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Theorem dprdval 15566
Description: The domain of definition of the internal direct product, which states that  S is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdval.0  |-  .0.  =  ( 0g `  G )
dprdval.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
Assertion
Ref Expression
dprdval  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
Distinct variable groups:    f, h, i, I    S, f, h, i    f, G, h, i
Allowed substitution hints:    W( f, h, i)    .0. ( f, h, i)

Proof of Theorem dprdval
Dummy variables  g 
s  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 445 . 2  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  G dom DProd  S )
2 reldmdprd 15563 . . . . . 6  |-  Rel  dom DProd
32brrelex2i 4922 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
43adantr 453 . . . 4  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  S  e.  _V )
52brrelexi 4921 . . . . . 6  |-  ( G dom DProd  s  ->  G  e.  _V )
6 breq1 4218 . . . . . . . 8  |-  ( g  =  G  ->  (
g dom DProd  s  <->  G dom DProd  s ) )
7 oveq1 6091 . . . . . . . . 9  |-  ( g  =  G  ->  (
g DProd  s )  =  ( G DProd  s ) )
8 fveq2 5731 . . . . . . . . . . . . . . . . 17  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
9 dprdval.0 . . . . . . . . . . . . . . . . 17  |-  .0.  =  ( 0g `  G )
108, 9syl6eqr 2488 . . . . . . . . . . . . . . . 16  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
1110sneqd 3829 . . . . . . . . . . . . . . 15  |-  ( g  =  G  ->  { ( 0g `  g ) }  =  {  .0.  } )
1211difeq2d 3467 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  ( _V  \  { ( 0g
`  g ) } )  =  ( _V 
\  {  .0.  }
) )
1312imaeq2d 5206 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  =  ( `' h " ( _V 
\  {  .0.  }
) ) )
1413eleq1d 2504 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
( `' h "
( _V  \  {
( 0g `  g
) } ) )  e.  Fin  <->  ( `' h " ( _V  \  {  .0.  } ) )  e.  Fin ) )
1514rabbidv 2950 . . . . . . . . . . 11  |-  ( g  =  G  ->  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin }  =  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin } )
16 oveq1 6091 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  gsumg  f )  =  ( G  gsumg  f ) )
1715, 16mpteq12dv 4290 . . . . . . . . . 10  |-  ( g  =  G  ->  (
f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  =  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) ) )
1817rneqd 5100 . . . . . . . . 9  |-  ( g  =  G  ->  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) ) )
197, 18eqeq12d 2452 . . . . . . . 8  |-  ( g  =  G  ->  (
( g DProd  s )  =  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  <->  ( G DProd  s )  =  ran  (
f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin } 
|->  ( G  gsumg  f ) ) ) )
206, 19imbi12d 313 . . . . . . 7  |-  ( g  =  G  ->  (
( g dom DProd  s  -> 
( g DProd  s )  =  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) ) )  <-> 
( G dom DProd  s  -> 
( G DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) ) ) ) )
21 df-br 4216 . . . . . . . . 9  |-  ( g dom DProd  s  <->  <. g ,  s >.  e.  dom DProd  )
22 fvex 5745 . . . . . . . . . . . . . . . . . 18  |-  ( s `
 i )  e. 
_V
2322rgenw 2775 . . . . . . . . . . . . . . . . 17  |-  A. i  e.  dom  s ( s `
 i )  e. 
_V
24 ixpexg 7089 . . . . . . . . . . . . . . . . 17  |-  ( A. i  e.  dom  s ( s `  i )  e.  _V  ->  X_ i  e.  dom  s ( s `
 i )  e. 
_V )
2523, 24ax-mp 5 . . . . . . . . . . . . . . . 16  |-  X_ i  e.  dom  s ( s `
 i )  e. 
_V
2625rabex 4357 . . . . . . . . . . . . . . 15  |-  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin }  e.  _V
2726mptex 5969 . . . . . . . . . . . . . 14  |-  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  e. 
_V
2827rnex 5136 . . . . . . . . . . . . 13  |-  ran  (
f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  e. 
_V
2928rgen2w 2776 . . . . . . . . . . . 12  |-  A. g  e.  Grp  A. s  e. 
{ h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) )  e. 
_V
30 df-dprd 15561 . . . . . . . . . . . . 13  |- DProd  =  ( g  e.  Grp , 
s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } 
|->  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) ) )
3130fmpt2x 6420 . . . . . . . . . . . 12  |-  ( A. g  e.  Grp  A. s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) )  e. 
_V 
<-> DProd  : U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) --> _V )
3229, 31mpbi 201 . . . . . . . . . . 11  |- DProd  : U_ g  e.  Grp  ( { g }  X.  {
h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) --> _V
3332fdmi 5599 . . . . . . . . . 10  |-  dom DProd  =  U_ g  e.  Grp  ( { g }  X.  {
h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } )
3433eleq2i 2502 . . . . . . . . 9  |-  ( <.
g ,  s >.  e.  dom DProd 
<-> 
<. g ,  s >.  e.  U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) )
35 opeliunxp 4932 . . . . . . . . 9  |-  ( <.
g ,  s >.  e.  U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } )  <->  ( g  e. 
