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Theorem dprdfaddOLD 16649
Description: Take the sum of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.) Obsolete version of dprdfadd 16642 as of 14-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
eldprdiOLD.0  |-  .0.  =  ( 0g `  G )
eldprdiOLD.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
eldprdiOLD.1  |-  ( ph  ->  G dom DProd  S )
eldprdiOLD.2  |-  ( ph  ->  dom  S  =  I )
eldprdiOLD.3  |-  ( ph  ->  F  e.  W )
dprdfaddOLD.4  |-  ( ph  ->  H  e.  W )
dprdfaddOLD.b  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
dprdfaddOLD  |-  ( ph  ->  ( ( F  oF  .+  H )  e.  W  /\  ( G 
gsumg  ( F  oF  .+  H ) )  =  ( ( G  gsumg  F ) 
.+  ( G  gsumg  H ) ) ) )
Distinct variable groups:    .+ , h    h, F    h, H    h, i, G    h, I, i    .0. , h    S, h, i
Allowed substitution hints:    ph( h, i)    .+ ( i)    F( i)    H( i)    W( h, i)    .0. ( i)

Proof of Theorem dprdfaddOLD
Dummy variables  k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldprdiOLD.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
2 eldprdiOLD.1 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
3 reldmdprd 16611 . . . . . . 7  |-  Rel  dom DProd
43brrelex2i 4991 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
5 dmexg 6622 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
62, 4, 53syl 20 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
71, 6eqeltrrd 2543 . . . 4  |-  ( ph  ->  I  e.  _V )
8 eldprdiOLD.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
9 eldprdiOLD.3 . . . . 5  |-  ( ph  ->  F  e.  W )
108, 2, 1, 9dprdfclOLD 16635 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
11 dprdfaddOLD.4 . . . . 5  |-  ( ph  ->  H  e.  W )
128, 2, 1, 11dprdfclOLD 16635 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  ( S `  x
) )
13 eqid 2454 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
148, 2, 1, 9, 13dprdffOLD 16634 . . . . 5  |-  ( ph  ->  F : I --> ( Base `  G ) )
1514feqmptd 5856 . . . 4  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
168, 2, 1, 11, 13dprdffOLD 16634 . . . . 5  |-  ( ph  ->  H : I --> ( Base `  G ) )
1716feqmptd 5856 . . . 4  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
187, 10, 12, 15, 17offval2 6449 . . 3  |-  ( ph  ->  ( F  oF  .+  H )  =  ( x  e.  I  |->  ( ( F `  x )  .+  ( H `  x )
) ) )
192, 1dprdf2 16623 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
2019ffvelrnda 5955 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
21 dprdfaddOLD.b . . . . . 6  |-  .+  =  ( +g  `  G )
2221subgcl 15814 . . . . 5  |-  ( ( ( S `  x
)  e.  (SubGrp `  G )  /\  ( F `  x )  e.  ( S `  x
)  /\  ( H `  x )  e.  ( S `  x ) )  ->  ( ( F `  x )  .+  ( H `  x
) )  e.  ( S `  x ) )
2320, 10, 12, 22syl3anc 1219 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
)  .+  ( H `  x ) )  e.  ( S `  x
) )
248, 2, 1, 9dprdffiOLD 16636 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
258, 2, 1, 11dprdffiOLD 16636 . . . . . 6  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
26 unfi 7693 . . . . . 6  |-  ( ( ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' H " ( _V  \  {  .0.  } ) )  e.  Fin )  -> 
( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )  e.  Fin )
2724, 25, 26syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )  e.  Fin )
28 ssun1 3630 . . . . . . . . . 10  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )
2928a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) )
3014, 29suppssrOLD 5949 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  ( F `  x )  =  .0.  )
31 ssun2 3631 . . . . . . . . . 10  |-  ( `' H " ( _V 
\  {  .0.  }
) )  C_  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )
3231a1i 11 . . . . . . . . 9  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  C_  ( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) )
3316, 32suppssrOLD 5949 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  ( H `  x )  =  .0.  )
3430, 33oveq12d 6221 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F `  x
)  .+  ( H `  x ) )  =  (  .0.  .+  .0.  ) )
35 dprdgrp 16621 . . . . . . . . . 10  |-  ( G dom DProd  S  ->  G  e. 
