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Theorem dprddisj 17641
Description: The function  S is a family having trivial intersections. (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 )
42, 3dprddomcld 17633 . . . . . 6  |-  ( ph  ->  I  e.  _V )
5 eqid 2451 . . . . . . 7  |-  (Cntz `  G )  =  (Cntz `  G )
6 dprddisj.0 . . . . . . 7  |-  .0.  =  ( 0g `  G )
7 dprddisj.k . . . . . . 7  |-  K  =  (mrCls `  (SubGrp `  G
) )
85, 6, 7dmdprd 17630 . . . . . 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.  } ) ) ) )
94, 3, 8syl2anc 667 . . . . 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.  }
) ) ) )
102, 9mpbid 214 . . . 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.  } ) ) )
1110simp3d 1022 . . 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.  }
) )
12 simpr 463 . . . 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.  } )
1312ralimi 2781 . . 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.  }
)
1411, 13syl 17 . 2  |-  ( ph  ->  A. x  e.  I 
( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  } )
15 fveq2 5865 . . . . 5  |-  ( x  =  X  ->  ( S `  x )  =  ( S `  X ) )
16 sneq 3978 . . . . . . . . 9  |-  ( x  =  X  ->  { x }  =  { X } )
1716difeq2d 3551 . . . . . . . 8  |-  ( x  =  X  ->  (
I  \  { x } )  =  ( I  \  { X } ) )
1817imaeq2d 5168 . . . . . . 7  |-  ( x  =  X  ->  ( S " ( I  \  { x } ) )  =  ( S
" ( I  \  { X } ) ) )
1918unieqd 4208 . . . . . 6  |-  ( x  =  X  ->  U. ( S " ( I  \  { x } ) )  =  U. ( S " ( I  \  { X } ) ) )
2019fveq2d 5869 . . . . 5  |-  ( x  =  X  ->  ( K `  U. ( S
" ( I  \  { x } ) ) )  =  ( K `  U. ( S " ( I  \  { X } ) ) ) )
2115, 20ineq12d 3635 . . . 4  |-  ( x  =  X  ->  (
( S `  x
)  i^i  ( K `  U. ( S "
( I  \  {
x } ) ) ) )  =  ( ( S `  X
)  i^i  ( K `  U. ( S "
( I  \  { X } ) ) ) ) )
2221eqeq1d 2453 . . 3  |-  ( x  =  X  ->  (
( ( S `  x )  i^i  ( K `  U. ( S
" ( I  \  { x } ) ) ) )  =  {  .0.  }  <->  ( ( S `  X )  i^i  ( K `  U. ( S " ( I 
\  { X }
) ) ) )  =  {  .0.  }
) )
2322rspcv 3146 . 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.  } ) )
241, 14, 23sylc 62 1  |-  ( ph  ->  ( ( S `  X )  i^i  ( K `  U. ( S
" ( I  \  { X } ) ) ) )  =  {  .0.  } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045    \ cdif 3401    i^i cin 3403    C_ wss 3404   {csn 3968   U.cuni 4198   class class class wbr 4402   dom cdm 4834   "cima 4837   -->wf 5578   ` cfv 5582   0gc0g 15338  mrClscmrc 15489   Grpcgrp 16669  SubGrpcsubg 16811  Cntzccntz 16969   DProd cdprd 17625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-ixp 7523  df-dprd 17627
This theorem is referenced by:  dprdfeq0  17655  dprdres  17661  dprdss  17662  dprdf1o  17665  dprd2da  17675  dmdprdsplit2lem  17678
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