Grp  /\  s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) )
3621, 34, 353bitri 264 . . . . . . . 8  |-  ( g dom DProd  s  <->  ( g  e.  Grp  /\  s  e. 
{ h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h
( A. y  e.  ( dom  h  \  { i } ) ( h `  i
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) )
3730ovmpt4g 6199 . . . . . . . . 9  |-  ( ( g  e.  Grp  /\  s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) }  /\  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  e. 
_V )  ->  (
g DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) ) )
3828, 37mp3an3 1269 . . . . . . . 8  |-  ( ( g  e.  Grp  /\  s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. i  e.  dom  h ( A. y  e.  ( dom  h  \  {
i } ) ( h `  i ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  i )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { i } ) ) ) )  =  { ( 0g `  g ) } ) ) } )  ->  ( g DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) ) )
3936, 38sylbi 189 . . . . . . 7  |-  ( g dom DProd  s  ->  (
g DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) ) )
4020, 39vtoclg 3013 . . . . . 6  |-  ( G  e.  _V  ->  ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) ) ) )
415, 40mpcom 35 . . . . 5  |-  ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) ) )
4241sbcth 3177 . . . 4  |-  ( S  e.  _V  ->  [. S  /  s ]. ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) ) ) )
434, 42syl 16 . . 3  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  [. S  /  s ]. ( G dom DProd  s  -> 
( G DProd  s )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) ) ) )
44 simpr 449 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  s  =  S )
4544breq2d 4227 . . . . 5  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G dom DProd  s  <->  G dom DProd  S )
)
4644oveq2d 6100 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G DProd  s )  =  ( G DProd 
S ) )
4744dmeqd 5075 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  s  =  dom  S )
48 simplr 733 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  S  =  I )
4947, 48eqtrd 2470 . . . . . . . . . . . 12  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  s  =  I )
5049ixpeq1d 7077 . . . . . . . . . . 11  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e. 
dom  s ( s `
 i )  = 
X_ i  e.  I 
( s `  i
) )
5144fveq1d 5733 . . . . . . . . . . . 12  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( s `  i )  =  ( S `  i ) )
5251ixpeq2dv 7081 . . . . . . . . . . 11  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e.  I  ( s `  i )  =  X_ i  e.  I  ( S `  i )
)
5350, 52eqtrd 2470 . . . . . . . . . 10  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e. 
dom  s ( s `
 i )  = 
X_ i  e.  I 
( S `  i
) )
54 biidd 230 . . . . . . . . . 10  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin  <->  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin ) )
5553, 54rabeqbidv 2953 . . . . . . . . 9  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin }  =  { h  e.  X_ i  e.  I 
( S `  i
)  |  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin } )
56 dprdval.w . . . . . . . . 9  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
5755, 56syl6eqr 2488 . . . . . . . 8  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin }  =  W )
58 eqidd 2439 . . . . . . . 8  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G  gsumg  f )  =  ( G 
gsumg  f ) )
5957, 58mpteq12dv 4290 . . . . . . 7  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) )  =  ( f  e.  W  |->  ( G  gsumg  f ) ) )
6059rneqd 5100 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ran  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin } 
|->  ( G  gsumg  f ) )  =  ran  ( f  e.  W  |->  ( G  gsumg  f ) ) )
6146, 60eqeq12d 2452 . . . . 5  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) )  <->  ( G DProd  S )  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
6245, 61imbi12d 313 . . . 4  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) ) )  <-> 
( G dom DProd  S  -> 
( G DProd  S )  =  ran  ( f  e.  W  |->  ( G  gsumg  f ) ) ) ) )
634, 62sbcied 3199 . . 3  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( [. S  /  s ]. ( G dom DProd  s  ->  ( G DProd  s )  =  ran  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {  .0.  } ) )  e. 
Fin }  |->  ( G 
gsumg  f ) ) )  <-> 
( G dom DProd  S  -> 
( G DProd  S )  =  ran  ( f  e.  W  |->  ( G  gsumg  f ) ) ) ) )
6443, 63mpbid 203 . 2  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G dom DProd  S  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
651, 64mpd 15 1  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   {crab 2711   _Vcvv 2958   [.wsbc 3163    \ cdif 3319    i^i cin 3321    C_ wss 3322   {csn 3816   <.cop 3819   U.cuni 4017   U_ciun 4095   class class class wbr 4215    e. cmpt 4269    X. cxp 4879   `'ccnv 4880   dom cdm 4881   ran crn 4882   "cima 4884   -->wf 5453   ` cfv 5457  (class class class)co 6084   X_cixp 7066   Fincfn 7112   0gc0g 13728    gsumg cgsu 13729  mrClscmrc 13813   Grpcgrp 14690  SubGrpcsubg 14943  Cntzccntz 15119   DProd cdprd 15559
This theorem is referenced by:  eldprd  15567  dprdlub  15589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-ixp 7067  df-dprd 15561
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