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Theorem dpjidcl 15571
Description: The key property of projections: the sum of all the projections of  A is  A. (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
dpjfval.1  |-  ( ph  ->  G dom DProd  S )
dpjfval.2  |-  ( ph  ->  dom  S  =  I )
dpjfval.p  |-  P  =  ( GdProj S )
dpjidcl.3  |-  ( ph  ->  A  e.  ( G DProd 
S ) )
dpjidcl.0  |-  .0.  =  ( 0g `  G )
dpjidcl.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
Assertion
Ref Expression
dpjidcl  |-  ( 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 dpjidcl
Dummy variables  k 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dpjidcl.3 . . . 4  |-  ( ph  ->  A  e.  ( G DProd 
S ) )
2 dpjfval.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
3 dpjidcl.0 . . . . . 6  |-  .0.  =  ( 0g `  G )
4 dpjidcl.w . . . . . 6  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
53, 4eldprd 15517 . . . . 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 202 . . 3  |-  ( ph  ->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
87simprd 450 . 2  |-  ( ph  ->  E. f  e.  W  A  =  ( G  gsumg  f ) )
9 dpjfval.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
109adantr 452 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  G dom DProd  S )
112adantr 452 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  dom  S  =  I )
129ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G dom DProd  S )
132ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  dom  S  =  I )
14 dpjfval.p . . . . . 6  |-  P  =  ( GdProj S )
15 simpr 448 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  x  e.  I )
1612, 13, 14, 15dpjf 15570 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( P `  x
) : ( G DProd 
S ) --> ( S `
 x ) )
171ad2antrr 707 . . . . 5  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  e.  ( G DProd 
S ) )
1816, 17ffvelrnd 5830 . . . 4  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  e.  ( S `
 x ) )
19 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  e.  W
)
204, 10, 11, 19dprdffi 15527 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' f
" ( _V  \  {  .0.  } ) )  e.  Fin )
21 eldifi 3429 . . . . . . . 8  |-  ( x  e.  ( I  \ 
( `' f "
( _V  \  {  .0.  } ) ) )  ->  x  e.  I
)
22 eqid 2404 . . . . . . . . . 10  |-  ( proj
1 `  G )  =  ( proj 1 `  G )
2312, 13, 14, 22, 15dpjval 15569 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( P `  x
)  =  ( ( S `  x ) ( proj 1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) )
2423fveq1d 5689 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  =  ( ( ( S `  x
) ( proj 1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A ) )
2521, 24sylan2 461 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
( P `  x
) `  A )  =  ( ( ( S `  x ) ( proj 1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A ) )
26 simplrr 738 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  =  ( G  gsumg  f ) )
27 eqid 2404 . . . . . . . . . . 11  |-  ( Base `  G )  =  (
Base `  G )
28 eqid 2404 . . . . . . . . . . 11  |-  (Cntz `  G )  =  (Cntz `  G )
29 dprdgrp 15518 . . . . . . . . . . . . 13  |-  ( G dom DProd  S  ->  G  e. 
Grp )
30 grpmnd 14772 . . . . . . . . . . . . 13  |-  ( G  e.  Grp  ->  G  e.  Mnd )
3110, 29, 303syl 19 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  G  e.  Mnd )
3231adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  G  e.  Mnd )
33 reldmdprd 15513 . . . . . . . . . . . . . . 15  |-  Rel  dom DProd
3433brrelex2i 4878 . . . . . . . . . . . . . 14  |-  ( G dom DProd  S  ->  S  e. 
