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Theorem dmdprd 16903
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.) (Proof shortened by AV, 11-Jul-2019.)
Hypotheses
Ref Expression
dmdprd.z  |-  Z  =  (Cntz `  G )
dmdprd.0  |-  .0.  =  ( 0g `  G )
dmdprd.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
Assertion
Ref Expression
dmdprd  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( G dom DProd  S  <-> 
( G  e.  Grp  /\  S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) ) ) )
Distinct variable groups:    x, y, G    x, I, y    x, S, y    x, V, y
Allowed substitution hints:    K( x, y)    .0. ( x, y)    Z( x, y)

Proof of Theorem dmdprd
Dummy variables  g  h  f  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3104 . . . . 5  |-  ( S  e.  { h  |  ( h : dom  h
--> (SubGrp `  G )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) }  ->  S  e.  _V )
21a1i 11 . . . 4  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( S  e. 
{ h  |  ( h : dom  h --> (SubGrp `  G )  /\  A. x  e.  dom  h
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) }  ->  S  e.  _V ) )
3 fex 6130 . . . . . . 7  |-  ( ( S : I --> (SubGrp `  G )  /\  I  e.  V )  ->  S  e.  _V )
43expcom 435 . . . . . 6  |-  ( I  e.  V  ->  ( S : I --> (SubGrp `  G )  ->  S  e.  _V ) )
54adantr 465 . . . . 5  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( S :
I --> (SubGrp `  G )  ->  S  e.  _V )
)
65adantrd 468 . . . 4  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( ( S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) )  ->  S  e.  _V ) )
7 df-sbc 3314 . . . . . 6  |-  ( [. S  /  h ]. (
h : dom  h --> (SubGrp `  G )  /\  A. x  e.  dom  h
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) )  <->  S  e.  { h  |  ( h : dom  h --> (SubGrp `  G )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) } )
8 simpr 461 . . . . . . 7  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  S  e. 
_V )  ->  S  e.  _V )
9 simpr 461 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  h  =  S )
109dmeqd 5195 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  h  =  dom  S )
11 simplr 755 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  S  =  I )
1210, 11eqtrd 2484 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  h  =  I )
139, 12feq12d 5710 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h : dom  h --> (SubGrp `  G )  <->  S :
I --> (SubGrp `  G )
) )
1412difeq1d 3606 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( dom  h  \  { x } )  =  ( I  \  { x } ) )
159fveq1d 5858 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h `  x )  =  ( S `  x ) )
169fveq1d 5858 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h `  y )  =  ( S `  y ) )
1716fveq2d 5860 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( Z `  ( h `  y ) )  =  ( Z `  ( S `  y )
) )
1815, 17sseq12d 3518 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
( h `  x
)  C_  ( Z `  ( h `  y
) )  <->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
1914, 18raleqbidv 3054 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( A. y  e.  ( dom  h  \  { x } ) ( h `
 x )  C_  ( Z `  ( h `
 y ) )  <->  A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) ) ) )
209, 14imaeq12d 5328 . . . . . . . . . . . . . . 15  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h " ( dom  h  \  { x } ) )  =  ( S " (
I  \  { x } ) ) )
2120unieqd 4244 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  U. (
h " ( dom  h  \  { x } ) )  = 
U. ( S "
( I  \  {
x } ) ) )
2221fveq2d 5860 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( K `  U. ( h
" ( dom  h  \  { x } ) ) )  =  ( K `  U. ( S " ( I  \  { x } ) ) ) )
2315, 22ineq12d 3686 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  ( ( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) ) )
2423eqeq1d 2445 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
( ( h `  x )  i^i  ( K `  U. ( h
" ( dom  h  \  { x } ) ) ) )  =  {  .0.  }  <->  ( ( S `  x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) )
2519, 24anbi12d 710 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } )  <->  ( A. y  e.  ( I  \  { x } ) ( S `  x
)  C_  ( Z `  ( S `  y
) )  /\  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } ) ) )
2612, 25raleqbidv 3054 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( A. x  e.  dom  h ( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } )  <->  A. x  e.  I  ( A. y  e.  ( I  \  { x } ) ( S `  x
)  C_  ( Z `  ( S `  y
) )  /\  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } ) ) )
2713, 26anbi12d 710 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
( h : dom  h
--> (SubGrp `  G )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) )  <->  ( S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) ) ) )
2827adantlr 714 . . . . . . 7  |-  ( ( ( ( I  e.  V  /\  dom  S  =  I )  /\  S  e.  _V )  /\  h  =  S )  ->  (
( h : dom  h
--> (SubGrp `  G )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) )  <->  ( S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) ) ) )
298, 28sbcied 3350 . . . . . 6  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  S  e. 
_V )  ->  ( [. S  /  h ]. ( h : dom  h
--> (SubGrp `  G )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) )  <->  ( S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) ) ) )
307, 29syl5bbr 259 . . . . 5  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  S  e. 
