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Theorem dprdfadd 15255
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 15235 . . . . . . 7  |-  Rel  dom DProd
43brrelex2i 4730 . . . . . 6  |-  ( G dom DProd  S  ->  S  e. 
_V )
5 dmexg 4939 . . . . . 6  |-  ( S  e.  _V  ->  dom  S  e.  _V )
62, 4, 53syl 18 . . . . 5  |-  ( ph  ->  dom  S  e.  _V )
71, 6eqeltrrd 2358 . . . 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 15248 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( F `  x )  e.  ( S `  x
) )
11 dprdfadd.4 . . . . 5  |-  ( ph  ->  H  e.  W )
128, 2, 1, 11dprdfcl 15248 . . . 4  |-  ( (
ph  /\  x  e.  I )  ->  ( H `  x )  e.  ( S `  x
) )
13 eqid 2283 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
148, 2, 1, 9, 13dprdff 15247 . . . . 5  |-  ( ph  ->  F : I --> ( Base `  G ) )
1514feqmptd 5575 . . . 4  |-  ( ph  ->  F  =  ( x  e.  I  |->  ( F `
 x ) ) )
168, 2, 1, 11, 13dprdff 15247 . . . . 5  |-  ( ph  ->  H : I --> ( Base `  G ) )
1716feqmptd 5575 . . . 4  |-  ( ph  ->  H  =  ( x  e.  I  |->  ( H `
 x ) ) )
187, 10, 12, 15, 17offval2 6095 . . 3  |-  ( ph  ->  ( F  o F 
.+  H )  =  ( x  e.  I  |->  ( ( F `  x )  .+  ( H `  x )
) ) )
192, 1dprdf2 15242 . . . . . 6  |-  ( ph  ->  S : I --> (SubGrp `  G ) )
20 ffvelrn 5663 . . . . . 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 14631 . . . . 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 15249 . . . . . 6  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
268, 2, 1, 11dprdffi 15249 . . . . . 6  |-  ( ph  ->  ( `' H "
( _V  \  {  .0.  } ) )  e. 
Fin )
27 unfi 7124 . . . . . 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 3338 . . . . . . . . . 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 5659 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  ( F `  x )  =  .0.  )
32 ssun2 3339 . . . . . . . . . 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 5659 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  ( H `  x )  =  .0.  )
3531, 34oveq12d 5876 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F `  x
)  .+  ( H `  x ) )  =  (  .0.  .+  .0.  ) )
36 dprdgrp 15240 . . . . . . . . . 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 14510 . . . . . . . . . 10  |-  ( G  e.  Grp  ->  .0.  e.  ( Base `  G
) )
4037, 39syl 15 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  ( Base `  G ) )
4113, 22, 38grplid 14512 . . . . . . . . 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 2315 . . . . . 6  |-  ( (
ph  /\  x  e.  ( I  \  (
( `' F "
( _V  \  {  .0.  } ) )  u.  ( `' H "
( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F `  x
)  .+  ( H `  x ) )  =  .0.  )
4544suppss2 6073 . . . . 5  |-  ( ph  ->  ( `' ( x  e.  I  |->  ( ( F `  x ) 
.+  ( H `  x ) ) )
" ( _V  \  {  .0.  } ) ) 
C_  ( ( `' F " ( _V 
\  {  .0.  }
) )  u.  ( `' H " ( _V 
\  {  .0.  }
) ) ) )
46 ssfi 7083 . . . . 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 15246 . . 3  |-  ( ph  ->  ( x  e.  I  |->  ( ( F `  x )  .+  ( H `  x )
) )  e.  W
)
4918, 48eqeltrd 2357 . 2  |-  ( ph  ->  ( F  o F 
.+  H )  e.  W )
50 eqid 2283 . . 3  |-  (Cntz `  G )  =  (Cntz `  G )
51 grpmnd 14494 . . . 4  |-  ( G  e.  Grp  ->  G  e.  Mnd )
5237, 51syl 15 . . 3  |-  ( ph  ->  G  e.  Mnd )
53 eqid 2283 . . 3  |-  ( `' ( F  u.  H
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  u.  H ) " ( _V  \  {  .0.  }
) )
548, 2, 1, 9, 50dprdfcntz 15250 . . 3  |-  ( ph  ->  ran  F  C_  (
(Cntz `  G ) `  ran  F ) )
558, 2, 1, 11, 50dprdfcntz 15250 . . 3  |-  ( ph  ->  ran  H  C_  (
(Cntz `  G ) `  ran  H ) )
568, 2, 1, 49, 50dprdfcntz 15250 . . 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 2791 . . . . . . . 8  |-  x  e. 