_V )
35 dmexg 5089 . . . . . . . . . . . . . 14  |-  ( S  e.  _V  ->  dom  S  e.  _V )
3610, 34, 353syl 19 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  dom  S  e.  _V )
3711, 36eqeltrrd 2479 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  I  e.  _V )
3837adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  I  e.  _V )
394, 10, 11, 19, 27dprdff 15525 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f : I --> ( Base `  G
) )
4039adantr 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  f : I --> ( Base `  G ) )
4119adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f  e.  W )
424, 12, 13, 41, 28dprdfcntz 15528 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ran  f  C_  (
(Cntz `  G ) `  ran  f ) )
4321, 42sylan2 461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ran  f  C_  ( (Cntz `  G ) `  ran  f ) )
44 snssi 3902 . . . . . . . . . . . . 13  |-  ( x  e.  ( I  \ 
( `' f "
( _V  \  {  .0.  } ) ) )  ->  { x }  C_  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) ) )
4544adantl 453 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  { x }  C_  ( I  \ 
( `' f "
( _V  \  {  .0.  } ) ) ) )
4645difss2d 3437 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  { x }  C_  I )
47 cnvimass 5183 . . . . . . . . . . . . . . 15  |-  ( `' f " ( _V 
\  {  .0.  }
) )  C_  dom  f
48 fdm 5554 . . . . . . . . . . . . . . . 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 3356 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' f
" ( _V  \  {  .0.  } ) ) 
C_  I )
5150adantr 452 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( `' f " ( _V  \  {  .0.  }
) )  C_  I
)
52 ssconb 3440 . . . . . . . . . . . . 13  |-  ( ( { x }  C_  I  /\  ( `' f
" ( _V  \  {  .0.  } ) ) 
C_  I )  -> 
( { x }  C_  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) )  <->  ( `' f " ( _V  \  {  .0.  } ) ) 
C_  ( I  \  { x } ) ) )
5346, 51, 52syl2anc 643 . . . . . . . . . . . 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 202 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( `' f " ( _V  \  {  .0.  }
) )  C_  (
I  \  { x } ) )
5520adantr 452 . . . . . . . . . . 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, 55gsumzres 15472 . . . . . . . . . 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 2439 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  =  ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) )
58 eqid 2404 . . . . . . . . . . 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 3434 . . . . . . . . . . . . . 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 15541 . . . . . . . . . . . 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 446 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G dom DProd  ( S  |`  ( I  \  {
x } ) ) )
6312, 13dprdf2 15520 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
64 fssres 5569 . . . . . . . . . . . . 13  |-  ( ( S : I --> (SubGrp `  G )  /\  (
I  \  { x } )  C_  I
)  ->  ( S  |`  ( I  \  {
x } ) ) : ( I  \  { x } ) --> (SubGrp `  G )
)
6563, 59, 64sylancl 644 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S  |`  (
I  \  { x } ) ) : ( I  \  {
x } ) --> (SubGrp `  G ) )
66 fdm 5554 . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f : I --> ( Base `  G ) )
6968feqmptd 5738 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f  =  ( k  e.  I  |->  ( f `
 k ) ) )
7069reseq1d 5104 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  =  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) ) )
71 resmpt 5150 . . . . . . . . . . . . . 14  |-  ( ( I  \  { x } )  C_  I  ->  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) )  =  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) )
7259, 71ax-mp 8 . . . . . . . . . . . . 13  |-  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) )  =  ( k  e.  ( I  \  { x } ) 
|->  ( f `  k
) )
7370, 72syl6eq 2452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  =  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) )
74 eldifi 3429 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( I  \  { x } )  ->  k  e.  I
)
754, 12, 13, 41dprdfcl 15526 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  I )  ->  (
f `  k )  e.  ( S `  k
) )
7674, 75sylan2 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( f `  k
)  e.  ( S `
 k ) )
77 fvres 5704 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( I  \  { x } )  ->  ( ( S  |`  ( I  \  {
x } ) ) `
 k )  =  ( S `  k
) )
7877adantl 453 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( ( S  |`  ( I  \  { x } ) ) `  k )  =  ( S `  k ) )
7976, 78eleqtrrd 2481 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( f `  k
)  e.  ( ( S  |`  ( I  \  { x } ) ) `  k ) )
8020adantr 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin )
81 ssdif 3442 . . . . . . . . . . . . . . . . . 18  |-  ( ( I  \  { x } )  C_  I  ->  ( ( I  \  { x } ) 
\  ( `' f
" ( _V  \  {  .0.  } ) ) )  C_  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )
8259, 81ax-mp 8 . . . . . . . . . . . . . . . . 17  |-  ( ( I  \  { x } )  \  ( `' f " ( _V  \  {  .0.  }
) ) )  C_  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) )
8382sseli 3304 . . . . . . . . . . . . . . . 16  |-  ( k  e.  ( ( I 
\  { x }
)  \  ( `' f " ( _V  \  {  .0.  } ) ) )  ->  k  e.  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) ) )
84 ssid 3327 . . . . . . . . . . . . . . . . . 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, 85suppssr 5823 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) ) )  ->  ( f `  k )  =  .0.  )
8783, 86sylan2 461 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( ( I  \  { x } ) 
\  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
f `  k )  =  .0.  )
8887suppss2 6259 . . . . . . . . . . . . . 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 7288 . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . 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, 90dprdwd 15524 . . . . . . . . . . . 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 2478 . . . . . . . . . . 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, 92eldprdi 15531 . . . . . . . . . 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 461 . . . . . . . . 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 2478 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  e.  ( G DProd  ( S  |`  ( I  \  {
x } ) ) ) )
96 eqid 2404 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
97 eqid 2404 . . . . . . . . . 10  |-  ( LSSum `  G )  =  (
LSSum `  G )
