MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dpjidcl Unicode version

Theorem dpjidcl 15293
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 15239 . . . . 5  |-  ( dom 
S  =  I  -> 
( A  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) ) )
62, 5syl 15 . . . 4  |-  ( ph  ->  ( A  e.  ( G DProd  S )  <->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) ) )
71, 6mpbid 201 . . 3  |-  ( ph  ->  ( G dom DProd  S  /\  E. f  e.  W  A  =  ( G  gsumg  f ) ) )
87simprd 449 . 2  |-  ( ph  ->  E. f  e.  W  A  =  ( G  gsumg  f ) )
9 dpjfval.1 . . . . . . 7  |-  ( ph  ->  G dom DProd  S )
109adantr 451 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  G dom DProd  S )
112adantr 451 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  dom  S  =  I )
129ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G dom DProd  S )
132ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  dom  S  =  I )
14 dpjfval.p . . . . . . . 8  |-  P  =  ( GdProj S )
15 simpr 447 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  x  e.  I )
1612, 13, 14, 15dpjf 15292 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( P `  x
) : ( G DProd 
S ) --> ( S `
 x ) )
171ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  e.  ( G DProd 
S ) )
18 ffvelrn 5663 . . . . . . 7  |-  ( ( ( P `  x
) : ( G DProd 
S ) --> ( S `
 x )  /\  A  e.  ( G DProd  S ) )  ->  (
( P `  x
) `  A )  e.  ( S `  x
) )
1916, 17, 18syl2anc 642 . . . . . 6  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  e.  ( S `
 x ) )
20 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  e.  W
)
214, 10, 11, 20dprdffi 15249 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' f
" ( _V  \  {  .0.  } ) )  e.  Fin )
22 eldifi 3298 . . . . . . . . . 10  |-  ( x  e.  ( I  \ 
( `' f "
( _V  \  {  .0.  } ) ) )  ->  x  e.  I
)
23 eqid 2283 . . . . . . . . . . . 12  |-  ( proj
1 `  G )  =  ( proj 1 `  G )
2412, 13, 14, 23, 15dpjval 15291 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( P `  x
)  =  ( ( S `  x ) ( proj 1 `  G ) ( G DProd 
( S  |`  (
I  \  { x } ) ) ) ) )
2524fveq1d 5527 . . . . . . . . . 10  |-  ( ( ( 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 ) )
2622, 25sylan2 460 . . . . . . . . 9  |-  ( ( ( 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 ) )
27 simplrr 737 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  =  ( G  gsumg  f ) )
28 eqid 2283 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
29 eqid 2283 . . . . . . . . . . . . 13  |-  (Cntz `  G )  =  (Cntz `  G )
30 dprdgrp 15240 . . . . . . . . . . . . . . 15  |-  ( G dom DProd  S  ->  G  e. 
Grp )
31 grpmnd 14494 . . . . . . . . . . . . . . 15  |-  ( G  e.  Grp  ->  G  e.  Mnd )
3210, 30, 313syl 18 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  G  e.  Mnd )
3332adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  G  e.  Mnd )
34 reldmdprd 15235 . . . . . . . . . . . . . . . . 17  |-  Rel  dom DProd
3534brrelex2i 4730 . . . . . . . . . . . . . . . 16  |-  ( G dom DProd  S  ->  S  e. 
