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Theorem dprdfadd 15271
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 15251 . . . . . . 7  |-  Rel  dom DProd
43brrelex2i 4746 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
5 dmexg 4955 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
62, 4, 53syl 18 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
71, 6eqeltrrd 2371 . . . 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 15264 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
11 dprdfadd.4 . . . . 5  |-  ( ph  ->  H  e.  W )
128, 2, 1, 11dprdfcl 15264 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  ( S `  x
) )
13 eqid 2296 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
148, 2, 1, 9, 13dprdff 15263 . . . . 5  |-  ( ph  ->  F : I --> ( Base `  G ) )
1514feqmptd 5591 . . . 4  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
168, 2, 1, 11, 13dprdff 15263 . . . . 5  |-  ( ph  ->  H : I --> ( Base `  G ) )
1716feqmptd 5591 . . . 4  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
187, 10, 12, 15, 17offval2 6111 . . 3  |-  ( ph  ->  ( F  o F 
.+  H )  =  ( x  e.  I  |->  ( ( F `  x )  .+  ( H `  x )
) ) )
192, 1dprdf2 15258 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
20 ffvelrn 5679 . . . . . 6  |-  ( ( S : I --> (SubGrp `  G )  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
2119, 20sylan 457 . . . . 5  |-  ( (
ph  /\  x  e.  I )  ->  ( S `  x )  e.  (SubGrp `  G )
)
22 dprdfadd.b . . . . . 6  |-  .+  =  ( +g  `  G )
2322subgcl 14647 . . . . 5  |-  ( ( ( S `  x
)  e.  (SubGrp `  G )  /\  ( F `  x )  e.  ( S `  x
)  /\  ( H `  x )  e.  ( S `  x ) )  ->  ( ( F `  x )  .+  ( H `  x
) )  e.  ( S `  x ) )
2421, 10, 12, 23syl3anc 1182 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  (
( F `  x
)  .+  ( H `  x ) )  e.  ( S `  x
) )
258, 2, 1, 9dprdffi 15265 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
268, 2, 1, 11dprdffi 15265 . . . . . 6  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
27 unfi 7140 . . . . . 6  |-  ( ( ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' H " ( _V  \  {  .0.  } ) )  e.  Fin )  -> 
( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )  e.  Fin )
2825, 26, 27syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )  e.  Fin )
29 ssun1 3351 . . . . . . . . . 10  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )
3029a1i 10 . . . . . . . . 9  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) )
3114, 30suppssr 5675 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  ( F `  x )  =  .0.  )
32 ssun2 3352 . . . . . . . . . 10  |-  ( `' H " ( _V 
\  {  .0.  }
) )  C_  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) )
3332a1i 10 . . . . . . . . 9  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  C_  ( ( `' F " ( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) )
3416, 33suppssr 5675 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  ( H `  x )  =  .0.  )
3531, 34oveq12d 5892 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F `  x
)  .+  ( H `  x ) )  =  (  .0.  .+  .0.  ) )
36 dprdgrp 15256 . . . . . . . . . 10  |-  ( G dom DProd  S  ->  G  e. 
Grp )
372, 36syl 15 . . . . . . . . 9  |-  ( ph  ->  G  e.  Grp )
38 eldprdi.0 . . . . . . . . . . 11  |-  .0.  =  ( 0g `  G )
3913, 38grpidcl 14526 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
4037, 39syl 15 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
4113, 22, 38grplid 14528 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  .0.  e.  ( Base `  G
) )  ->  (  .0.  .+  .0.  )  =  .0.  )
4237, 40, 41syl2anc 642 . . . . . . . 8  |-  ( ph  ->  (  .0.  .+  .0.  )  =  .0.  )
4342adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (  .0.  .+  .0.  )  =  .0.  )
4435, 43eqtrd 2328 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F `  x
)  .+  ( H `  x ) )  =  .0.  )
4544suppss2 6089 . . . . 5  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( ( F `  x ) 
.+  ( H `  x ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  ( ( `' F " ( _V 
\  {  .0.  }
) )  u.  ( `' H " ( _V 
\  {  .0.  }
) ) ) )
46 ssfi 7099 . . . . 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 )
4728, 45, 46syl2anc 642 . . . 4  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( ( F `  x ) 
.+  ( H `  x ) ) )
" ( _V  \  {  .0.  } ) )  e.  Fin )
488, 2, 1, 24, 47dprdwd 15262 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x )  .+  ( H `  x )
) )  e.  W
)
4918, 48eqeltrd 2370 . 2  |-  ( ph  ->  ( F  o F 
.+  H )  e.  W )
50 eqid 2296 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
51 grpmnd 14510 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5237, 51syl 15 . . 3  |-  ( ph  ->  G  e.  Mnd )
53 eqid 2296 . . 3  |-  ( `' ( F  u.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  u.  H ) " ( _V  \  {  .0.  }
) )
548, 2, 1, 9, 50dprdfcntz 15266 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
558, 2, 1, 11, 50dprdfcntz 15266 . . 3  |-  ( ph  ->  ran  H  C_  (
(Cntz `  G ) `  ran  H ) )
568, 2, 1, 49, 50dprdfcntz 15266 . . 3  |-  ( ph  ->  ran  ( F  o F  .+  H )  C_  ( (Cntz `  G ) `  ran  ( F  o F  .+  H ) ) )
5752adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  G  e.  Mnd )
58 vex 2804 . . . . . . . 8  |-  x  e. 