Grp )
362, 35syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
37 eldprdiOLD.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
3813, 37grpidcl 15689 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
3936, 38syl 16 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
4013, 21, 37grplid 15691 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  .+  .0.  )  =  .0.  )
4136, 39, 40syl2anc 661 . . . . . . . 8  |-  ( ph  ->  (  .0.  .+  .0.  )  =  .0.  )
4241adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (  .0.  .+  .0.  )  =  .0.  )
4334, 42eqtrd 2495 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F `  x
)  .+  ( H `  x ) )  =  .0.  )
4443suppss2OLD 6428 . . . . 5  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( ( F `  x ) 
.+  ( H `  x ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  ( ( `' F " ( _V 
\  {  .0.  }
) )  u.  ( `' H " ( _V 
\  {  .0.  }
) ) ) )
45 ssfi 7647 . . . . 5  |-  ( ( ( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )  e.  Fin  /\  ( `' ( x  e.  I  |->  ( ( F `
 x )  .+  ( H `  x ) ) ) " ( _V  \  {  .0.  }
) )  C_  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) )  ->  ( `' ( x  e.  I  |->  ( ( F `  x )  .+  ( H `  x )
) ) " ( _V  \  {  .0.  }
) )  e.  Fin )
4627, 44, 45syl2anc 661 . . . 4  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( ( F `  x ) 
.+  ( H `  x ) ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
478, 2, 1, 23, 46dprdwdOLD 16633 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x )  .+  ( H `  x )
) )  e.  W
)
4818, 47eqeltrd 2542 . 2  |-  ( ph  ->  ( F  oF  .+  H )  e.  W )
49 eqid 2454 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
50 grpmnd 15673 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5136, 50syl 16 . . 3  |-  ( ph  ->  G  e.  Mnd )
52 eqid 2454 . . 3  |-  ( `' ( F  u.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  u.  H ) " ( _V  \  {  .0.  }
) )
538, 2, 1, 9, 49dprdfcntzOLD 16637 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
548, 2, 1, 11, 49dprdfcntzOLD 16637 . . 3  |-  ( ph  ->  ran  H  C_  (
(Cntz `  G ) `  ran  H ) )
558, 2, 1, 48, 49dprdfcntzOLD 16637 . . 3  |-  ( ph  ->  ran  ( F  oF  .+  H )  C_  ( (Cntz `  G ) `  ran  ( F  oF  .+  H ) ) )
5651adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  G  e.  Mnd )
57 vex 3081 . . . . . . . 8  |-  x  e. 
_V
5857a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  x  e.  _V )
59 eldifi 3589 . . . . . . . . . . 11  |-  ( k  e.  ( I  \  x )  ->  k  e.  I )
6059adantl 466 . . . . . . . . . 10  |-  ( ( x  C_  I  /\  k  e.  ( I  \  x ) )  -> 
k  e.  I )
61 ffvelrn 5953 . . . . . . . . . 10  |-  ( ( F : I --> ( Base `  G )  /\  k  e.  I )  ->  ( F `  k )  e.  ( Base `  G
) )
6214, 60, 61syl2an 477 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( F `  k
)  e.  ( Base `  G ) )
6362snssd 4129 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( F `  k ) }  C_  ( Base `  G )
)
6413, 49cntzsubm 15976 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  { ( F `  k
) }  C_  ( Base `  G ) )  ->  ( (Cntz `  G ) `  {
( F `  k
) } )  e.  (SubMnd `  G )
)
6556, 63, 64syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( (Cntz `  G
) `  { ( F `  k ) } )  e.  (SubMnd `  G ) )
6616adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  H : I --> ( Base `  G ) )
67 ffn 5670 . . . . . . . . . 10  |-  ( H : I --> ( Base `  G )  ->  H  Fn  I )
6866, 67syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  H  Fn  I )
69 simprl 755 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  x  C_  I )
70 fnssres 5635 . . . . . . . . 9  |-  ( ( H  Fn  I  /\  x  C_  I )  -> 
( H  |`  x
)  Fn  x )
7168, 69, 70syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
)  Fn  x )
72 fvres 5816 . . . . . . . . . . 11  |-  ( y  e.  x  ->  (
( H  |`  x
) `  y )  =  ( H `  y ) )