_V )  ->  ( S  e.  { h  |  ( h : dom  h --> (SubGrp `  G )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) }  <-> 
( S : I --> (SubGrp `  G )  /\  A. x  e.  I 
( A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( Z `  ( S `  y ) )  /\  ( ( S `  x )  i^i  ( K `  U. ( S " (
I  \  { x } ) ) ) )  =  {  .0.  } ) ) ) )
3130ex 434 . . . 4  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( S  e. 
_V  ->  ( S  e. 
{ h  |  ( h : dom  h --> (SubGrp `  G )  /\  A. x  e.  dom  h
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) }  <-> 
( S : I --> (SubGrp `  G )  /\  A. x  e.  I 
( A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( Z `  ( S `  y ) )  /\  ( ( S `  x )  i^i  ( K `  U. ( S " (
I  \  { x } ) ) ) )  =  {  .0.  } ) ) ) ) )
322, 6, 31pm5.21ndd 354 . . 3  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( S  e. 
{ h  |  ( h : dom  h --> (SubGrp `  G )  /\  A. x  e.  dom  h
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) }  <-> 
( S : I --> (SubGrp `  G )  /\  A. x  e.  I 
( A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( Z `  ( S `  y ) )  /\  ( ( S `  x )  i^i  ( K `  U. ( S " (
I  \  { x } ) ) ) )  =  {  .0.  } ) ) ) )
3332anbi2d 703 . 2  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( ( G  e.  Grp  /\  S  e.  { h  |  ( h : dom  h --> (SubGrp `  G )  /\  A. x  e.  dom  h
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) } )  <->  ( G  e. 
Grp  /\  ( S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) ) ) ) )
34 df-br 4438 . . 3  |-  ( G dom DProd  S  <->  <. G ,  S >.  e.  dom DProd  )
35 fvex 5866 . . . . . . . . . . 11  |-  ( s `
 x )  e. 
_V
3635rgenw 2804 . . . . . . . . . 10  |-  A. x  e.  dom  s ( s `
 x )  e. 
_V
37 ixpexg 7495 . . . . . . . . . 10  |-  ( A. x  e.  dom  s ( s `  x )  e.  _V  ->  X_ x  e.  dom  s ( s `
 x )  e. 
_V )
3836, 37ax-mp 5 . . . . . . . . 9  |-  X_ x  e.  dom  s ( s `
 x )  e. 
_V
3938mptrabex 6129 . . . . . . . 8  |-  ( f  e.  { h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  e. 
_V
4039rnex 6719 . . . . . . 7  |-  ran  (
f  e.  { h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  e. 
_V
4140rgen2w 2805 . . . . . 6  |-  A. g  e.  Grp  A. s  e. 
{ h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) } ran  ( f  e. 
{ h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) )  e. 
_V
42 df-dprd 16900 . . . . . . 7  |- DProd  =  ( g  e.  Grp , 
s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) } 
|->  ran  ( f  e. 
{ h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) ) )
4342fmpt2x 6851 . . . . . 6  |-  ( A. g  e.  Grp  A. s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) } ran  ( f  e. 
{ h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp  ( 0g `  g ) }  |->  ( g  gsumg  f ) )  e. 
_V 
<-> DProd  : U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) --> _V )
4441, 43mpbi 208 . . . . 5  |- DProd  : U_ g  e.  Grp  ( { g }  X.  {
h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) --> _V
4544fdmi 5726 . . . 4  |-  dom DProd  =  U_ g  e.  Grp  ( { g }  X.  {
h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) } )
4645eleq2i 2521 . . 3  |-  ( <. G ,  S >.  e. 
dom DProd 
<-> 
<. G ,  S >.  e. 
