818   {crab 2821   _Vcvv 3118    \ cdif 3478    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   {csn 4033   class class class wbr 4453    |-> cmpt 4511   `'ccnv 5004   dom cdm 5005   ran crn 5006    |` cres 5007   "cima 5008   -->wf 5590   ` cfv 5594  (class class class)co 6295   X_cixp 7481   Fincfn 7528   Basecbs 14506   +g cplusg 14571   0gc0g 14711    gsumg cgsu 14712   Mndcmnd 15792   Grpcgrp 15924  SubGrpcsubg 16066  Cntzccntz 16224   LSSumclsm 16525   proj1cpj1 16526   DProd cdprd 16895  dProjcdpj 16896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-tpos 6967  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-seq 12088  df-hash 12386  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-0g 14713  df-gsum 14714  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15745  df-sgrp 15784  df-mnd 15794  df-mhm 15838  df-submnd 15839  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mulg 15931  df-subg 16069  df-ghm 16136  df-gim 16178  df-cntz 16226  df-oppg 16252  df-lsm 16527  df-pj1 16528  df-cmn 16671  df-dprd 16897  df-dpj 16898
This theorem is referenced by:  dpjeqOLD  16985
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