_V )
36 dmexg 4939 . . . . . . . . . . . . . . . 16  |-  ( S  e.  _V  ->  dom  S  e.  _V )
3710, 35, 363syl 18 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  dom  S  e.  _V )
3811, 37eqeltrrd 2358 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  I  e.  _V )
3938adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  I  e.  _V )
404, 10, 11, 20, 28dprdff 15247 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f : I --> ( Base `  G
) )
4140adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  f : I --> ( Base `  G ) )
4220adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f  e.  W )
434, 12, 13, 42, 29dprdfcntz 15250 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ran  f  C_  (
(Cntz `  G ) `  ran  f ) )
4422, 43sylan2 460 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ran  f  C_  ( (Cntz `  G ) `  ran  f ) )
45 snssi 3759 . . . . . . . . . . . . . . 15  |-  ( x  e.  ( I  \ 
( `' f "
( _V  \  {  .0.  } ) ) )  ->  { x }  C_  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) ) )
4645adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  { x }  C_  ( I  \ 
( `' f "
( _V  \  {  .0.  } ) ) ) )
47 difss 3303 . . . . . . . . . . . . . . . 16  |-  ( I 
\  ( `' f
" ( _V  \  {  .0.  } ) ) )  C_  I
4846, 47syl6ss 3191 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  { x }  C_  I )
49 cnvimass 5033 . . . . . . . . . . . . . . . . 17  |-  ( `' f " ( _V 
\  {  .0.  }
) )  C_  dom  f
50 fdm 5393 . . . . . . . . . . . . . . . . . 18  |-  ( f : I --> ( Base `  G )  ->  dom  f  =  I )
5140, 50syl 15 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  dom  f  =  I )
5249, 51syl5sseq 3226 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' f
" ( _V  \  {  .0.  } ) ) 
C_  I )
5352adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( `' f " ( _V  \  {  .0.  }
) )  C_  I
)
54 ssconb 3309 . . . . . . . . . . . . . . 15  |-  ( ( { x }  C_  I  /\  ( `' f
" ( _V  \  {  .0.  } ) ) 
C_  I )  -> 
( { x }  C_  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) )  <->  ( `' f " ( _V  \  {  .0.  } ) ) 
C_  ( I  \  { x } ) ) )
5548, 53, 54syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( 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 } ) ) )
5646, 55mpbid 201 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( `' f " ( _V  \  {  .0.  }
) )  C_  (
I  \  { x } ) )
5721adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( `' f " ( _V  \  {  .0.  }
) )  e.  Fin )
5828, 3, 29, 33, 39, 41, 44, 56, 57gsumzres 15194 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  ( G  gsumg  ( f  |`  (
I  \  { x } ) ) )  =  ( G  gsumg  f ) )
5927, 58eqtr4d 2318 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  =  ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) )
60 eqid 2283 . . . . . . . . . . . . 13  |-  { 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 }
61 difss 3303 . . . . . . . . . . . . . . . 16  |-  ( I 
\  { x }
)  C_  I
6261a1i 10 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( I  \  {
x } )  C_  I )
6312, 13, 62dprdres 15263 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) )
6463simpld 445 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G dom DProd  ( S  |`  ( I  \  {
x } ) ) )
6512, 13dprdf2 15242 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  S : I --> (SubGrp `  G ) )
66 fssres 5408 . . . . . . . . . . . . . . 15  |-  ( ( S : I --> (SubGrp `  G )  /\  (
I  \  { x } )  C_  I
)  ->  ( S  |`  ( I  \  {
x } ) ) : ( I  \  { x } ) --> (SubGrp `  G )
)
6765, 61, 66sylancl 643 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S  |`  (
I  \  { x } ) ) : ( I  \  {
x } ) --> (SubGrp `  G ) )
68 fdm 5393 . . . . . . . . . . . . . 14  |-  ( ( S  |`  ( I  \  { x } ) ) : ( I 
\  { x }
) --> (SubGrp `  G )  ->  dom  ( S  |`  ( I  \  { x } ) )  =  ( I  \  {
x } ) )
