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Theorem dprdvalOLD 17162
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.) Obsolete version of dprdval 17160 as of 11-Jul-2019. Proof adapted to use the new definition df-dprd 17152. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
dprdvalOLD.0  |-  .0.  =  ( 0g `  G )
dprdvalOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
Assertion
Ref Expression
dprdvalOLD  |-  ( ( 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 dprdvalOLD
Dummy variables  g 
s  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . 2  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  G dom DProd  S )
2 reldmdprd 17154 . . . . . 6  |-  Rel  dom DProd
32brrelex2i 5050 . . . . 5  |-  ( G dom DProd  S  ->  S  e. 
_V )
43adantr 465 . . . 4  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  S  e.  _V )
52brrelexi 5049 . . . . . 6  |-  ( G dom DProd  s  ->  G  e.  _V )
6 breq1 4459 . . . . . . . 8  |-  ( g  =  G  ->  (
g dom DProd  s  <->  G dom DProd  s ) )
7 oveq1 6303 . . . . . . . . 9  |-  ( g  =  G  ->  (
g DProd  s )  =  ( G DProd  s ) )
8 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
9 dprdvalOLD.0 . . . . . . . . . . . . . . . . 17  |-  .0.  =  ( 0g `  G )
108, 9syl6eqr 2516 . . . . . . . . . . . . . . . 16  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
1110sneqd 4044 . . . . . . . . . . . . . . 15  |-  ( g  =  G  ->  { ( 0g `  g ) }  =  {  .0.  } )
1211difeq2d 3618 . . . . . . . . . . . . . 14  |-  ( g  =  G  ->  ( _V  \  { ( 0g
`  g ) } )  =  ( _V 
\  {  .0.  }
) )
1312imaeq2d 5347 . . . . . . . . . . . . 13  |-  ( g  =  G  ->  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  =  ( `' h " ( _V 
\  {  .0.  }
) ) )
1413eleq1d 2526 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (
( `' h "
( _V  \  {
( 0g `  g
) } ) )  e.  Fin  <->  ( `' h " ( _V  \  {  .0.  } ) )  e.  Fin ) )
1514rabbidv 3101 . . . . . . . . . . 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 6303 . . . . . . . . . . 11  |-  ( g  =  G  ->  (
g  gsumg  f )  =  ( G  gsumg  f ) )
1715, 16mpteq12dv 4535 . . . . . . . . . 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 5240 . . . . . . . . 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 2479 . . . . . . . 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 320 . . . . . . 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 4457 . . . . . . . . 9  |-  ( g dom DProd  s  <->  <. g ,  s >.  e.  dom DProd  )
22 fvex 5882 . . . . . . . . . . . . . . . . . 18  |-  ( s `
 i )  e. 
_V
2322rgenw 2818 . . . . . . . . . . . . . . . . 17  |-  A. i  e.  dom  s ( s `
 i )  e. 
_V
24 ixpexg 7512 . . . . . . . . . . . . . . . . 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 4607 . . . . . . . . . . . . . . 15  |-  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin }  e.  _V
2726mptex 6144 . . . . . . . . . . . . . 14  |-  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  e. 
_V
2827rnex 6733 . . . . . . . . . . . . 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 2819 . . . . . . . . . . . 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 17152 . . . . . . . . . . . . . 14  |- 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 finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
31 eqid 2457 . . . . . . . . . . . . . . 15  |-  Grp  =  Grp
32 eqid 2457 . . . . . . . . . . . . . . 15  |-  { 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 ) } ) ) }  =  { 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 ) } ) ) }
33 ixpfn 7494 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( h  e.  X_ i  e.  dom  s ( s `  i )  ->  h  Fn  dom  s )
34 fnfun 5684 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( h  Fn  dom  s  ->  Fun  h )
3533, 34syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( h  e.  X_ i  e.  dom  s ( s `  i )  ->  Fun  h )
36 id 22 . . . . . . . . . . . . . . . . . . . . 21  |-  ( h  e.  X_ i  e.  dom  s ( s `  i )  ->  h  e.  X_ i  e.  dom  s ( s `  i ) )
37 fvex 5882 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 0g
`  g )  e. 
