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Theorem dmdprd 16879
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 3127 . . . . 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 6143 . . . . . . 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 3337 . . . . . 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 5210 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  h  =  dom  S )
11 simplr 754 . . . . . . . . . . 11  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  S  =  I )
1210, 11eqtrd 2508 . . . . . . . . . 10  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  dom  h  =  I )
139, 12feq12d 5725 . . . . . . . . 9  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h : dom  h --> (SubGrp `  G )  <->  S :
I --> (SubGrp `  G )
) )
1412difeq1d 3626 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( dom  h  \  { x } )  =  ( I  \  { x } ) )
159fveq1d 5873 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h `  x )  =  ( S `  x ) )
169fveq1d 5873 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h `  y )  =  ( S `  y ) )
1716fveq2d 5875 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( Z `  ( h `  y ) )  =  ( Z `  ( S `  y )
) )
1815, 17sseq12d 3538 . . . . . . . . . . . 12  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
( h `  x
)  C_  ( Z `  ( h `  y
) )  <->  ( S `  x )  C_  ( Z `  ( S `  y ) ) ) )
1914, 18raleqbidv 3077 . . . . . . . . . . 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 5343 . . . . . . . . . . . . . . 15  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  (
h " ( dom  h  \  { x } ) )  =  ( S " (
I  \  { x } ) ) )
2120unieqd 4260 . . . . . . . . . . . . . 14  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  U. (
h " ( dom  h  \  { x } ) )  = 
U. ( S "
( I  \  {
x } ) ) )
2221fveq2d 5875 . . . . . . . . . . . . 13  |-  ( ( ( I  e.  V  /\  dom  S  =  I )  /\  h  =  S )  ->  ( K `  U. ( h
" ( dom  h  \  { x } ) ) )  =  ( K `  U. ( S " ( I  \  { x } ) ) ) )
2315, 22ineq12d 3706 . . . . . . . . . . . 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 2469 . . . . . . . . . . 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 3077 . . . . . . . . 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 3373 . . . . . 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 4453 . . 3  |-  ( G dom DProd  S  <->  <. G ,  S >.  e.  dom DProd  )
35 fvex 5881 . . . . . . . . . . . 12  |-  ( s `
 x )  e. 
_V
3635rgenw 2828 . . . . . . . . . . 11  |-  A. x  e.  dom  s ( s `
 x )  e. 
_V
37 ixpexg 7503 . . . . . . . . . . 11  |-  ( A. x  e.  dom  s ( s `  x )  e.  _V  ->  X_ x  e.  dom  s ( s `
 x )  e. 
_V )
3836, 37ax-mp 5 . . . . . . . . . 10  |-  X_ x  e.  dom  s ( s `
 x )  e. 
_V
3938rabex 4603 . . . . . . . . 9  |-  { h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp 
( 0g `  g
) }  e.  _V
4039mptex 6141 . . . . . . . 8  |-  ( f  e.  { h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  e. 
_V
4140rnex 6728 . . . . . . 7  |-  ran  (
f  e.  { h  e.  X_ x  e.  dom  s ( s `  x )  |  h finSupp 
( 0g `  g
) }  |->  ( g 
gsumg  f ) )  e. 
_V
4241rgen2w 2829 . . . . . 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
43 df-dprd 16876 . . . . . . 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 ) ) )
4443fmpt2x 6860 . . . . . 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 )
4542, 44mpbi 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
4645fdmi 5741 . . . 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 2545 . . 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 5871 . . . . . . 7  |-  ( g  =  G  ->  (SubGrp `  g )  =  (SubGrp `  G ) )
49 feq3 5720 . . . . . . 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 5871 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (Cntz `  g )  =  (Cntz `  G ) )
52 dmdprd.z . . . . . . . . . . . 12  |-  Z  =  (Cntz `  G )
5351, 52syl6eqr 2526 . . . . . . . . . . 11  |-  ( g  =  G  ->  (Cntz `  g )  =  Z )
5453fveq1d 5873 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(Cntz `  g ) `  ( h `  y
) )  =  ( Z `  ( h `
 y ) ) )
5554sseq2d 3537 . . . . . . . . 9  |-  ( g  =  G  ->  (
( h `  x
)  C_  ( (Cntz `  g ) `  (
h `  y )
)  <->  ( h `  x )  C_  ( Z `  ( h `  y ) ) ) )
5655ralbidv 2906 . . . . . . . 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 5875 . . . . . . . . . . . 12  |-  ( g  =  G  ->  (mrCls `  (SubGrp `  g )
)  =  (mrCls `  (SubGrp `  G ) ) )
58 dmdprd.k . . . . . . . . . . . 12  |-  K  =  (mrCls `  (SubGrp `  G
) )
5957, 58syl6eqr 2526 . . . . . . . . . . 11  |-  ( g  =  G  ->  (mrCls `  (SubGrp `  g )
)  =  K )
6059fveq1d 5873 . . . . . . . . . 10  |-  ( g  =  G  ->  (
(mrCls `  (SubGrp `  g
) ) `  U. ( h " ( dom  h  \  { x } ) ) )  =  ( K `  U. ( h " ( dom  h  \  { x } ) ) ) )
6160ineq2d 3705 . . . . . . . . 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 5871 . . . . . . . . . . 11  |-  ( g  =  G  ->  ( 0g `  g )  =  ( 0g `  G
) )
63 dmdprd.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
6462, 63syl6eqr 2526 . . . . . . . . . 10  |-  ( g  =  G  ->  ( 0g `  g )  =  .0.  )
6564sneqd 4044 . . . . . . . . 9  |-  ( g  =  G  ->  { ( 0g `  g ) }  =  {  .0.  } )
6661, 65eqeq12d 2489 . . . . . . . 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 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.  } ) ) )
6867ralbidv 2906 . . . . . 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 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.  } ) ) ) )
7069abbidv 2603 . . . 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 5146 . . 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 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.  } ) ) } ) )
73 3anass 977 . 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 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 973    = wceq 1379    e. wcel 1767   {cab 2452   A.wral 2817   {crab 2821   _Vcvv 3118   [.wsbc 3336    \ cdif 3478    i^i cin 3480    C_ wss 3481   {csn 4032   <.cop 4038   U.cuni 4250   U_ciun 4330   class class class wbr 4452    |-> cmpt 4510    X. cxp 5002   dom cdm 5004   ran crn 5005   "cima 5007   -->wf 5589   ` cfv 5593  (class class class)co 6294   X_cixp 7479   finSupp cfsupp 7839   0gc0g 14707    gsumg cgsu 14708  mrClscmrc 14850   Grpcgrp 15902  SubGrpcsubg 16044  Cntzccntz 16202   DProd cdprd 16874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-oprab 6298  df-mpt2 6299  df-1st 6794  df-2nd 6795  df-ixp 7480  df-dprd 16876
This theorem is referenced by:  dmdprdd  16880  dprdgrp  16888  dprdf  16889  dprdcntz  16891  dprddisj  16892  dprdres  16924  subgdmdprd  16930
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