6967, 68syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  dom  ( S  |`  ( I  \  { x } ) )  =  ( I  \  {
x } ) )
7040adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f : I --> ( Base `  G ) )
7170feqmptd 5575 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  f  =  ( k  e.  I  |->  ( f `
 k ) ) )
7271reseq1d 4954 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  =  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) ) )
73 resmpt 5000 . . . . . . . . . . . . . . . 16  |-  ( ( I  \  { x } )  C_  I  ->  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) )  =  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) )
7461, 73ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  I  |->  ( f `  k ) )  |`  ( I  \  { x } ) )  =  ( k  e.  ( I  \  { x } ) 
|->  ( f `  k
) )
7572, 74syl6eq 2331 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  (
I  \  { x } ) )  =  ( k  e.  ( I  \  { x } )  |->  ( f `
 k ) ) )
76 eldifi 3298 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( I  \  { x } )  ->  k  e.  I
)
774, 12, 13, 42dprdfcl 15248 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  I )  ->  (
f `  k )  e.  ( S `  k
) )
7876, 77sylan2 460 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( f `  k
)  e.  ( S `
 k ) )
79 fvres 5542 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  ( I  \  { x } )  ->  ( ( S  |`  ( I  \  {
x } ) ) `
 k )  =  ( S `  k
) )
8079adantl 452 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( ( S  |`  ( I  \  { x } ) ) `  k )  =  ( S `  k ) )
8178, 80eleqtrrd 2360 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  { x } ) )  -> 
( f `  k
)  e.  ( ( S  |`  ( I  \  { x } ) ) `  k ) )
8221adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin )
83 ssdif 3311 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( I  \  { x } )  C_  I  ->  ( ( I  \  { x } ) 
\  ( `' f
" ( _V  \  {  .0.  } ) ) )  C_  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )
8461, 83ax-mp 8 . . . . . . . . . . . . . . . . . . 19  |-  ( ( I  \  { x } )  \  ( `' f " ( _V  \  {  .0.  }
) ) )  C_  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) )
8584sseli 3176 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  ( ( I 
\  { x }
)  \  ( `' f " ( _V  \  {  .0.  } ) ) )  ->  k  e.  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) ) )
86 ssid 3197 . . . . . . . . . . . . . . . . . . . 20  |-  ( `' f " ( _V 
\  {  .0.  }
) )  C_  ( `' f " ( _V  \  {  .0.  }
) )
8786a1i 10 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' f "
( _V  \  {  .0.  } ) )  C_  ( `' f " ( _V  \  {  .0.  }
) ) )
8870, 87suppssr 5659 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( I  \  ( `' f " ( _V  \  {  .0.  }
) ) ) )  ->  ( f `  k )  =  .0.  )
8985, 88sylan2 460 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I
)  /\  k  e.  ( ( I  \  { x } ) 
\  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
f `  k )  =  .0.  )
9089suppss2 6073 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' ( k  e.  ( I  \  { x } ) 
|->  ( f `  k
) ) " ( _V  \  {  .0.  }
) )  C_  ( `' f " ( _V  \  {  .0.  }
) ) )
91 ssfi 7083 . . . . . . . . . . . . . . . 16  |-  ( ( ( `' 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 )
9282, 90, 91syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' ( k  e.  ( I  \  { x } ) 
|->  ( f `  k
) ) " ( _V  \  {  .0.  }
) )  e.  Fin )
9360, 64, 69, 81, 92dprdwd 15246 . . . . . . . . . . . . . 14  |-  ( ( ( 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 } )
9475, 93eqeltrd 2357 . . . . . . . . . . . . 13  |-  ( ( ( 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 } )
953, 60, 64, 69, 94eldprdi 15253 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  (
I  \  { x } ) ) )  e.  ( G DProd  ( S  |`  ( I  \  { x } ) ) ) )
9622, 95sylan2 460 . . . . . . . . . . 11  |-  ( ( ( 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 } ) ) ) )