7372adantl 466 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( ( H  |`  x ) `  y )  =  ( H `  y ) )
742ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  G dom DProd  S )
751ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  dom  S  =  I )
7674, 75dprdf2 16623 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  S :
I --> (SubGrp `  G )
)
7760ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  k  e.  I )
7876, 77ffvelrnd 5956 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( S `  k )  e.  (SubGrp `  G ) )
7913subgss 15805 . . . . . . . . . . . . 13  |-  ( ( S `  k )  e.  (SubGrp `  G
)  ->  ( S `  k )  C_  ( Base `  G ) )
8078, 79syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( S `  k )  C_  ( Base `  G ) )
819ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  F  e.  W )
828, 74, 75, 81dprdfclOLD 16635 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  /\  k  e.  I )  ->  ( F `  k )  e.  ( S `  k
) )
8377, 82mpdan 668 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( F `  k )  e.  ( S `  k ) )
8483snssd 4129 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  { ( F `  k ) }  C_  ( S `  k ) )
8513, 49cntz2ss 15973 . . . . . . . . . . . 12  |-  ( ( ( S `  k
)  C_  ( Base `  G )  /\  {
( F `  k
) }  C_  ( S `  k )
)  ->  ( (Cntz `  G ) `  ( S `  k )
)  C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
8680, 84, 85syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( (Cntz `  G ) `  ( S `  k )
)  C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
8769sselda 3467 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  e.  I )
88 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  e.  x )
89 simplrr 760 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  k  e.  ( I  \  x
) )
9089eldifbd 3452 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  -.  k  e.  x )
91 nelne2 2782 . . . . . . . . . . . . . 14  |-  ( ( y  e.  x  /\  -.  k  e.  x
)  ->  y  =/=  k )
9288, 90, 91syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  =/=  k )
9374, 75, 87, 77, 92, 49dprdcntz 16624 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( S `  y )  C_  (
(Cntz `  G ) `  ( S `  k
) ) )
9411ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  H  e.  W )
958, 74, 75, 94dprdfclOLD 16635 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  /\  y  e.  I )  ->  ( H `  y )  e.  ( S `  y
) )
9687, 95mpdan 668 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( S `  y ) )
9793, 96sseldd 3468 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( (Cntz `  G ) `  ( S `  k
) ) )
9886, 97sseldd 3468 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
9973, 98eqeltrd 2542 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( ( H  |`  x ) `  y )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
10099ralrimiva 2830 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  A. y  e.  x  ( ( H  |`  x ) `  y
)  e.  ( (Cntz `  G ) `  {
( F `  k
) } ) )
101 ffnfv 5981 . . . . . . . 8  |-  ( ( H  |`  x ) : x --> ( (Cntz `  G ) `  {
( F `  k
) } )  <->  ( ( H  |`  x )  Fn  x  /\  A. y  e.  x  ( ( H  |`  x ) `  y )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) ) )
10271, 100, 101sylanbrc 664 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
) : x --> ( (Cntz `  G ) `  {
( F `  k
) } ) )
103 resss 5245 . . . . . . . . . 10  |-  ( H  |`  x )  C_  H
104 rnss 5179 . . . . . . . . . 10  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
105103, 104ax-mp 5 . . . . . . . . 9  |-  ran  ( H  |`  x )  C_  ran  H
10649cntzidss 15978 . . . . . . . . 9  |-  ( ( ran  H  C_  (
(Cntz `  G ) `  ran  H )  /\  ran  ( H  |`  x
)  C_  ran  H )  ->  ran  ( H  |`  x )  C_  (
(Cntz `  G ) `  ran  ( H  |`  x ) ) )
10754, 105, 106sylancl 662 . . . . . . . 8  |-  ( ph  ->  ran  ( H  |`  x )  C_  (
(Cntz `  G ) `  ran  ( H  |`  x ) ) )
108107adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  ran  ( H  |`  x