_V
3837a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( h  e.  X_ i  e.  dom  s ( s `  i )  ->  ( 0g `  g )  e. 
_V )
3935, 36, 383jca 1176 . . . . . . . . . . . . . . . . . . . 20  |-  ( h  e.  X_ i  e.  dom  s ( s `  i )  ->  ( Fun  h  /\  h  e.  X_ i  e.  dom  s ( s `  i )  /\  ( 0g `  g )  e. 
_V ) )
40 funisfsupp 7852 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( Fun  h  /\  h  e.  X_ i  e.  dom  s ( s `  i )  /\  ( 0g `  g )  e. 
_V )  ->  (
h finSupp  ( 0g `  g
)  <->  ( h supp  ( 0g `  g ) )  e.  Fin ) )
4139, 40syl 16 . . . . . . . . . . . . . . . . . . 19  |-  ( h  e.  X_ i  e.  dom  s ( s `  i )  ->  (
h finSupp  ( 0g `  g
)  <->  ( h supp  ( 0g `  g ) )  e.  Fin ) )
4236, 38jca 532 . . . . . . . . . . . . . . . . . . . . 21  |-  ( h  e.  X_ i  e.  dom  s ( s `  i )  ->  (
h  e.  X_ i  e.  dom  s ( s `
 i )  /\  ( 0g `  g )  e.  _V ) )
43 suppimacnv 6928 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( h  e.  X_ i  e.  dom  s ( s `
 i )  /\  ( 0g `  g )  e.  _V )  -> 
( h supp  ( 0g
`  g ) )  =  ( `' h " ( _V  \  {
( 0g `  g
) } ) ) )
4442, 43syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( h  e.  X_ i  e.  dom  s ( s `  i )  ->  (
h supp  ( 0g `  g ) )  =  ( `' h "
( _V  \  {
( 0g `  g
) } ) ) )
4544eleq1d 2526 . . . . . . . . . . . . . . . . . . 19  |-  ( h  e.  X_ i  e.  dom  s ( s `  i )  ->  (
( h supp  ( 0g
`  g ) )  e.  Fin  <->  ( `' h " ( _V  \  { ( 0g `  g ) } ) )  e.  Fin )
)
4641, 45bitrd 253 . . . . . . . . . . . . . . . . . 18  |-  ( h  e.  X_ i  e.  dom  s ( s `  i )  ->  (
h finSupp  ( 0g `  g
)  <->  ( `' h " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin ) )
4746rabbiia 3098 . . . . . . . . . . . . . . . . 17  |-  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  =  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {
( 0g `  g
) } ) )  e.  Fin }
48 eqid 2457 . . . . . . . . . . . . . . . . 17  |-  ( g 
gsumg  f )  =  ( g  gsumg  f )
4947, 48mpteq12i 4541 . . . . . . . . . . . . . . . 16  |-  ( f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  =  ( f  e.  {
h  e.  X_ i  e.  dom  s ( s `
 i )  |  ( `' h "
( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) )
5049rneqi 5239 . . . . . . . . . . . . . . 15  |-  ran  (
f  e.  { h  e.  X_ i  e.  dom  s ( s `  i )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  =  ran  ( f  e. 
