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Theorem dprddisj 15572
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 15563 . . . . . . . . 9  |-  Rel  dom DProd
54brrelex2i 4922 . . . . . . . 8  |-  ( G dom DProd  S  ->  S  e. 
_V )
6 dmexg 5133 . . . . . . . 8  |-  ( S  e.  _V  ->  dom  S  e.  _V )
72, 5, 63syl 19 . . . . . . 7  |-  ( ph  ->  dom  S  e.  _V )
83, 7eqeltrrd 2513 . . . . . 6  |-  ( ph  ->  I  e.  _V )
9 eqid 2438 . . . . . . 7  |-  (Cntz `  G )  =  (Cntz `  G )
10 dprddisj.0 . . . . . . 7  |-  .0.  =  ( 0g `  G )
11 dprddisj.k . . . . . . 7  |-  K  =  (mrCls `  (SubGrp `  G
) )
129, 10, 11dmdprd 15564 . . . . . 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 644 . . . . 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 203 . . . 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 972 . . 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 449 . . . 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 2783 . . 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 5731 . . . . 5  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
20 sneq 3827 . . . . . . . . 9  |-  ( x  =  X  ->  { x }  =  { X } )
2120difeq2d 3467 . . . . . . . 8  |-  ( x  =  X  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
2221imaeq2d 5206 . . . . . . 7  |-  ( x  =  X  ->  ( S " ( I  \  { x } ) )  =  ( S
" ( I  \  { X } ) ) )
2322unieqd 4028 . . . . . 6  |-  ( x  =  X  ->  U. ( S " ( I  \  { x } ) )  =  U. ( S " ( I  \  { X } ) ) )
2423fveq2d 5735 . . . . 5  |-  ( x  =  X  ->  ( K `  U. ( S
" ( I  \  { x } ) ) )  =  ( K `  U. ( S " ( I  \  { X } ) ) ) )
2519, 24ineq12d 3545 . . . 4  |-  ( x  =  X  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  ( ( S `  X
)  i^i  ( K `  U. ( S "
( I  \  { X } ) ) ) ) )
2625eqeq1d 2446 . . 3  |-  ( x  =  X  ->  (
( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  }  <->  ( ( S `  X )  i^i  ( K `  U. ( S " ( I 
\  { X }
) ) ) )  =  {  .0.  }
) )
2726rspcv 3050 . 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 59 1  |-  ( ph  ->  ( ( S `  X )  i^i  ( K `  U. ( S
" ( I  \  { X } ) ) ) )  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   {csn 3816   U.cuni 4017   class class class wbr 4215   dom cdm 4881   "cima 4884   -->wf 5453   ` cfv 5457   0gc0g 13728  mrClscmrc 13813   Grpcgrp 14690  SubGrpcsubg 14943  Cntzccntz 15119   DProd cdprd 15559
This theorem is referenced by:  dprdfeq0  15585  dprdres  15591  dprdss  15592  dprdf1o  15595  dprd2da  15605  dmdprdsplit2lem  15608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-ixp 7067  df-dprd 15561
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