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Theorem dmdprd 15236
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 2796 . . . . 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 10 . . . 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 5749 . . . . . . 7  |-  ( ( S : I --> (SubGrp `  G )  /\  I  e.  V )  ->  S  e.  _V )
43expcom 424 . . . . . 6  |-  ( I  e.  V  ->  ( S : I --> (SubGrp `  G )  ->  S  e.  _V ) )
54adantr 451 . . . . 5  |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( S :
I --> (SubGrp `  G )  ->  S  e.  _V )
)
65adantrd 454 . . . 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 2992 . . . . . 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 447 . . . . . . 7  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  S  e. 
_V )  ->  S  e.  _V )
9 simpr 447 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  h  =  S )
109dmeqd 4881 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  h  =  dom  S )
11 simplr 731 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  S  =  I )
1210, 11eqtrd 2315 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  h  =  I )
139, 12feq12d 5381 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h : dom  h --> (SubGrp `  G )  <->  S :
I --> (SubGrp `  G )
) )
1412difeq1d 3293 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( dom  h  \  { x } )  =  ( I  \  { x } ) )
159fveq1d 5527 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h `  x )  =  ( S `  x ) )
169fveq1d 5527 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h `  y )  =  ( S `  y ) )
1716fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( Z `  ( h `  y ) )  =  ( Z `  ( S `  y )
) )
1815, 17sseq12d 3207 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
( h `  x
)  C_  ( Z `  ( h `  y
) )  <->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
1914, 18raleqbidv 2748 . . . . . . . . . . 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 ) ) ) )
209imaeq1d 5011 . . . . . . . . . . . . . . . 16  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h " ( dom  h  \  { x } ) )  =  ( S " ( dom  h  \  { x } ) ) )
2114imaeq2d 5012 . . . . . . . . . . . . . . . 16  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( S " ( dom  h  \  { x } ) )  =  ( S
" ( I  \  { x } ) ) )
2220, 21eqtrd 2315 . . . . . . . . . . . . . . 15  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h " ( dom  h  \  { x } ) )  =  ( S " (
I  \  { x } ) ) )
2322unieqd 3838 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  U. (
h " ( dom  h  \  { x } ) )  = 
U. ( S "
( I  \  {
x } ) ) )
2423fveq2d 5529 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( K `  U. ( h
" ( dom  h  \  { x } ) ) )  =  ( K `  U. ( S " ( I  \  { x } ) ) ) )
2515, 24ineq12d 3371 . . . . . . . . . . . 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 } ) ) ) ) )
2625eqeq1d 2291 . . . . . . . . . . 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.  }
) )
2719, 26anbi12d 691 . . . . . . . . . 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.  } ) ) )
2812, 27raleqbidv 2748 . . . . . . . . 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.  } ) ) )
2913, 28anbi12d 691 . . . . . . . 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.  }
) ) ) )
3029adantlr 695 . . . . . . 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.  }
) ) ) )
318, 30sbcied 3027 . . . . . 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.  }
) ) ) )
327, 31syl5bbr 250 . . . . 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.  } ) ) ) )
3332ex 423 . . . 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.  } ) ) ) ) )
342, 6, 33pm5.21ndd 343 . . 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.  } ) ) ) )
3534anbi2d 684 . 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.  }
) ) ) ) )
36 df-br 4024 . . 3  |-  ( G dom DProd  S  <->  <. G ,  S >.  e.  dom DProd  )
37 fvex 5539 . . . . . . . . . . . 12  |-  ( s `
 x )  e. 
_V
3837rgenw 2610 . . . . . . . . . . 11  |-  A. x  e.  dom  s ( s `
 x )  e. 
_V
39 ixpexg 6840 . . . . . . . . . . 11  |-  ( A. x  e.  dom  s ( s `  x )  e.  _V  ->  X_ x  e.  dom  s ( s `
 x )  e. 
_V )
4038, 39ax-mp 8 . . . . . . . . . 10  |-  X_ x  e.  dom  s ( s `
 x )  e. 
