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Theorem dprdfaddOLD 17262
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 17255 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 17223 . . . . . . 7  |-  Rel  dom DProd
43brrelex2i 5030 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
5 dmexg 6704 . . . . . 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 17248 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
11 dprdfaddOLD.4 . . . . 5  |-  ( ph  ->  H  e.  W )
128, 2, 1, 11dprdfclOLD 17248 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  ( S `  x
) )
13 eqid 2454 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
148, 2, 1, 9, 13dprdffOLD 17247 . . . . 5  |-  ( ph  ->  F : I --> ( Base `  G ) )
1514feqmptd 5901 . . . 4  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
168, 2, 1, 11, 13dprdffOLD 17247 . . . . 5  |-  ( ph  ->  H : I --> ( Base `  G ) )
1716feqmptd 5901 . . . 4  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
187, 10, 12, 15, 17offval2 6529 . . 3  |-  ( ph  ->  ( F  oF  .+  H )  =  ( x  e.  I  |->  ( ( F `  x )  .+  ( H `  x )
) ) )
192, 1dprdf2 17235 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
2019ffvelrnda 6007 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
21 dprdfaddOLD.b . . . . . 6  |-  .+  =  ( +g  `  G )
2221subgcl 16410 . . . . 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 1226 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
)  .+  ( H `  x ) )  e.  ( S `  x
) )
248, 2, 1, 9dprdffiOLD 17249 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
258, 2, 1, 11dprdffiOLD 17249 . . . . . 6  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
26 unfi 7779 . . . . . 6  |-  ( ( ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' H " ( _V  \  {  .0.  } ) )  e.  Fin )  -> 
( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )  e.  Fin )
2724, 25, 26syl2anc 659 . . . . 5  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )  e.  Fin )
28 ssun1 3653 . . . . . . . . . 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 5997 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  ( F `  x )  =  .0.  )
31 ssun2 3654 . . . . . . . . . 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 5997 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  ( H `  x )  =  .0.  )
3430, 33oveq12d 6288 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F `  x
)  .+  ( H `  x ) )  =  (  .0.  .+  .0.  ) )
35 dprdgrp 17233 . . . . . . . . . 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 16277 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
3936, 38syl 16 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
4013, 21, 37grplid 16279 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  .+  .0.  )  =  .0.  )
4136, 39, 40syl2anc 659 . . . . . . . 8  |-  ( ph  ->  (  .0.  .+  .0.  )  =  .0.  )
4241adantr 463 . . . . . . 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 6503 . . . . 5  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( ( F `  x ) 
.+  ( H `  x ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  ( ( `' F " ( _V 
\  {  .0.  }
) )  u.  ( `' H " ( _V 
\  {  .0.  }
) ) ) )
45 ssfi 7733 . . . . 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 659 . . . 4  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( ( F `  x ) 
.+  ( H `  x ) ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
478, 2, 1, 23, 46dprdwdOLD 17246 . . 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 16261 . . . 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 17250 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
548, 2, 1, 11, 49dprdfcntzOLD 17250 . . 3  |-  ( ph  ->  ran  H  C_  (
(Cntz `  G ) `  ran  H ) )
558, 2, 1, 48, 49dprdfcntzOLD 17250 . . 3  |-  ( ph  ->  ran  ( F  oF  .+  H )  C_  ( (Cntz `  G ) `  ran  ( F  oF  .+  H ) ) )
5651adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  G  e.  Mnd )
57 vex 3109 . . . . . . . 8  |-  x  e. 
_V
5857a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  x  e.  _V )
59 eldifi 3612 . . . . . . . . . . 11  |-  ( k  e.  ( I  \  x )  ->  k  e.  I )
6059adantl 464 . . . . . . . . . 10  |-  ( ( x  C_  I  /\  k  e.  ( I  \  x ) )  -> 
k  e.  I )
61 ffvelrn 6005 . . . . . . . . . 10  |-  ( ( F : I --> ( Base `  G )  /\  k  e.  I )  ->  ( F `  k )  e.  ( Base `  G
) )
6214, 60, 61syl2an 475 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( F `  k
)  e.  ( Base `  G ) )
6362snssd 4161 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( F `  k ) }  C_  ( Base `  G )
)
6413, 49cntzsubm 16572 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  { ( F `  k
) }  C_  ( Base `  G ) )  ->  ( (Cntz `  G ) `  {
( F `  k
) } )  e.  (SubMnd `  G )
)
6556, 63, 64syl2anc 659 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( (Cntz `  G
) `  { ( F `  k ) } )  e.  (SubMnd `  G ) )
6616adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  H : I --> ( Base `  G ) )
67 ffn 5713 . . . . . . . . . 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 754 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  x  C_  I )
70 fnssres 5676 . . . . . . . . 9  |-  ( ( H  Fn  I  /\  x  C_  I )  -> 
( H  |`  x
)  Fn  x )
7168, 69, 70syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
)  Fn  x )
72 fvres 5862 . . . . . . . . . . 11  |-  ( y  e.  x  ->  (
( H  |`  x
) `  y )  =  ( H `  y ) )
7372adantl 464 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( ( H  |`  x ) `  y )  =  ( H `  y ) )
742ad2antrr 723 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  G dom DProd  S )