9759, 96eqeltrd 2357 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  A  e.  ( G DProd  ( S  |`  ( I  \  {
x } ) ) ) )
98 eqid 2283 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
99 eqid 2283 . . . . . . . . . . . 12  |-  ( LSSum `  G )  =  (
LSSum `  G )
100 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( S : I --> (SubGrp `  G )  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
10165, 15, 100syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S `  x
)  e.  (SubGrp `  G ) )
102 dprdsubg 15259 . . . . . . . . . . . . 13  |-  ( G dom DProd  ( S  |`  ( I  \  { x } ) )  -> 
( G DProd  ( S  |`  ( I  \  {
x } ) ) )  e.  (SubGrp `  G ) )
10364, 102syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G DProd  ( S  |`  ( I  \  {
x } ) ) )  e.  (SubGrp `  G ) )
10412, 13, 15, 3dpjdisj 15288 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( S `  x )  i^i  ( G DProd  ( S  |`  (
I  \  { x } ) ) ) )  =  {  .0.  } )
10512, 13, 15, 29dpjcntz 15287 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( S `  x
)  C_  ( (Cntz `  G ) `  ( G DProd  ( S  |`  (
I  \  { x } ) ) ) ) )
10698, 99, 3, 29, 101, 103, 104, 105, 23pj1rid 15011 . . . . . . . . . . 11  |-  ( ( ( ( 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.  )
10722, 106sylanl2 632 . . . . . . . . . 10  |-  ( ( ( ( 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.  )
10897, 107mpdan 649 . . . . . . . . 9  |-  ( ( ( 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.  )
10926, 108eqtrd 2315 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  ( I  \  ( `' f
" ( _V  \  {  .0.  } ) ) ) )  ->  (
( P `  x
) `  A )  =  .0.  )
110109suppss2 6073 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' ( x  e.  I  |->  ( ( P `  x
) `  A )
) " ( _V 
\  {  .0.  }
) )  C_  ( `' f " ( _V  \  {  .0.  }
) ) )
111 ssfi 7083 . . . . . . 7  |-  ( ( ( `' 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 )
11221, 110, 111syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( `' ( x  e.  I  |->  ( ( P `  x
) `  A )
) " ( _V 
\  {  .0.  }
) )  e.  Fin )
1134, 10, 11, 19, 112dprdwd 15246 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  e.  W )
114 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  A  =  ( G  gsumg  f ) )
11540feqmptd 5575 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  =  ( x  e.  I  |->  ( f `  x ) ) )
116 simplrr 737 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  =  ( G 
gsumg  f ) )
11712, 30, 313syl 18 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  G  e.  Mnd )
11812, 35, 363syl 18 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  dom  S  e.  _V )
11913, 118eqeltrrd 2358 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  I  e.  _V )
1204, 12, 13, 42dprdffi 15249 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( `' f "
( _V  \  {  .0.  } ) )  e. 
Fin )
121 disjdif 3526 . . . . . . . . . . . . . . 15  |-  ( { x }  i^i  (
I  \  { x } ) )  =  (/)
122121a1i 10 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( { x }  i^i  ( I  \  {
x } ) )  =  (/) )
123 undif2 3530 . . . . . . . . . . . . . . 15  |-  ( { x }  u.  (
I  \  { x } ) )  =  ( { x }  u.  I )
12415snssd 3760 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  { x }  C_  I )
125 ssequn1 3345 . . . . . . . . . . . . . . . 16  |-  ( { x }  C_  I  <->  ( { x }  u.  I )  =  I )
126124, 125sylib 188 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( { x }  u.  I )  =  I )
127123, 126syl5req 2328 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  I  =  ( { x }  u.  (
I  \  { x } ) ) )
12828, 3, 98, 29, 117, 119, 70, 43, 120, 122, 127gsumzsplit 15206 . . . . . . . . . . . . 13  |-  ( ( ( 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 } ) ) ) ) )
12970, 124feqresmpt 5576 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f  |`  { x } )  =  ( k  e.  { x }  |->  ( f `  k ) ) )