)  C_  ( (Cntz `  G ) `  ran  ( H  |`  x ) ) )
109 cnvss 5123 . . . . . . . . . 10  |-  ( ( H  |`  x )  C_  H  ->  `' ( H  |`  x )  C_  `' H )
110 imass1 5314 . . . . . . . . . 10  |-  ( `' ( H  |`  x
)  C_  `' H  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) ) )
111103, 109, 110mp2b 10 . . . . . . . . 9  |-  ( `' ( H  |`  x
) " ( _V 
\  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) )
112 ssfi 7647 . . . . . . . . 9  |-  ( ( ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( H  |`  x )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  } ) )  e.  Fin )
11325, 111, 112sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
114113adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
11537, 49, 56, 58, 65, 102, 108, 114gsumzsubmclOLD 16528 . . . . . 6  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
116115snssd 4129 . . . . 5  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
117 fssres 5689 . . . . . . . . 9  |-  ( ( H : I --> ( Base `  G )  /\  x  C_  I )  ->  ( H  |`  x ) : x --> ( Base `  G
) )
11866, 69, 117syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
) : x --> ( Base `  G ) )
11913, 37, 49, 56, 58, 118, 108, 114gsumzclOLD 16518 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  (
Base `  G )
)
120119snssd 4129 . . . . . 6  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( Base `  G
) )
12113, 49cntzrec 15974 . . . . . 6  |-  ( ( { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( Base `  G
)  /\  { ( F `  k ) }  C_  ( Base `  G
) )  ->  ( { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( (Cntz `  G ) `  {
( F `  k
) } )  <->  { ( F `  k ) }  C_  ( (Cntz `  G ) `  {
( G  gsumg  ( H  |`  x
) ) } ) ) )
122120, 63, 121syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( { ( G 
gsumg  ( H  |`  x ) ) }  C_  (
(Cntz `  G ) `  { ( F `  k ) } )  <->  { ( F `  k ) }  C_  ( (Cntz `  G ) `  { ( G  gsumg  ( H  |`  x ) ) } ) ) )
123116, 122mpbid 210 . . . 4  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( F `  k ) }  C_  ( (Cntz `  G ) `  { ( G  gsumg  ( H  |`  x ) ) } ) )
124 fvex 5812 . . . . 5  |-  ( F `
 k )  e. 
_V
125124snss 4110 . . . 4  |-  ( ( F `  k )  e.  ( (Cntz `  G ) `  {
( G  gsumg  ( H  |`  x
) ) } )  <->  { ( F `  k ) }  C_  ( (Cntz `  G ) `  { ( G  gsumg  ( H  |`  x ) ) } ) )
126123, 125sylibr 212 . . 3  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( F `  k
)  e.  ( (Cntz `  G ) `  {
( G  gsumg  ( H  |`  x
) ) } ) )
12713, 37, 21, 49, 51, 7, 24, 25, 52, 14, 16, 53, 54, 55, 126gsumzaddlemOLD 16535 . 2  |-  ( ph  ->  ( G  gsumg  ( F  oF  .+  H ) )  =  ( ( G 
gsumg  F )  .+  ( G  gsumg  H ) ) )
12848, 127jca 532 1  |-  ( ph  ->  ( ( F  oF  .+  H )  e.  W  /\  ( G 
gsumg  ( F  oF  .+  H ) )  =  ( ( G  gsumg  F ) 
.+  ( G  gsumg  H ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   {crab 2803   _Vcvv 3078    \ cdif 3436    u. cun 3437    C_ wss 3439   {csn 3988   class class class wbr 4403    |-> cmpt 4461   `'ccnv 4950   dom cdm 4951   ran crn 4952    |` cres 4953   "cima 4954    Fn wfn 5524   -->wf 5525   ` cfv 5529  (class class class)co 6203    oFcof 6431   X_cixp 7376   Fincfn 7423   Basecbs 14296   +g cplusg 14361   0gc0g 14501    gsumg cgsu 14502   Mndcmnd 15532   Grpcgrp 15533  SubMndcsubmnd 15586  SubGrpcsubg 15798  Cntzccntz 15956   DProd cdprd 16607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-ixp 7377  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-n0 10695  df-z 10762  df-uz 10977  df-fz 11559  df-fzo 11670  df-seq 11928  df-hash 12225  df-ndx 14299  df-slot 14300  df-base 14301  df-sets 14302  df-ress 14303  df-plusg 14374  df-0g 14503  df-gsum 14504  df-mnd 15538  df-submnd 15588  df-grp 15668  df-subg 15801  df-cntz 15958  df-dprd 16609
This theorem is referenced by:  dprdfsubOLD  16650
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