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Theorem dprddisj 15530
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 15521 . . . . . . . . 9  |-  Rel  dom DProd
54brrelex2i 4886 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
6 dmexg 5097 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
72, 5, 63syl 19 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
83, 7eqeltrrd 2487 . . . . . 6  |-  ( ph  ->  I  e.  _V )
9 eqid 2412 . . . . . . 7  |-  (Cntz `  G )  =  (Cntz `  G )
10 dprddisj.0 . . . . . . 7  |-  .0.  =  ( 0g `  G )
11 dprddisj.k . . . . . . 7  |-  K  =  (mrCls `  (SubGrp `  G
) )
129, 10, 11dmdprd 15522 . . . . . 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 643 . . . . 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 202 . . . 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 971 . . 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 448 . . . 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 2749 . . 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 16 . 2  |-  ( ph  ->  A. x  e.  I 
( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  } )
19 fveq2 5695 . . . . 5  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
20 sneq 3793 . . . . . . . . 9  |-  ( x  =  X  ->  { x }  =  { X } )
2120difeq2d 3433 . . . . . . . 8  |-  ( x  =  X  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
2221imaeq2d 5170 . . . . . . 7  |-  ( x  =  X  ->  ( S " ( I  \  { x } ) )  =  ( S
" ( I  \  { X } ) ) )
2322unieqd 3994 . . . . . 6  |-  ( x  =  X  ->  U. ( S " ( I  \  { x } ) )  =  U. ( S " ( I  \  { X } ) ) )
2423fveq2d 5699 . . . . 5  |-  ( x  =  X  ->  ( K `  U. ( S
" ( I  \  { x } ) ) )  =  ( K `  U. ( S " ( I  \  { X } ) ) ) )
2519, 24ineq12d 3511 . . . 4  |-  ( x  =  X  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  ( ( S `  X
)  i^i  ( K `  U. ( S "
( I  \  { X } ) ) ) ) )
2625eqeq1d 2420 . . 3  |-  ( x  =  X  ->  (
( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  }  <->  ( ( S `  X )  i^i  ( K `  U. ( S " ( I 
\  { X }
) ) ) )  =  {  .0.  }
) )
2726rspcv 3016 . 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 58 1  |-  ( ph  ->  ( ( 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 1721   A.wral 2674   _Vcvv 2924    \ cdif 3285    i^i cin 3287    C_ wss 3288   {csn 3782   U.cuni 3983   class class class wbr 4180   dom cdm 4845   "cima 4848   -->wf 5417   ` cfv 5421   0gc0g 13686  mrClscmrc 13771   Grpcgrp 14648  SubGrpcsubg 14901  Cntzccntz 15077   DProd cdprd 15517
This theorem is referenced by:  dprdfeq0  15543  dprdres  15549  dprdss  15550  dprdf1o  15553  dprd2da  15563  dmdprdsplit2lem  15566
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-ixp 7031  df-dprd 15519
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