_V
5958a1i 10 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  x  e.  _V )
60 eldifi 3311 . . . . . . . . . . 11  |-  ( k  e.  ( I  \  x )  ->  k  e.  I )
6160adantl 452 . . . . . . . . . 10  |-  ( ( x  C_  I  /\  k  e.  ( I  \  x ) )  -> 
k  e.  I )
62 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( F : I --> ( Base `  G )  /\  k  e.  I )  ->  ( F `  k )  e.  ( Base `  G
) )
6314, 61, 62syl2an 463 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( F `  k
)  e.  ( Base `  G ) )
6463snssd 3776 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( F `  k ) }  C_  ( Base `  G )
)
6513, 50cntzsubm 14827 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  { ( F `  k
) }  C_  ( Base `  G ) )  ->  ( (Cntz `  G ) `  {
( F `  k
) } )  e.  (SubMnd `  G )
)
6657, 64, 65syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( (Cntz `  G
) `  { ( F `  k ) } )  e.  (SubMnd `  G ) )
6716adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  H : I --> ( Base `  G ) )
68 ffn 5405 . . . . . . . . . 10  |-  ( H : I --> ( Base `  G )  ->  H  Fn  I )
6967, 68syl 15 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  H  Fn  I )
70 simprl 732 . . . . . . . . 9  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  x  C_  I )
71 fnssres 5373 . . . . . . . . 9  |-  ( ( H  Fn  I  /\  x  C_  I )  -> 
( H  |`  x
)  Fn  x )
7269, 70, 71syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
)  Fn  x )
73 fvres 5558 . . . . . . . . . . 11  |-  ( y  e.  x  ->  (
( H  |`  x
) `  y )  =  ( H `  y ) )
7473adantl 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( ( H  |`  x ) `  y )  =  ( H `  y ) )
752ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  G dom DProd  S )
761ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  dom  S  =  I )
7775, 76dprdf2 15258 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  S :
I --> (SubGrp `  G )
)
7861ad2antlr 707 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  k  e.  I )
79 ffvelrn 5679 . . . . . . . . . . . . . 14  |-  ( ( S : I --> (SubGrp `  G )  /\  k  e.  I )  ->  ( S `  k )  e.  (SubGrp `  G )
)
8077, 78, 79syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( S `  k )  e.  (SubGrp `  G ) )
8113subgss 14638 . . . . . . . . . . . . 13  |-  ( ( S `  k )  e.  (SubGrp `  G
)  ->  ( S `  k )  C_  ( Base `  G ) )
8280, 81syl 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( S `  k )  C_  ( Base `  G ) )
839ad2antrr 706 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  F  e.  W )
848, 75, 76, 83dprdfcl 15264 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  /\  k  e.  I )  ->  ( F `  k )  e.  ( S `  k
) )
8578, 84mpdan 649 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( F `  k )  e.  ( S `  k ) )
8685snssd 3776 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  { ( F `  k ) }  C_  ( S `  k ) )
8713, 50cntz2ss 14824 . . . . . . . . . . . 12  |-  ( ( ( S `  k
)  C_  ( Base `  G )  /\  {
( F `  k
) }  C_  ( S `  k )
)  ->  ( (Cntz `  G ) `  ( S `  k )
)  C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
8882, 86, 87syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( (Cntz `  G ) `  ( S `  k )
)  C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
8970sselda 3193 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  e.  I )
90 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  e.  x )
91 simplrr 737 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  k  e.  ( I  \  x
) )
92 eldifn 3312 . . . . . . . . . . . . . . 15  |-  ( k  e.  ( I  \  x )  ->  -.  k  e.  x )
9391, 92syl 15 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  -.  k  e.  x )
94 nelne2 2549 . . . . . . . . . . . . . 14  |-  ( ( y  e.  x  /\  -.  k  e.  x
)  ->  y  =/=  k )
9590, 93, 94syl2anc 642 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  y  =/=  k )
9675, 76, 89, 78, 95, 50dprdcntz 15259 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( S `  y )  C_  (
(Cntz `  G ) `  ( S `  k
) ) )
9711ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  H  e.  W )
988, 75, 76, 97dprdfcl 15264 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  /\  y  e.  I )  ->  ( H `  y )  e.  ( S `  y
) )
9989, 98mpdan 649 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( S `  y ) )
10096, 99sseldd 3194 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( (Cntz `  G ) `  ( S `  k
) ) )
