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Theorem dprddisj 15454
Description: The function  S is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
dprdcntz.1  |-  ( ph  ->  G dom DProd  S )
dprdcntz.2  |-  ( ph  ->  dom  S  =  I )
dprdcntz.3  |-  ( ph  ->  X  e.  I )
dprddisj.0  |-  .0.  =  ( 0g `  G )
dprddisj.k  |-  K  =  (mrCls `  (SubGrp `  G
) )
Assertion
Ref Expression
dprddisj  |-  ( ph  ->  ( ( S `  X )  i^i  ( K `  U. ( S
" ( I  \  { X } ) ) ) )  =  {  .0.  } )

Proof of Theorem dprddisj
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dprdcntz.3 . 2  |-  ( ph  ->  X  e.  I )
2 dprdcntz.1 . . . . 5  |-  ( ph  ->  G dom DProd  S )
3 dprdcntz.2 . . . . . . 7  |-  ( ph  ->  dom  S  =  I )
4 reldmdprd 15445 . . . . . . . . 9  |-  Rel  dom DProd
54brrelex2i 4833 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
6 dmexg 5042 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
72, 5, 63syl 18 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
83, 7eqeltrrd 2441 . . . . . 6  |-  ( ph  ->  I  e.  _V )
9 eqid 2366 . . . . . . 7  |-  (Cntz `  G )  =  (Cntz `  G )
10 dprddisj.0 . . . . . . 7  |-  .0.  =  ( 0g `  G )
11 dprddisj.k . . . . . . 7  |-  K  =  (mrCls `  (SubGrp `  G
) )
129, 10, 11dmdprd 15446 . . . . . 6  |-  ( ( 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_  ( (Cntz `  G ) `  ( S `  y
) )  /\  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } ) ) ) )
138, 3, 12syl2anc 642 . . . . 5  |-  ( ph  ->  ( G dom DProd  S  <->  ( G  e.  Grp  /\  S :
I --> (SubGrp `  G )  /\  A. x  e.  I 
( A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) ) ) )
142, 13mpbid 201 . . . 4  |-  ( ph  ->  ( G  e.  Grp  /\  S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( (Cntz `  G ) `  ( S `  y
) )  /\  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } ) ) )
1514simp3d 970 . . 3  |-  ( ph  ->  A. x  e.  I 
( A. y  e.  ( I  \  {
x } ) ( S `  x ) 
C_  ( (Cntz `  G ) `  ( S `  y )
)  /\  ( ( S `  x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
) )
16 simpr 447 . . . 4  |-  ( ( A. y  e.  ( I  \  { x } ) ( S `
 x )  C_  ( (Cntz `  G ) `  ( S `  y
) )  /\  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } )  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } )
1716ralimi 2703 . . 3  |-  ( A. x  e.  I  ( A. y  e.  (
I  \  { x } ) ( S `
 x )  C_  ( (Cntz `  G ) `  ( S `  y
) )  /\  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  {  .0.  } )  ->  A. x  e.  I  ( ( S `  x )  i^i  ( K `  U. ( S " ( I 
\  { x }
) ) ) )  =  {  .0.  }
)
1815, 17syl 15 . 2  |-  ( ph  ->  A. x  e.  I 
( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  } )
19 fveq2 5632 . . . . 5  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
20 sneq 3740 . . . . . . . . 9  |-  ( x  =  X  ->  { x }  =  { X } )
2120difeq2d 3381 . . . . . . . 8  |-  ( x  =  X  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
2221imaeq2d 5115 . . . . . . 7  |-  ( x  =  X  ->  ( S " ( I  \  { x } ) )  =  ( S
" ( I  \  { X } ) ) )
2322unieqd 3940 . . . . . 6  |-  ( x  =  X  ->  U. ( S " ( I  \  { x } ) )  =  U. ( S " ( I  \  { X } ) ) )
2423fveq2d 5636 . . . . 5  |-  ( x  =  X  ->  ( K `  U. ( S
" ( I  \  { x } ) ) )  =  ( K `  U. ( S " ( I  \  { X } ) ) ) )
2519, 24ineq12d 3459 . . . 4  |-  ( x  =  X  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  ( ( S `  X
)  i^i  ( K `  U. ( S "
( I  \  { X } ) ) ) ) )
2625eqeq1d 2374 . . 3  |-  ( x  =  X  ->  (
( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  }  <->  ( ( S `  X )  i^i  ( K `  U. ( S " ( I 
\  { X }
) ) ) )  =  {  .0.  }
) )
2726rspcv 2965 . 2  |-  ( X  e.  I  ->  ( A. x  e.  I 
( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  }  ->  ( ( S `  X
)  i^i  ( K `  U. ( S "
( I  \  { X } ) ) ) )  =  {  .0.  } ) )
281, 18, 27sylc 56 1  |-  ( ph  ->  ( ( 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 935    = wceq 1647    e. wcel 1715   A.wral 2628   _Vcvv 2873    \ cdif 3235    i^i cin 3237    C_ wss 3238   {csn 3729   U.cuni 3929   class class class wbr 4125   dom cdm 4792   "cima 4795   -->wf 5354   ` cfv 5358   0gc0g 13610  mrClscmrc 13695   Grpcgrp 14572  SubGrpcsubg 14825  Cntzccntz 15001   DProd cdprd 15441
This theorem is referenced by:  dprdfeq0  15467  dprdres  15473  dprdss  15474  dprdf1o  15477  dprd2da  15487  dmdprdsplit2lem  15490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-ixp 6961  df-dprd 15443
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