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Theorem dmdprd 15486
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
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 2907 . . . . 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 5908 . . . . . . 7  |-  ( ( S : I --> (SubGrp `  G )  /\  I  e.  V )  ->  S  e.  _V )
43expcom 425 . . . . . 6  |-  ( I  e.  V  ->  ( S : I --> (SubGrp `  G )  ->  S  e.  _V ) )
54adantr 452 . . . . 5  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( S :
I --> (SubGrp `  G )  ->  S  e.  _V )
)
65adantrd 455 . . . 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 3105 . . . . . 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 448 . . . . . . 7  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  S  e. 
_V )  ->  S  e.  _V )
9 simpr 448 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  h  =  S )
109dmeqd 5012 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  h  =  dom  S )
11 simplr 732 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  S  =  I )
1210, 11eqtrd 2419 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  h  =  I )
139, 12feq12d 5522 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h : dom  h --> (SubGrp `  G )  <->  S :
I --> (SubGrp `  G )
) )
1412difeq1d 3407 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( dom  h  \  { x } )  =  ( I  \  { x } ) )
159fveq1d 5670 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h `  x )  =  ( S `  x ) )
169fveq1d 5670 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h `  y )  =  ( S `  y ) )
1716fveq2d 5672 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( Z `  ( h `  y ) )  =  ( Z `  ( S `  y )
) )
1815, 17sseq12d 3320 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
( h `  x
)  C_  ( Z `  ( h `  y
) )  <->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
1914, 18raleqbidv 2859 . . . . . . . . . . 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 5144 . . . . . . . . . . . . . . 15  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h " ( dom  h  \  { x } ) )  =  ( S " (
I  \  { x } ) ) )
2120unieqd 3968 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  U. (
h " ( dom  h  \  { x } ) )  = 
U. ( S "
( I  \  {
x } ) ) )
2221fveq2d 5672 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( K `  U. ( h
" ( dom  h  \  { x } ) ) )  =  ( K `  U. ( S " ( I  \  { x } ) ) ) )
2315, 22ineq12d 3486 . . . . . . . . . . . 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 2395 . . . . . . . . . . 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 692 . . . . . . . . . 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 2859 . . . . . . . . 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 692 . . . . . . . 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 696 . . . . . . 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 3140 . . . . . 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 251 . . . . 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 424 . . . 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 344 . . 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 685 . 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 4154 . . 3  |-  ( G dom DProd  S  <->  <. G ,  S >.  e.  dom DProd  )
35 fvex 5682 . . . . . . . . . . . 12  |-  ( s `
 x )  e. 
_V
3635rgenw 2716 . . . . . . . . . . 11  |-  A. x  e.  dom  s ( s `
 x )  e. 
_V
37 ixpexg 7022 . . . . . . . . . . 11  |-  ( A. x  e.  dom  s ( s `  x )  e.  _V  ->  X_ x  e.  dom  s ( s `
 x )  e. 
_V )
3836, 37ax-mp 8 . . . . . . . . . 10  |-  X_ x  e.  dom  s ( s `
 x )  e. 
_V
3938rabex 4295 . . . . . . . . 9  |-  { h  e.  X_ x  e.  dom  s ( s `  x )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin }  e.  _V
4039mptex 5905 . . . . . . . 8  |-  ( f  e.  { h  e.  X_ x  e.  dom  s ( s `  x )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  e. 
_V
4140rnex 5073 . . . . . . 7  |-  ran  (
f  e.  { h  e.  X_ x  e.  dom  s ( s `  x )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  e. 
_V
4241rgen2w 2717 . . . . . 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 " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) )  e. 