U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) } ) )
47 fveq2 5856 . . . . . . 7  |-  ( g  =  G  ->  (SubGrp `  g )  =  (SubGrp `  G ) )
4847feq3d 5709 . . . . . 6  |-  ( g  =  G  ->  (
h : dom  h --> (SubGrp `  g )  <->  h : dom  h --> (SubGrp `  G )
) )
49 fveq2 5856 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (Cntz `  g )  =  (Cntz `  G ) )
50 dmdprd.z . . . . . . . . . . . 12  |-  Z  =  (Cntz `  G )
5149, 50syl6eqr 2502 . . . . . . . . . . 11  |-  ( g  =  G  ->  (Cntz `  g )  =  Z )
5251fveq1d 5858 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(Cntz `  g ) `  ( h `  y
) )  =  ( Z `  ( h `
 y ) ) )
5352sseq2d 3517 . . . . . . . . 9  |-  ( g  =  G  ->  (
( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  <->  ( h `  x )  C_  ( Z `  ( h `  y ) ) ) )
5453ralbidv 2882 . . . . . . . 8  |-  ( g  =  G  ->  ( A. y  e.  ( dom  h  \  { x } ) ( h `
 x )  C_  ( (Cntz `  g ) `  ( h `  y
) )  <->  A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) ) ) )
5547fveq2d 5860 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (mrCls `  (SubGrp `  g )
)  =  (mrCls `  (SubGrp `  G ) ) )
56 dmdprd.k . . . . . . . . . . . 12  |-  K  =  (mrCls `  (SubGrp `  G
) )
5755, 56syl6eqr 2502 . . . . . . . . . . 11  |-  ( g  =  G  ->  (mrCls `  (SubGrp `  g )
)  =  K )
5857fveq1d 5858 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(mrCls `  (SubGrp `  g
) ) `  U. ( h " ( dom  h  \  { x } ) ) )  =  ( K `  U. ( h " ( dom  h  \  { x } ) ) ) )
5958ineq2d 3685 . . . . . . . . 9  |-  ( g  =  G  ->  (
( h `  x
)  i^i  ( (mrCls `  (SubGrp `  g )
) `  U. ( h
" ( dom  h  \  { x } ) ) ) )  =  ( ( h `  x )  i^i  ( K `  U. ( h
" ( dom  h  \  { x } ) ) ) ) )
60 fveq2 5856 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
61 dmdprd.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
6260, 61syl6eqr 2502 . . . . . . . . . 10  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
6362sneqd 4026 . . . . . . . . 9  |-  ( g  =  G  ->  { ( 0g `  g ) }  =  {  .0.  } )
6459, 63eqeq12d 2465 . . . . . . . 8  |-  ( g  =  G  ->  (
( ( h `  x )  i^i  (
(mrCls `  (SubGrp `  g
) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) }  <->  ( ( h `
 x )  i^i  ( K `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  {  .0.  } ) )
6554, 64anbi12d 710 . . . . . . 7  |-  ( g  =  G  ->  (
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } )  <->  ( A. y  e.  ( dom  h  \  { x }
) ( h `  x )  C_  ( Z `  ( h `  y ) )  /\  ( ( h `  x )  i^i  ( K `  U. ( h
" ( dom  h  \  { x } ) ) ) )  =  {  .0.  } ) ) )
6665ralbidv 2882 . . . . . 6  |-  ( g  =  G  ->  ( A. x  e.  dom  h ( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } )  <->  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  { x }
) ( h `  x )  C_  ( Z `  ( h `  y ) )  /\  ( ( h `  x )  i^i  ( K `  U. ( h
" ( dom  h  \  { x } ) ) ) )  =  {  .0.  } ) ) )
6748, 66anbi12d 710 . . . . 5  |-  ( g  =  G  ->  (
( h : dom  h
--> (SubGrp `  g )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) )  <->  ( h : dom  h --> (SubGrp `  G )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) ) )
6867abbidv 2579 . . . 4  |-  ( g  =  G  ->  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) }  =  { h  |  ( h : dom  h
--> (SubGrp `  G )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) } )
6968opeliunxp2 5131 . . 3  |-  ( <. G ,  S >.  e. 
U_ g  e.  Grp  ( { g }  X.  { h  |  (
h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h
( A. y  e.  ( dom  h  \  { x } ) ( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  /\  ( (
h `  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) } ) ) } )  <->  ( G  e. 
Grp  /\  S  e.  { h  |  ( h : dom  h --> (SubGrp `  G )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) } ) )
7034, 46, 693bitri 271 . 2  |-  ( G dom DProd  S  <->  ( G  e. 
Grp  /\  S  e.  { h  |  ( h : dom  h --> (SubGrp `  G )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  {
x } ) ( h `  x ) 
C_  ( Z `  ( h `  y
) )  /\  (
( h `  x
)  i^i  ( K `  U. ( h "
( dom  h  \  {
x } ) ) ) )  =  {  .0.  } ) ) } ) )
71 3anass 978 . 2  |-  ( ( G  e.  Grp  /\  S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) )  <->  ( G  e.  Grp  /\  ( S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) ) ) )
7233, 70, 713bitr4g 288 1  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( G dom DProd  S  <-> 
( G  e.  Grp  /\  S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( Z `  ( S `
 y ) )  /\  ( ( S `
 x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   {cab 2428   A.wral 2793   {crab 2797   _Vcvv 3095   [.wsbc 3313    \ cdif 3458    i^i cin 3460    C_ wss 3461   {csn 4014   <.cop 4020   U.cuni 4234   U_ciun 4315   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   dom cdm 4989   ran crn 4990   "cima 4992   -->wf 5574   ` cfv 5578  (class class class)co 6281   X_cixp 7471   finSupp cfsupp 7831   0gc0g 14714    gsumg cgsu 14715  mrClscmrc 14857   Grpcgrp 15927  SubGrpcsubg 16069  Cntzccntz 16227   DProd cdprd 16898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-ixp 7472  df-dprd 16900
This theorem is referenced by:  dmdprdd  16904  dprdgrp  16912  dprdf  16913  dprdcntz  16915  dprddisj  16916  dprdres  16949  subgdmdprd  16955
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