10188, 100sseldd 3194 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
10274, 101eqeltrd 2370 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( ( H  |`  x ) `  y )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
103102ralrimiva 2639 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  A. y  e.  x  ( ( H  |`  x ) `  y
)  e.  ( (Cntz `  G ) `  {
( F `  k
) } ) )
104 ffnfv 5701 . . . . . . . 8  |-  ( ( H  |`  x ) : x --> ( (Cntz `  G ) `  {
( F `  k
) } )  <->  ( ( H  |`  x )  Fn  x  /\  A. y  e.  x  ( ( H  |`  x ) `  y )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) ) )
10572, 103, 104sylanbrc 645 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
) : x --> ( (Cntz `  G ) `  {
( F `  k
) } ) )
106 resss 4995 . . . . . . . . . 10  |-  ( H  |`  x )  C_  H
107 rnss 4923 . . . . . . . . . 10  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
108106, 107ax-mp 8 . . . . . . . . 9  |-  ran  ( H  |`  x )  C_  ran  H
10950cntzidss 14829 . . . . . . . . 9  |-  ( ( ran  H  C_  (
(Cntz `  G ) `  ran  H )  /\  ran  ( H  |`  x
)  C_  ran  H )  ->  ran  ( H  |`  x )  C_  (
(Cntz `  G ) `  ran  ( H  |`  x ) ) )
11055, 108, 109sylancl 643 . . . . . . . 8  |-  ( ph  ->  ran  ( H  |`  x )  C_  (
(Cntz `  G ) `  ran  ( H  |`  x ) ) )
111110adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  ran  ( H  |`  x
)  C_  ( (Cntz `  G ) `  ran  ( H  |`  x ) ) )
112 cnvss 4870 . . . . . . . . . 10  |-  ( ( H  |`  x )  C_  H  ->  `' ( H  |`  x )  C_  `' H )
113 imass1 5064 . . . . . . . . . 10  |-  ( `' ( H  |`  x
)  C_  `' H  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) ) )
114106, 112, 113mp2b 9 . . . . . . . . 9  |-  ( `' ( H  |`  x
) " ( _V 
\  {  .0.  }
) )  C_  ( `' H " ( _V 
\  {  .0.  }
) )
115 ssfi 7099 . . . . . . . . 9  |-  ( ( ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin  /\  ( `' ( H  |`  x )
" ( _V  \  {  .0.  } ) ) 
C_  ( `' H " ( _V  \  {  .0.  } ) ) )  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  } ) )  e.  Fin )
11626, 114, 115sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
117116adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( `' ( H  |`  x ) " ( _V  \  {  .0.  }
) )  e.  Fin )
11838, 50, 57, 59, 66, 105, 111, 117gsumzsubmcl 15216 . . . . . 6  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
119118snssd 3776 . . . . 5  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
120 fssres 5424 . . . . . . . . 9  |-  ( ( H : I --> ( Base `  G )  /\  x  C_  I )  ->  ( H  |`  x ) : x --> ( Base `  G
) )
12167, 70, 120syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( H  |`  x
) : x --> ( Base `  G ) )
12213, 38, 50, 57, 59, 121, 111, 117gsumzcl 15211 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  (
Base `  G )
)
123122snssd 3776 . . . . . 6  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( Base `  G
) )
12413, 50cntzrec 14825 . . . . . 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
) ) } ) ) )
125123, 64, 124syl2anc 642 . . . . 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 ) ) } ) ) )
126119, 125mpbid 201 . . . 4  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( F `  k ) }  C_  ( (Cntz `  G ) `  { ( G  gsumg  ( H  |`  x ) ) } ) )
127 fvex 5555 . . . . 5  |-  ( F `
 k )  e. 
_V
128127snss 3761 . . . 4  |-  ( ( F `  k )  e.  ( (Cntz `  G ) `  {
( G  gsumg  ( H  |`  x
) ) } )  <->  { ( F `  k ) }  C_  ( (Cntz `  G ) `  { ( G  gsumg  ( H  |`  x ) ) } ) )
129126, 128sylibr 203 . . 3  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( F `  k
)  e.  ( (Cntz `  G ) `  {
( G  gsumg  ( H  |`  x
) ) } ) )
13013, 38, 22, 50, 52, 7, 25, 26, 53, 14, 16, 54, 55, 56, 129gsumzaddlem 15219 . 2  |-  ( ph  ->  ( G  gsumg  ( F  o F 
.+  H ) )  =  ( ( G 
gsumg  F )  .+  ( G  gsumg  H ) ) )
13149, 130jca 518 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   {csn 3653   class class class wbr 4039    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   X_cixp 6833   Fincfn 6879   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377   Grpcgrp 14378  SubMndcsubmnd 14430  SubGrpcsubg 14631  Cntzccntz 14807   DProd cdprd 15247
This theorem is referenced by:  dprdfsub  15272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-gsum 13421  df-mnd 14383  df-submnd 14432  df-grp 14505  df-subg 14634  df-cntz 14809  df-dprd 15249
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