130129oveq2d 5874 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  { x } ) )  =  ( G  gsumg  ( k  e.  {
x }  |->  ( f `
 k ) ) ) )
131 ffvelrn 5663 . . . . . . . . . . . . . . . . 17  |-  ( ( f : I --> ( Base `  G )  /\  x  e.  I )  ->  (
f `  x )  e.  ( Base `  G
) )
13270, 15, 131syl2anc 642 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f `  x
)  e.  ( Base `  G ) )
133 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( k  =  x  ->  (
f `  k )  =  ( f `  x ) )
13428, 133gsumsn 15220 . . . . . . . . . . . . . . . 16  |-  ( ( G  e.  Mnd  /\  x  e.  I  /\  ( f `  x
)  e.  ( Base `  G ) )  -> 
( G  gsumg  ( k  e.  {
x }  |->  ( f `
 k ) ) )  =  ( f `
 x ) )
135117, 15, 132, 134syl3anc 1182 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( k  e.  {
x }  |->  ( f `
 k ) ) )  =  ( f `
 x ) )
136130, 135eqtrd 2315 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G  gsumg  ( f  |`  { x } ) )  =  ( f `  x
) )
137136oveq1d 5873 . . . . . . . . . . . . 13  |-  ( ( ( 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 } ) ) ) ) )
138116, 128, 1373eqtrd 2319 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  =  ( ( f `  x ) ( +g  `  G
) ( G  gsumg  ( f  |`  ( I  \  {
x } ) ) ) ) )
13912, 13, 15, 99dpjlsm 15289 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( G DProd  S )  =  ( ( S `
 x ) (
LSSum `  G ) ( G DProd  ( S  |`  ( I  \  { x } ) ) ) ) )
14017, 139eleqtrd 2359 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  A  e.  ( ( S `  x ) ( LSSum `  G )
( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) )
1414, 10, 11, 20dprdfcl 15248 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( f `  x
)  e.  ( S `
 x ) )
14298, 99, 3, 29, 101, 103, 104, 105, 23, 140, 141, 95pj1eq 15009 . . . . . . . . . . . 12  |-  ( ( ( 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 } ) ) ) ) ) )
143138, 142mpbid 201 . . . . . . . . . . 11  |-  ( ( ( 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 } ) ) ) ) )
144143simpld 445 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( ( S `
 x ) (
proj 1 `  G ) ( G DProd  ( S  |`  ( I  \  {
x } ) ) ) ) `  A
)  =  ( f `
 x ) )
14525, 144eqtrd 2315 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  /\  x  e.  I )  ->  ( ( P `  x ) `  A
)  =  ( f `
 x ) )
146145mpteq2dva 4106 . . . . . . . 8  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( x  e.  I  |->  ( ( P `
 x ) `  A ) )  =  ( x  e.  I  |->  ( f `  x
) ) )
147115, 146eqtr4d 2318 . . . . . . 7  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  f  =  ( x  e.  I  |->  ( ( P `  x
) `  A )
) )
148147oveq2d 5874 . . . . . 6  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  ( G  gsumg  f )  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `
 A ) ) ) )
149114, 148eqtrd 2315 . . . . 5  |-  ( (
ph  /\  ( f  e.  W  /\  A  =  ( G  gsumg  f ) ) )  ->  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A
) ) ) )
150113, 149jca 518 . . . 4  |-  ( (
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 ) ) ) ) )
151150expr 598 . . 3  |-  ( (
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 )
) ) ) ) )
152151rexlimdva 2667 . 2  |-  ( ph  ->  ( E. f  e.  W  A  =  ( G  gsumg  f )  ->  (
( x  e.  I  |->  ( ( P `  x ) `  A
) )  e.  W  /\  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A
) ) ) ) ) )
1538, 152mpd 14 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   X_cixp 6817   Fincfn 6863   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   LSSumclsm 14945   proj
1cpj1 14946   DProd cdprd 15231  dProjcdpj 15232
This theorem is referenced by:  dpjeq  15294  dpjid  15295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-ghm 14681  df-gim 14723  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-pj1 14948  df-cmn 15091  df-dprd 15233  df-dpj 15234
  Copyright terms: Public domain W3C validator