_V
5958a1i 10 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  x  e.  _V )
60 eldifi 3298 . . . . . . . . . . 11  |-  ( k  e.  ( I  \  x )  ->  k  e.  I )
6160adantl 452 . . . . . . . . . 10  |-  ( ( x  C_  I  /\  k  e.  ( I  \  x ) )  -> 
k  e.  I )
62 ffvelrn 5663 . . . . . . . . . 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 3760 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( F `  k ) }  C_  ( Base `  G )
)
6513, 50cntzsubm 14811 . . . . . . . 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 5389 . . . . . . . . . 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 5357 . . . . . . . . 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 5542 . . . . . . . . . . 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 15242 . . . . . . . . . . . . . 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 5663 . . . . . . . . . . . . . 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 14622 . . . . . . . . . . . . 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 15248 . . . . . . . . . . . . . 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 3760 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  { ( F `  k ) }  C_  ( S `  k ) )
8713, 50cntz2ss 14808 . . . . . . . . . . . 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 3180 . . . . . . . . . . . . 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 3299 . . . . . . . . . . . . . . 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 2536 . . . . . . . . . . . . . 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 15243 . . . . . . . . . . . 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 15248 . . . . . . . . . . . . 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 3181 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( (Cntz `  G ) `  ( S `  k
) ) )
10188, 100sseldd 3181 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( H `  y )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
10274, 101eqeltrd 2357 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  C_  I  /\  k  e.  ( I  \  x ) ) )  /\  y  e.  x
)  ->  ( ( H  |`  x ) `  y )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
103102ralrimiva 2626 . . . . . . . 8  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  A. y  e.  x  ( ( H  |`  x ) `  y
)  e.  ( (Cntz `  G ) `  {
( F `  k
) } ) )
104 ffnfv 5685 . . . . . . . 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 4979 . . . . . . . . . 10  |-  ( H  |`  x )  C_  H
107 rnss 4907 . . . . . . . . . 10  |-  ( ( H  |`  x )  C_  H  ->  ran  ( H  |`  x )  C_  ran  H )
108106, 107ax-mp 8 . . . . . . . . 9  |-  ran  ( H  |`  x )  C_  ran  H
10950cntzidss 14813 . . . . . . . . 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 4854 . . . . . . . . . 10  |-  ( ( H  |`  x )  C_  H  ->  `' ( H  |`  x )  C_  `' H )
113 imass1 5048 . . . . . . . . . 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 7083 . . . . . . . . 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 15200 . . . . . 6  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  ( (Cntz `  G ) `  { ( F `  k ) } ) )
119118snssd 3760 . . . . 5  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( (Cntz `  G ) `  {
( F `  k
) } ) )
120 fssres 5408 . . . . . . . . 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 15195 . . . . . . 7  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  -> 
( G  gsumg  ( H  |`  x
) )  e.  (
Base `  G )
)
123122snssd 3760 . . . . . 6  |-  ( (
ph  /\  ( x  C_  I  /\  k  e.  ( I  \  x
) ) )  ->  { ( G  gsumg  ( H  |`  x ) ) } 
C_  ( Base `  G
) )
12413, 50cntzrec 14809 . . . . . 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 5539 . . . . 5  |-  ( F `
 k )  e. 
_V
128127snss 3748 . . . 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 15203 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149    u. cun 3150    C_ wss 3152   {csn 3640   class class class wbr 4023    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   X_cixp 6817   Fincfn 6863   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361   Grpcgrp 14362  SubMndcsubmnd 14414  SubGrpcsubg 14615  Cntzccntz 14791   DProd cdprd 15231
This theorem is referenced by:  dprdfsub  15256
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mnd 14367  df-submnd 14416  df-grp 14489  df-subg 14618  df-cntz 14793  df-dprd 15233
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