MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dprdfadd Structured version   Unicode version

Theorem dprdfadd 15578
Description: Take the sum of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
eldprdi.0  |-  .0.  =  ( 0g `  G )
eldprdi.w  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
eldprdi.1  |-  ( ph  ->  G dom DProd  S )
eldprdi.2  |-  ( ph  ->  dom  S  =  I )
eldprdi.3  |-  ( ph  ->  F  e.  W )
dprdfadd.4  |-  ( ph  ->  H  e.  W )
dprdfadd.b  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
dprdfadd  |-  ( ph  ->  ( ( F  o F  .+  H )  e.  W  /\  ( G 
gsumg  ( F  o F  .+  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 dprdfadd
Dummy variables  k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldprdi.2 . . . . 5  |-  ( ph  ->  dom  S  =  I )
2 eldprdi.1 . . . . . 6  |-  ( ph  ->  G dom DProd  S )
3 reldmdprd 15558 . . . . . . 7  |-  Rel  dom DProd
43brrelex2i 4919 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
5 dmexg 5130 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
62, 4, 53syl 19 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
71, 6eqeltrrd 2511 . . . 4  |-  ( ph  ->  I  e.  _V )
8 eldprdi.w . . . . 5  |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
Fin }
9 eldprdi.3 . . . . 5  |-  ( ph  ->  F  e.  W )
108, 2, 1, 9dprdfcl 15571 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
11 dprdfadd.4 . . . . 5  |-  ( ph  ->  H  e.  W )
128, 2, 1, 11dprdfcl 15571 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  ( S `  x
) )
13 eqid 2436 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
148, 2, 1, 9, 13dprdff 15570 . . . . 5  |-  ( ph  ->  F : I --> ( Base `  G ) )
1514feqmptd 5779 . . . 4  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
168, 2, 1, 11, 13dprdff 15570 . . . . 5  |-  ( ph  ->  H : I --> ( Base `  G ) )
1716feqmptd 5779 . . . 4  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
187, 10, 12, 15, 17offval2 6322 . . 3  |-  ( ph  ->  ( F  o F 
.+  H )  =  ( x  e.  I  |->  ( ( F `  x )  .+  ( H `  x )
) ) )
192, 1dprdf2 15565 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
2019ffvelrnda 5870 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
21 dprdfadd.b . . . . . 6  |-  .+  =  ( +g  `  G )
2221subgcl 14954 . . . . 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 1184 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
)  .+  ( H `  x ) )  e.  ( S `  x
) )
248, 2, 1, 9dprdffi 15572 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
258, 2, 1, 11dprdffi 15572 . . . . . 6  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
26 unfi 7374 . . . . . 6  |-  ( ( ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' H " ( _V  \  {  .0.  } ) )  e.  Fin )  -> 
( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )  e.  Fin )
2724, 25, 26syl2anc 643 . . . . 5  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )  e.  Fin )
28 ssun1 3510 . . . . . . . . . 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, 29suppssr 5864 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  ( F `  x )  =  .0.  )
31 ssun2 3511 . . . . . . . . . 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, 32suppssr 5864 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  ( H `  x )  =  .0.  )
3430, 33oveq12d 6099 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F `  x
)  .+  ( H `  x ) )  =  (  .0.  .+  .0.  ) )
35 dprdgrp 15563 . . . . . . . . . 10  |-  ( G dom DProd  S  ->  G  e. 