751ad2antrr 723 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  dom  S  =  I )
7674, 75dprdf2 17235 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  S :
I --> (SubGrp `  G )
)
7760ad2antlr 724 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  k  e.  I )
7876, 77ffvelrnd 6008 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( S `  k )  e.  (SubGrp `  G ) )
7913subgss 16401 . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  F  e.  W )
828, 74, 75, 81dprdfclOLD 17248 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  /\  k  e.  I )  ->  ( F `  k )  e.  ( S `  k
) )
8377, 82mpdan 666 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( F `  k )  e.  ( S `  k ) )
8483snssd 4161 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  { ( F `  k ) }  C_  ( S `  k ) )
8513, 49cntz2ss 16569 . . . . . . . . . . . 12  |-  ( ( ( S `  k
)  C_  ( Base `  G )  /\  {
( F `  k
) }  C_  ( S `  k )
)  ->  ( (Cntz `  G ) `  ( S `  k )
)  C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
8680, 84, 85syl2anc 659 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( (Cntz `  G ) `  ( S `  k )
)  C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
8769sselda 3489 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  e.  I )
88 simpr 459 . . . . . . . . . . . . . 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 3474 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  -.  k  e.  x )
91 nelne2 2784 . . . . . . . . . . . . . 14  |-  ( ( y  e.  x  /\  -.  k  e.  x
)  ->  y  =/=  k )
9288, 90, 91syl2anc 659 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  =/=  k )
9374, 75, 87, 77, 92, 49dprdcntz 17236 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( S `  y )  C_  (
(Cntz `  G ) `  ( S `  k
) ) )
9411ad2antrr 723 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  H  e.  W )
958, 74, 75, 94dprdfclOLD 17248 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  /\  y  e.  I )  ->  ( H `  y )  e.  ( S `  y
) )
9687, 95mpdan 666 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( S `  y ) )
9793, 96sseldd 3490 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( (Cntz `  G ) `  ( S `  k
) ) )
9886, 97sseldd 3490 . . . . . . . . . 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 2868 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  A. y  e.  x  ( ( H  |`  x ) `  y
)  e.  ( (Cntz `  G ) `  {
( F `  k
) } ) )
101 ffnfv 6033 . . . . . . . 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 662 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
) : x --> ( (Cntz `  G ) `  {
( F `  k
) } ) )
103 resss 5285 . . . . . . . . . 10  |-  ( H  |`  x )  C_  H
104 rnss 5220 . . . . . . . . . 10  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
105103, 104ax-mp 5 . . . . . . . . 9  |-  ran  ( H  |`  x )  C_  ran  H
10649cntzidss 16574 . . . . . . . . 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 660 . . . . . . . 8  |-  ( ph  ->  ran  ( H  |`  x )  C_  (
(Cntz `  G ) `  ran  ( H  |`  x ) ) )
108107adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  ran  ( H  |`  x
)  C_  ( (Cntz `  G ) `  ran  ( H  |`  x ) ) )
109 cnvss 5164 . . . . . . . . . 10  |-  ( ( H  |`  x )  C_  H  ->  `' ( H  |`  x )  C_  `' H )
110 imass1 5359 . . . . . . . . . 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 7733 . . . . . . . . 9  |-  ( ( ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( H  |`  x )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  } ) )  e.  Fin )
11325, 111, 112sylancl 660 . . . . . . . 8  |-  ( ph  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
114113adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
11537, 49, 56, 58, 65, 102, 108, 114gsumzsubmclOLD 17128 . . . . . 6  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
116115snssd 4161 . . . . 5  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
117 fssres 5733 . . . . . . . . 9  |-  ( ( H : I --> ( Base `  G )  /\  x  C_  I )  ->  ( H  |`  x ) : x --> ( Base `  G
) )
11866, 69, 117syl2anc 659 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
) : x --> ( Base `  G ) )
11913, 37, 49, 56, 58, 118, 108, 114gsumzclOLD 17118 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  (
Base `  G )
)
120119snssd 4161 . . . . . 6  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( Base `  G
) )
12113, 49cntzrec 16570 . . . . . 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 659 . . . . 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 5858 . . . . 5  |-  ( F `
 k )  e. 
_V
125124snss 4140 . . . 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 17135 . 2  |-  ( ph  ->  ( G  gsumg  ( F  oF  .+  H ) )  =  ( ( G 
gsumg  F )  .+  ( G  gsumg  H ) ) )
12848, 127jca 530 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {crab 2808   _Vcvv 3106    \ cdif 3458    u. cun 3459    C_ wss 3461   {csn 4016   class class class wbr 4439    |-> cmpt 4497   `'ccnv 4987   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511   X_cixp 7462   Fincfn 7509   Basecbs 14716   +g cplusg 14784   0gc0g 14929    gsumg cgsu 14930   Mndcmnd 16118  SubMndcsubmnd 16164   Grpcgrp 16252  SubGrpcsubg 16394  Cntzccntz 16552   DProd cdprd 17219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-seq 12090  df-hash 12388  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-0g 14931  df-gsum 14932  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-subg 16397  df-cntz 16554  df-dprd 17221
This theorem is referenced by:  dprdfsubOLD  17263
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