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Theorem dprdval 15238
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 443 . 2  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  G dom DProd  S )
2 reldmdprd 15235 . . . . . 6  |-  Rel  dom DProd
32brrelex2i 4730 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
43adantr 451 . . . 4  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  S  e.  _V )
52brrelexi 4729 . . . . . 6  |-  ( G dom DProd  s  ->  G  e.  _V )
6 breq1 4026 . . . . . . . 8  |-  ( g  =  G  ->  (
g dom DProd  s  <->  G dom DProd  s ) )
7 oveq1 5865 . . . . . . . . 9  |-  ( g  =  G  ->  (
g DProd  s )  =  ( G DProd  s ) )
8 fveq2 5525 . . . . . . . . . . . . . . . . 17  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
9 dprdval.0 . . . . . . . . . . . . . . . . 17  |-  .0.  =  ( 0g `  G )
108, 9syl6eqr 2333 . . . . . . . . . . . . . . . 16  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
1110sneqd 3653 . . . . . . . . . . . . . . 15  |-  ( g  =  G  ->  { ( 0g `  g ) }  =  {  .0.  } )
1211difeq2d 3294 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  ( _V  \  { ( 0g
`  g ) } )  =  ( _V 
\  {  .0.  }
) )
1312imaeq2d 5012 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  =  ( `' h " ( _V 
\  {  .0.  }
) ) )
1413eleq1d 2349 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
( `' h "
( _V  \  {
( 0g `  g
) } ) )  e.  Fin  <->  ( `' h " ( _V  \  {  .0.  } ) )  e.  Fin ) )
1514rabbidv 2780 . . . . . . . . . . 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 5865 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  gsumg  f )  =  ( G  gsumg  f ) )
1715, 16mpteq12dv 4098 . . . . . . . . . 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 4906 . . . . . . . . 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 2297 . . . . . . . 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 311 . . . . . . 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 4024 . . . . . . . . 9  |-  ( g dom DProd  s  <->  <. g ,  s >.  e.  dom DProd  )
22 fvex 5539 . . . . . . . . . . . . . . . . . 18  |-  ( s `
 i )  e. 
_V
2322rgenw 2610 . . . . . . . . . . . . . . . . 17  |-  A. i  e.  dom  s ( s `
 i )  e. 
_V
24 ixpexg 6840 . . . . . . . . . . . . . . . . 17  |-  ( A. i  e.  dom  s ( s `  i )  e.  _V  ->  X_ i  e.  dom  s ( s `
 i )  e. 
_V )
2523, 24ax-mp 8 . . . . . . . . . . . . . . . 16  |-  X_ i  e.  dom  s ( s `
 i )  e. 
_V
2625rabex 4165 . . . . . . . . . . . . . . 15  |-  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin }  e.  _V
2726mptex 5746 . . . . . . . . . . . . . 14  |-  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  e. 
_V
2827rnex 4942 . . . . . . . . . . . . 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 2611 . . . . . . . . . . . 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 15233 . . . . . . . . . . . . 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 6190 . . . . . . . . . . . 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 199 . . . . . . . . . . 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 5394 . . . . . . . . . 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 2347 . . . . . . . . 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 4740 . . . . . . . . 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 262 . . . . . . . 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 5970 . . . . . . . . 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 1266 . . . . . . . 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 187 . . . . . . 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 2843 . . . . . 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 32 . . . . 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 3005 . . . 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 15 . . 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 447 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  s  =  S )
4544breq2d 4035 . . . . 5  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G dom DProd  s  <->  G dom DProd  S )
)
4644oveq2d 5874 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G DProd  s )  =  ( G DProd 
S ) )
4744dmeqd 4881 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  s  =  dom  S )
48 simplr 731 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  S  =  I )
4947, 48eqtrd 2315 . . . . . . . . . . . 12  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  s  =  I )
50 ixpeq1 6827 . . . . . . . . . . . 12  |-  ( dom  s  =  I  ->  X_ i  e.  dom  s
( s `  i
)  =  X_ i  e.  I  ( s `  i ) )
5149, 50syl 15 . . . . . . . . . . 11  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e. 
dom  s ( s `
 i )  = 
X_ i  e.  I 
( s `  i
) )
5244fveq1d 5527 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( s `  i )  =  ( S `  i ) )
5352ralrimivw 2627 . . . . . . . . . . . 12  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  A. i  e.  I  ( s `  i )  =  ( S `  i ) )
54 ixpeq2 6830 . . . . . . . . . . . 12  |-  ( A. i  e.  I  (
s `  i )  =  ( S `  i )  ->  X_ i  e.  I  ( s `  i )  =  X_ i  e.  I  ( S `  i )
)
5553, 54syl 15 . . . . . . . . . . 11  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e.  I  ( s `  i )  =  X_ i  e.  I  ( S `  i )
)
5651, 55eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e. 
dom  s ( s `
 i )  = 
X_ i  e.  I 
( S `  i
) )
57 biidd 228 . . . . . . . . . 10  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin  <->  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin ) )
5856, 57rabeqbidv 2783 . . . . . . . . 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 } )
59 dprdval.w . . . . . . . . 9  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
6058, 59syl6eqr 2333 . . . . . . . 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 )
61 eqidd 2284 . . . . . . . 8  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G  gsumg  f )  =  ( G 
gsumg  f ) )
6260, 61mpteq12dv 4098 . . . . . . 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 ) ) )
6362rneqd 4906 . . . . . 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 ) ) )
6446, 63eqeq12d 2297 . . . . 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 ) ) ) )
6545, 64imbi12d 311 . . . 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 ) ) ) ) )
664, 65sbcied 3027 . . 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 ) ) ) ) )
6743, 66mpbid 201 . 2  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G dom DProd  S  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
681, 67mpd 14 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 358    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   {crab 2547   _Vcvv 2788   [.wsbc 2991    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640   <.cop 3643   U.cuni 3827   U_ciun 3905   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   X_cixp 6817   Fincfn 6863   0gc0g 13400    gsumg cgsu 13401  mrClscmrc 13485   Grpcgrp 14362  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  eldprd  15239  dprdlub  15261
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-reu 2550  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-ixp 6818  df-dprd 15233
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