Grp )
362, 35syl 16 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
37 eldprdi.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
3813, 37grpidcl 14833 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
3936, 38syl 16 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
4013, 21, 37grplid 14835 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  .+  .0.  )  =  .0.  )
4136, 39, 40syl2anc 643 . . . . . . . 8  |-  ( ph  ->  (  .0.  .+  .0.  )  =  .0.  )
4241adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (  .0.  .+  .0.  )  =  .0.  )
4334, 42eqtrd 2468 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F `  x
)  .+  ( H `  x ) )  =  .0.  )
4443suppss2 6300 . . . . 5  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( ( F `  x ) 
.+  ( H `  x ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  ( ( `' F " ( _V 
\  {  .0.  }
) )  u.  ( `' H " ( _V 
\  {  .0.  }
) ) ) )
45 ssfi 7329 . . . . 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 643 . . . 4  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( ( F `  x ) 
.+  ( H `  x ) ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
478, 2, 1, 23, 46dprdwd 15569 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x )  .+  ( H `  x )
) )  e.  W
)
4818, 47eqeltrd 2510 . 2  |-  ( ph  ->  ( F  o F 
.+  H )  e.  W )
49 eqid 2436 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
50 grpmnd 14817 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5136, 50syl 16 . . 3  |-  ( ph  ->  G  e.  Mnd )
52 eqid 2436 . . 3  |-  ( `' ( F  u.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  u.  H ) " ( _V  \  {  .0.  }
) )
538, 2, 1, 9, 49dprdfcntz 15573 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
548, 2, 1, 11, 49dprdfcntz 15573 . . 3  |-  ( ph  ->  ran  H  C_  (
(Cntz `  G ) `  ran  H ) )
558, 2, 1, 48, 49dprdfcntz 15573 . . 3  |-  ( ph  ->  ran  ( F  o F  .+  H )  C_  ( (Cntz `  G ) `  ran  ( F  o F  .+  H ) ) )
5651adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  G  e.  Mnd )
57 vex 2959 . . . . . . . 8  |-  x  e. 
_V
5857a1i 11 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  x  e.  _V )
59 eldifi 3469 . . . . . . . . . . 11  |-  ( k  e.  ( I  \  x )  ->  k  e.  I )
6059adantl 453 . . . . . . . . . 10  |-  ( ( x  C_  I  /\  k  e.  ( I  \  x ) )  -> 
k  e.  I )
61 ffvelrn 5868 . . . . . . . . . 10  |-  ( ( F : I --> ( Base `  G )  /\  k  e.  I )  ->  ( F `  k )  e.  ( Base `  G
) )
6214, 60, 61syl2an 464 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( F `  k
)  e.  ( Base `  G ) )
6362snssd 3943 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( F `  k ) }  C_  ( Base `  G )
)
6413, 49cntzsubm 15134 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  { ( F `  k
) }  C_  ( Base `  G ) )  ->  ( (Cntz `  G ) `  {
( F `  k
) } )  e.  (SubMnd `  G )
)
6556, 63, 64syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( (Cntz `  G
) `  { ( F `  k ) } )  e.  (SubMnd `  G ) )
6616adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  H : I --> ( Base `  G ) )
67 ffn 5591 . . . . . . . . . 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 733 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  x  C_  I )
70 fnssres 5558 . . . . . . . . 9  |-  ( ( H  Fn  I  /\  x  C_  I )  -> 
( H  |`  x
)  Fn  x )
7168, 69, 70syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
)  Fn  x )
72 fvres 5745 . . . . . . . . . . 11  |-  ( y  e.  x  ->  (
( H  |`  x
) `  y )  =  ( H `  y ) )
7372adantl 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( ( H  |`  x ) `  y )  =  ( H `  y ) )
742ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  G dom DProd  S )
751ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  dom  S  =  I )
7674, 75dprdf2 15565 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  S :
I --> (SubGrp `  G )
)
7760ad2antlr 708 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  k  e.  I )
7876, 77ffvelrnd 5871 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( S `  k )  e.  (SubGrp `  G ) )
7913subgss 14945 . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  F  e.  W )
828, 74, 75, 81dprdfcl 15571 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  /\  k  e.  I )  ->  ( F `  k )  e.  ( S `  k
) )
8377, 82mpdan 650 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( F `  k )  e.  ( S `  k ) )
8483snssd 3943 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  { ( F `  k ) }  C_  ( S `  k ) )
8513, 49cntz2ss 15131 . . . . . . . . . . . 12  |-  ( ( ( S `  k
)  C_  ( Base `  G )  /\  {
( F `  k
) }  C_  ( S `  k )
)  ->  ( (Cntz `  G ) `  ( S `  k )
)  C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
8680, 84, 85syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( (Cntz `  G ) `  ( S `  k )
)  C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