{ h  e.  X_ i  e.  dom  s ( s `  i )  |  ( `' h " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) )
5131, 32, 50mpt2eq123i 6359 . . . . . . . . . . . . . 14  |-  ( 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 finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )  =  ( 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 ) ) )
5230, 51eqtri 2486 . . . . . . . . . . . . 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 ) ) )
5352fmpt2x 6865 . . . . . . . . . . . 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 )
5429, 53mpbi 208 . . . . . . . . . . 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
5554fdmi 5742 . . . . . . . . . 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 ) } ) ) } )
5655eleq2i 2535 . . . . . . . . 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 ) } ) ) } ) )
57 opeliunxp 5060 . . . . . . . . 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 ) } ) ) } ) )
5821, 56, 573bitri 271 . . . . . . . 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 ) } ) ) } ) )
5952ovmpt4g 6424 . . . . . . . . 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 ) ) )
6028, 59mp3an3 1313 . . . . . . . 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 ) ) )
6158, 60sylbi 195 . . . . . . 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 ) ) )
6220, 61vtoclg 3167 . . . . . 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 ) ) ) )
635, 62mpcom 36 . . . . 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 ) ) )
6463sbcth 3342 . . . 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 ) ) ) )
654, 64syl 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 ) ) ) )
66 simpr 461 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  s  =  S )
6766breq2d 4468 . . . . 5  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G dom DProd  s  <->  G dom DProd  S )
)
6866oveq2d 6312 . . . . . 6  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G DProd  s )  =  ( G DProd 
S ) )
6966dmeqd 5215 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  s  =  dom  S )
70 simplr 755 . . . . . . . . . . . . 13  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  S  =  I )
7169, 70eqtrd 2498 . . . . . . . . . . . 12  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  dom  s  =  I )
7271ixpeq1d 7500 . . . . . . . . . . 11  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e. 
dom  s ( s `
 i )  = 
X_ i  e.  I 
( s `  i
) )
7366fveq1d 5874 . . . . . . . . . . . 12  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( s `  i )  =  ( S `  i ) )
7473ixpeq2dv 7504 . . . . . . . . . . 11  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e.  I  ( s `  i )  =  X_ i  e.  I  ( S `  i )
)
7572, 74eqtrd 2498 . . . . . . . . . 10  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  X_ i  e. 
dom  s ( s `
 i )  = 
X_ i  e.  I 
( S `  i
) )
76 biidd 237 . . . . . . . . . 10  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin  <->  ( `' h " ( _V 
\  {  .0.  }
) )  e.  Fin ) )
7775, 76rabeqbidv 3104 . . . . . . . . 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 } )
78 dprdvalOLD.w . . . . . . . . 9  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
7977, 78syl6eqr 2516 . . . . . . . 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 )
80 eqidd 2458 . . . . . . . 8  |-  ( ( ( G dom DProd  S  /\  dom  S  =  I )  /\  s  =  S )  ->  ( G  gsumg  f )  =  ( G 
gsumg  f ) )
8179, 80mpteq12dv 4535 . . . . . . 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 ) ) )
8281rneqd 5240 . . . . . 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 ) ) )
8368, 82eqeq12d 2479 . . . . 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 ) ) ) )
8467, 83imbi12d 320 . . . 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 ) ) ) ) )
854, 84sbcied 3364 . . 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 ) ) ) ) )
8665, 85mpbid 210 . 2  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G dom DProd  S  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) ) )
871, 86mpd 15 1  |-  ( ( G dom DProd  S  /\  dom  S  =  I )  ->  ( G DProd  S
)  =  ran  (
f  e.  W  |->  ( G  gsumg  f ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   {crab 2811   _Vcvv 3109   [.wsbc 3327    \ cdif 3468    i^i cin 3470    C_ wss 3471   {csn 4032   <.cop 4038   U.cuni 4251   U_ciun 4332   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011   Fun wfun 5588    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   supp csupp 6917   X_cixp 7488   Fincfn 7535   finSupp cfsupp 7847   0gc0g 14856    gsumg cgsu 14857  mrClscmrc 14999   Grpcgrp 16179  SubGrpcsubg 16321  Cntzccntz 16479   DProd cdprd 17150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-supp 6918  df-ixp 7489  df-fsupp 7848  df-dprd 17152
This theorem is referenced by:  eldprdOLD  17163
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