_V
43 df-dprd 15483 . . . . . . 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 " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( g  gsumg  f ) ) )
4443fmpt2x 6356 . . . . . 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 " ( _V  \  {
( 0g `  g
) } ) )  e.  Fin }  |->  ( 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 )
4542, 44mpbi 200 . . . . 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
4645fdmi 5536 . . . 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 ) } ) ) } )
4746eleq2i 2451 . . 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 ) } ) ) } ) )
48 fveq2 5668 . . . . . . 7  |-  ( g  =  G  ->  (SubGrp `  g )  =  (SubGrp `  G ) )
49 feq3 5518 . . . . . . 7  |-  ( (SubGrp `  g )  =  (SubGrp `  G )  ->  (
h : dom  h --> (SubGrp `  g )  <->  h : dom  h --> (SubGrp `  G )
) )
5048, 49syl 16 . . . . . 6  |-  ( g  =  G  ->  (
h : dom  h --> (SubGrp `  g )  <->  h : dom  h --> (SubGrp `  G )
) )
51 fveq2 5668 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (Cntz `  g )  =  (Cntz `  G ) )
52 dmdprd.z . . . . . . . . . . . 12  |-  Z  =  (Cntz `  G )
5351, 52syl6eqr 2437 . . . . . . . . . . 11  |-  ( g  =  G  ->  (Cntz `  g )  =  Z )
5453fveq1d 5670 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(Cntz `  g ) `  ( h `  y
) )  =  ( Z `  ( h `
 y ) ) )
5554sseq2d 3319 . . . . . . . . 9  |-  ( g  =  G  ->  (
( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  <->  ( h `  x )  C_  ( Z `  ( h `  y ) ) ) )
5655ralbidv 2669 . . . . . . . 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
) ) ) )
5748fveq2d 5672 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (mrCls `  (SubGrp `  g )
)  =  (mrCls `  (SubGrp `  G ) ) )
58 dmdprd.k . . . . . . . . . . . 12  |-  K  =  (mrCls `  (SubGrp `  G
) )
5957, 58syl6eqr 2437 . . . . . . . . . . 11  |-  ( g  =  G  ->  (mrCls `  (SubGrp `  g )
)  =  K )
6059fveq1d 5670 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(mrCls `  (SubGrp `  g
) ) `  U. ( h " ( dom  h  \  { x } ) ) )  =  ( K `  U. ( h " ( dom  h  \  { x } ) ) ) )
6160ineq2d 3485 . . . . . . . . 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 } ) ) ) ) )
62 fveq2 5668 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
63 dmdprd.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
6462, 63syl6eqr 2437 . . . . . . . . . 10  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
6564sneqd 3770 . . . . . . . . 9  |-  ( g  =  G  ->  { ( 0g `  g ) }  =  {  .0.  } )
6661, 65eqeq12d 2401 . . . . . . . 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.  } ) )
6756, 66anbi12d 692 . . . . . . 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.  } ) ) )
6867ralbidv 2669 . . . . . 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.  } ) ) )
6950, 68anbi12d 692 . . . . 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.  } ) ) ) )
7069abbidv 2501 . . . 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.  } ) ) } )
7170opeliunxp2 4953 . . 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.  } ) ) } ) )
7234, 47, 713bitri 263 . 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.  } ) ) } ) )
73 3anass 940 . 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.  }
) ) ) )
7433, 72, 733bitr4g 280 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   {cab 2373   A.wral 2649   {crab 2653   _Vcvv 2899   [.wsbc 3104    \ cdif 3260    i^i cin 3262    C_ wss 3263   {csn 3757   <.cop 3760   U.cuni 3957   U_ciun 4035   class class class wbr 4153    e. cmpt 4207    X. cxp 4816   `'ccnv 4817   dom cdm 4818   ran crn 4819   "cima 4821   -->wf 5390   ` cfv 5394  (class class class)co 6020   X_cixp 6999   Fincfn 7045   0gc0g 13650    gsumg cgsu 13651  mrClscmrc 13735   Grpcgrp 14612  SubGrpcsubg 14865  Cntzccntz 15041   DProd cdprd 15481
This theorem is referenced by:  dmdprdd  15487  dprdgrp  15490  dprdf  15491  dprdcntz  15493  dprddisj  15494  dprdres  15513  subgdmdprd  15519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-ixp 7000  df-dprd 15483
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