9863, 15ffvelrnd 5830 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S `  x
)  e.  (SubGrp `  G ) )
99 dprdsubg 15537 . . . . . . . . . . 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 15566 . . . . . . . . . 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 15565 . . . . . . . . . 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 15289 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  A  e.  ( G DProd  ( S  |`  ( I  \  {
x } ) ) ) )  ->  (
( ( S `  x ) ( proj
1 `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) `  A
)  =  .0.  )
10421, 103sylanl2 633 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f " ( _V 
\  {  .0.  }
) ) ) )  /\  A  e.  ( G DProd  ( S  |`  ( I  \  { x } ) ) ) )  ->  ( (
( S `  x
) ( proj 1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A )  =  .0.  )
10595, 104mpdan 650 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
( ( S `  x ) ( proj
1 `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) `  A
)  =  .0.  )
10625, 105eqtrd 2436 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
( P `  x
) `  A )  =  .0.  )
107106suppss2 6259 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' ( x  e.  I  |->  ( ( P `  x
) `  A )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' f " ( _V  \  {  .0.  }
) ) )
108 ssfi 7288 . . . . 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 643 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' ( x  e.  I  |->  ( ( P `  x
) `  A )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
1104, 10, 11, 18, 109dprdwd 15524 . . 3  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e.  W )
111 simprr 734 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  A  =  ( G  gsumg  f ) )
11239feqmptd 5738 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  =  ( x  e.  I  |->  ( f `  x ) ) )
113 simplrr 738 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  =  ( G 
gsumg  f ) )
11412, 29, 303syl 19 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G  e.  Mnd )
11512, 34, 353syl 19 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  dom  S  e.  _V )
11613, 115eqeltrrd 2479 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  I  e.  _V )
1174, 12, 13, 41dprdffi 15527 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin )
118 disjdif 3660 . . . . . . . . . . . . 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 3664 . . . . . . . . . . . . 13  |-  ( { x }  u.  (
I  \  { x } ) )  =  ( { x }  u.  I )
12115snssd 3903 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  { x }  C_  I )
122 ssequn1 3477 . . . . . . . . . . . . . 14  |-  ( { x }  C_  I  <->  ( { x }  u.  I )  =  I )
123121, 122sylib 189 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( { x }  u.  I )  =  I )
124120, 123syl5req 2449 . . . . . . . . . . . 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, 124gsumzsplit 15484 . . . . . . . . . . 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 5739 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  { x } )  =  ( k  e.  { x }  |->  ( f `  k ) ) )
127126oveq2d 6056 . . . . . . . . . . . . 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 5830 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f `  x
)  e.  ( Base `  G ) )
129 fveq2 5687 . . . . . . . . . . . . . . 15  |-  ( k  =  x  ->  (
f `  k )  =  ( f `  x ) )
13027, 129gsumsn 15498 . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( k  e.  {
x }  |->  ( f `
 k ) ) )  =  ( f `
 x ) )
132127, 131eqtrd 2436 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  { x } ) )  =  ( f `  x
) )
133132oveq1d 6055 . . . . . . . . . . 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 2440 . . . . . . . . . 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 15567 . . . . . . . . . . . 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 2480 . . . . . . . . . . 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, 19dprdfcl 15526 . . . . . . . . . . 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 15287 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( A  =  ( ( f `  x
) ( +g  `  G
) ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) )  <->  ( (
( ( S `  x ) ( proj
1 `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) `  A
)  =  ( f `
 x )  /\  ( ( ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ( proj 1 `  G ) ( S `
 x ) ) `
 A )  =  ( G  gsumg  ( f  |`  (
I  \  { x } ) ) ) ) ) )
139134, 138mpbid 202 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( ( ( S `  x ) ( proj 1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) `  A )  =  ( f `  x )  /\  (
( ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ( proj
1 `  G )
( S `  x
) ) `  A
)  =  ( G 
gsumg  ( f  |`  (
I  \  { x } ) ) ) ) )
140139simpld 446 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( ( S `
 x ) (
proj 1 `  G ) ( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) `  A
)  =  ( f `
 x ) )
14124, 140eqtrd 2436 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  =  ( f `
 x ) )
142141mpteq2dva 4255 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  =  ( x  e.  I  |->  ( f `  x
) ) )
143112, 142eqtr4d 2439 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  =  ( x  e.  I  |->  ( ( P `  x
) `  A )
) )
144143oveq2d 6056 . . . 4  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( G  gsumg  f )  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) ) )
145111, 144eqtrd 2436 . . 3  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A
) ) ) )
146110, 145jca 519 . 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 2794 1  |-  ( ph  ->  ( ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e.  W  /\  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A
) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   {crab 2670   _Vcvv 2916    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   class class class wbr 4172    e. cmpt 4226   `'ccnv 4836   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   -->wf 5409   ` cfv 5413  (class class class)co 6040   X_cixp 7022   Fincfn 7068   Basecbs 13424   +g cplusg 13484   0gc0g 13678    gsumg cgsu 13679   Mndcmnd 14639   Grpcgrp 14640  SubGrpcsubg 14893  Cntzccntz 15069   LSSumclsm 15223   proj
1cpj1 15224   DProd cdprd 15509  dProjcdpj 15510
This theorem is referenced by:  dpjeq  15572  dpjid  15573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-gim 15001  df-cntz 15071  df-oppg 15097  df-lsm 15225  df-pj1 15226  df-cmn 15369  df-dprd 15511  df-dpj 15512
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