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Theorem dpjidclOLD 17240
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 17233 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 17163 . . . . 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 17232 . . . . 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 756 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  e.  W
)
204, 10, 11, 19dprdffiOLD 17180 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' f
" ( _V  \  {  .0.  } ) )  e.  Fin )
21 eldifi 3622 . . . . . . . 8  |-  ( x  e.  ( I  \ 
( `' f "
( _V  \  {  .0.  } ) ) )  ->  x  e.  I
)
22 eqid 2457 . . . . . . . . . 10  |-  ( proj1 `  G )  =  ( proj1 `  G )
2312, 13, 14, 22, 15dpjval 17231 . . . . . . . . 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 762 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  =  ( G  gsumg  f ) )
27 eqid 2457 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
28 eqid 2457 . . . . . . . . . . 11  |-  (Cntz `  G )  =  (Cntz `  G )
29 dprdgrp 17164 . . . . . . . . . . . . 13  |-  ( G dom DProd  S  ->  G  e. 
Grp )
30 grpmnd 16188 . . . . . . . . . . . . 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 17154 . . . . . . . . . . . . . . 15  |-  Rel  dom DProd
3433brrelex2i 5050 . . . . . . . . . . . . . 14  |-  ( G dom DProd  S  ->  S  e. 
_V )
35 dmexg 6730 . . . . . . . . . . . . . 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 2546 . . . . . . . . . . . 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 17178 . . . . . . . . . . . 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 17181 . . . . . . . . . . . 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 4176 . . . . . . . . . . . . 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 3630 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  { x }  C_  I )
47 cnvimass 5367 . . . . . . . . . . . . . . 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 3547 . . . . . . . . . . . . . 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 3633 . . . . . . . . . . . . 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 17044 . . . . . . . . . 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 2501 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  =  ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) )
58 eqid 2457 . . . . . . . . . . 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 3627 . . . . . . . . . . . . . 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 17201 . . . . . . . . . . . 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 17166 . . . . . . . . . . . . 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 5926 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f  =  ( k  e.  I  |->  ( f `
 k ) ) )
7069reseq1d 5282 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  =  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) ) )
71 resmpt 5333 . . . . . . . . . . . . . 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 2514 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  =  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) )
74 eldifi 3622 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( I  \  { x } )  ->  k  e.  I
)
754, 12, 13, 41dprdfclOLD 17179 . . . . . . . . . . . . . . 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 2548 . . . . . . . . . . . . 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 3635 . . . . . . . . . . . . . . . . . 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 3495 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( ( I 
\  { x }
)  \  ( `' f " ( _V  \  {  .0.  } ) ) )  ->  k  e.  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) ) )
84 ssid 3518 . . . . . . . . . . . . . . . . . 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 6529 . . . . . . . . . . . . . 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 7759 . . . . . . . . . . . . . 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 17177 . . . . . . . . . . . 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 2545 . . . . . . . . . . 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 17191 . . . . . . . . . 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 2545 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  e.  ( G DProd  ( S  |`  ( I  \  {
x } ) ) ) )
96 eqid 2457 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
97 eqid 2457 . . . . . . . . . 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 17197 . . . . . . . . . . 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 17228 . . . . . . . . . 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 17227 . . . . . . . . . 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 16846 . . . . . . . . 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 2498 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
( P `  x
) `  A )  =  .0.  )
107106suppss2OLD 6529 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' ( x  e.  I  |->  ( ( P `  x
) `  A )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' f " ( _V  \  {  .0.  }
) ) )
108 ssfi 7759 . . . . 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 17177 . . 3  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e.  W )
111 simprr 757 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  A  =  ( G  gsumg  f ) )
11239feqmptd 5926 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  =  ( x  e.  I  |->  ( f `  x ) ) )
113 simplrr 762 . . . . . . . . . . 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 2546 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  I  e.  _V )
1174, 12, 13, 41dprdffiOLD 17180 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin )
118 disjdif 3903 . . . . . . . . . . . . 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 3907 . . . . . . . . . . . . 13  |-  ( { x }  u.  (
I  \  { x } ) )  =  ( { x }  u.  I )
12115snssd 4177 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  { x }  C_  I )
122 ssequn1 3670 . . . . . . . . . . . . . 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 2511 . . . . . . . . . . . 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 17071 . . . . . . . . . . 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 5927 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  { x } )  =  ( k  e.  { x }  |->  ( f `  k ) ) )
127126oveq2d 6312 . . . . . . . . . . . . 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 17107 . . . . . . . . . . . . . 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 2498 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  { x } ) )  =  ( f `  x
) )
133132oveq1d 6311 . . . . . . . . . . 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 2502 . . . . . . . . . 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 17229 . . . . . . . . . . . 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 2547 . . . . . . . . . . 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 17179 . . . . . . . . . . 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 16844 . . . . . . . . . 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 2498 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  =  ( f `
 x ) )
142141mpteq2dva 4543 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  =  ( x  e.  I  |->  ( f `  x
) ) )
143112, 142eqtr4d 2501 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  =  ( x  e.  I  |->  ( ( P `  x
) `  A )
) )
144143oveq2d 6312 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( G  gsumg  f )  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) ) )
145111, 144eqtrd 2498 . . 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 2953 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 1395    e. wcel 1819   E.wrex 2808   {crab 2811   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   {csn 4032   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296   X_cixp 7488   Fincfn 7535   Basecbs 14643   +g cplusg 14711   0gc0g 14856    gsumg cgsu 14857   Mndcmnd 16045   Grpcgrp 16179  SubGrpcsubg 16321  Cntzccntz 16479   LSSumclsm 16780   proj1cpj1 16781   DProd cdprd 17150  dProjcdpj 17151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11821  df-seq 12110  df-hash 12408  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-0g 14858  df-gsum 14859  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-mhm 16092  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-mulg 16186  df-subg 16324  df-ghm 16391  df-gim 16433  df-cntz 16481  df-oppg 16507  df-lsm 16782  df-pj1 16783  df-cmn 16926  df-dprd 17152  df-dpj 17153
This theorem is referenced by:  dpjeqOLD  17241
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