_V
4140rabex 4165 . . . . . . . . 9  |-  { h  e.  X_ x  e.  dom  s ( s `  x )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin }  e.  _V
4241mptex 5746 . . . . . . . 8  |-  ( f  e.  { h  e.  X_ x  e.  dom  s ( s `  x )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  e. 
_V
4342rnex 4942 . . . . . . 7  |-  ran  (
f  e.  { h  e.  X_ x  e.  dom  s ( s `  x )  |  ( `' h " ( _V 
\  { ( 0g
`  g ) } ) )  e.  Fin } 
|->  ( g  gsumg  f ) )  e. 
_V
4443rgen2w 2611 . . . . . 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
45 df-dprd 15233 . . . . . . 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 ) ) )
4645fmpt2x 6190 . . . . . 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 )
4744, 46mpbi 199 . . . . 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
4847fdmi 5394 . . . 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 ) } ) ) } )
4948eleq2i 2347 . . 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 ) } ) ) } ) )
50 fveq2 5525 . . . . . . 7  |-  ( g  =  G  ->  (SubGrp `  g )  =  (SubGrp `  G ) )
51 feq3 5377 . . . . . . 7  |-  ( (SubGrp `  g )  =  (SubGrp `  G )  ->  (
h : dom  h --> (SubGrp `  g )  <->  h : dom  h --> (SubGrp `  G )
) )
5250, 51syl 15 . . . . . 6  |-  ( g  =  G  ->  (
h : dom  h --> (SubGrp `  g )  <->  h : dom  h --> (SubGrp `  G )
) )
53 fveq2 5525 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (Cntz `  g )  =  (Cntz `  G ) )
54 dmdprd.z . . . . . . . . . . . 12  |-  Z  =  (Cntz `  G )
5553, 54syl6eqr 2333 . . . . . . . . . . 11  |-  ( g  =  G  ->  (Cntz `  g )  =  Z )
5655fveq1d 5527 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(Cntz `  g ) `  ( h `  y
) )  =  ( Z `  ( h `
 y ) ) )
5756sseq2d 3206 . . . . . . . . 9  |-  ( g  =  G  ->  (
( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  <->  ( h `  x )  C_  ( Z `  ( h `  y ) ) ) )
5857ralbidv 2563 . . . . . . . 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
) ) ) )
5950fveq2d 5529 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (mrCls `  (SubGrp `  g )
)  =  (mrCls `  (SubGrp `  G ) ) )
60 dmdprd.k . . . . . . . . . . . 12  |-  K  =  (mrCls `  (SubGrp `  G
) )
6159, 60syl6eqr 2333 . . . . . . . . . . 11  |-  ( g  =  G  ->  (mrCls `  (SubGrp `  g )
)  =  K )
6261fveq1d 5527 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(mrCls `  (SubGrp `  g
) ) `  U. ( h " ( dom  h  \  { x } ) ) )  =  ( K `  U. ( h " ( dom  h  \  { x } ) ) ) )
6362ineq2d 3370 . . . . . . . . 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 } ) ) ) ) )
64 fveq2 5525 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
65 dmdprd.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
6664, 65syl6eqr 2333 . . . . . . . . . 10  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
6766sneqd 3653 . . . . . . . . 9  |-  ( g  =  G  ->  { ( 0g `  g ) }  =  {  .0.  } )
6863, 67eqeq12d 2297 . . . . . . . 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.  } ) )
6958, 68anbi12d 691 . . . . . . 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.  } ) ) )
7069ralbidv 2563 . . . . . 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.  } ) ) )
7152, 70anbi12d 691 . . . . 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.  } ) ) ) )
7271abbidv 2397 . . . 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.  } ) ) } )
7372opeliunxp2 4824 . . 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.  } ) ) } ) )
7436, 49, 733bitri 262 . 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.  } ) ) } ) )
75 3anass 938 . 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.  }
) ) ) )
7635, 74, 753bitr4g 279 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 176    /\ wa 358    /\ w3a 934    = 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:  dmdprdd  15237  dprdgrp  15240  dprdf  15241  dprdcntz  15243  dprddisj  15244  dprdres  15263  subgdmdprd  15269
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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-ixp 6818  df-dprd 15233
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