8769sselda 3348 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  e.  I )
88 simpr 448 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  e.  x )
89 simplrr 738 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  k  e.  ( I  \  x
) )
9089eldifbd 3333 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  -.  k  e.  x )
91 nelne2 2694 . . . . . . . . . . . . . 14  |-  ( ( y  e.  x  /\  -.  k  e.  x
)  ->  y  =/=  k )
9288, 90, 91syl2anc 643 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  =/=  k )
9374, 75, 87, 77, 92, 49dprdcntz 15566 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( S `  y )  C_  (
(Cntz `  G ) `  ( S `  k
) ) )
9411ad2antrr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  H  e.  W )
958, 74, 75, 94dprdfcl 15571 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  /\  y  e.  I )  ->  ( H `  y )  e.  ( S `  y
) )
9687, 95mpdan 650 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( S `  y ) )
9793, 96sseldd 3349 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( (Cntz `  G ) `  ( S `  k
) ) )
9886, 97sseldd 3349 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
9973, 98eqeltrd 2510 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( ( H  |`  x ) `  y )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
10099ralrimiva 2789 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  A. y  e.  x  ( ( H  |`  x ) `  y
)  e.  ( (Cntz `  G ) `  {
( F `  k
) } ) )
101 ffnfv 5894 . . . . . . . 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 646 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
) : x --> ( (Cntz `  G ) `  {
( F `  k
) } ) )
103 resss 5170 . . . . . . . . . 10  |-  ( H  |`  x )  C_  H
104 rnss 5098 . . . . . . . . . 10  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
105103, 104ax-mp 8 . . . . . . . . 9  |-  ran  ( H  |`  x )  C_  ran  H
10649cntzidss 15136 . . . . . . . . 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 644 . . . . . . . 8  |-  ( ph  ->  ran  ( H  |`  x )  C_  (
(Cntz `  G ) `  ran  ( H  |`  x ) ) )
108107adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  ran  ( H  |`  x
)  C_  ( (Cntz `  G ) `  ran  ( H  |`  x ) ) )
109 cnvss 5045 . . . . . . . . . 10  |-  ( ( H  |`  x )  C_  H  ->  `' ( H  |`  x )  C_  `' H )
110 imass1 5239 . . . . . . . . . 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 7329 . . . . . . . . 9  |-  ( ( ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( H  |`  x )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  } ) )  e.  Fin )
11325, 111, 112sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
114113adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
11537, 49, 56, 58, 65, 102, 108, 114gsumzsubmcl 15523 . . . . . 6  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
116115snssd 3943 . . . . 5  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
117 fssres 5610 . . . . . . . . 9  |-  ( ( H : I --> ( Base `  G )  /\  x  C_  I )  ->  ( H  |`  x ) : x --> ( Base `  G
) )
11866, 69, 117syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
) : x --> ( Base `  G ) )
11913, 37, 49, 56, 58, 118, 108, 114gsumzcl 15518 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  (
Base `  G )
)
120119snssd 3943 . . . . . 6  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( Base `  G
) )
12113, 49cntzrec 15132 . . . . . 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 643 . . . . 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 202 . . . 4  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( F `  k ) }  C_  ( (Cntz `  G ) `  { ( G  gsumg  ( H  |`  x ) ) } ) )
124 fvex 5742 . . . . 5  |-  ( F `
 k )  e. 
_V
125124snss 3926 . . . 4  |-  ( ( F `  k )  e.  ( (Cntz `  G ) `  {
( G  gsumg  ( H  |`  x
) ) } )  <->  { ( F `  k ) }  C_  ( (Cntz `  G ) `  { ( G  gsumg  ( H  |`  x ) ) } ) )
126123, 125sylibr 204 . . 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, 126gsumzaddlem 15526 . 2  |-  ( ph  ->  ( G  gsumg  ( F  o F 
.+  H ) )  =  ( ( G 
gsumg  F )  .+  ( G  gsumg  H ) ) )
12848, 127jca 519 1  |-  ( ph  ->  ( ( F  o F  .+  H )  e.  W  /\  ( G 
gsumg  ( F  o F  .+  H ) )  =  ( ( G  gsumg  F ) 
.+  ( G  gsumg  H ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   {crab 2709   _Vcvv 2956    \ cdif 3317    u. cun 3318    C_ wss 3320   {csn 3814   class class class wbr 4212    e. cmpt 4266   `'ccnv 4877   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   X_cixp 7063   Fincfn 7109   Basecbs 13469   +g cplusg 13529   0gc0g 13723    gsumg cgsu 13724   Mndcmnd 14684   Grpcgrp 14685  SubMndcsubmnd 14737  SubGrpcsubg 14938  Cntzccntz 15114   DProd cdprd 15554
This theorem is referenced by:  dprdfsub  15579
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-gsum 13728  df-mnd 14690  df-submnd 14739  df-grp 14812  df-subg 14941  df-cntz 15116  df-dprd 15556
  Copyright terms: Public domain W3C validator