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Theorem gsumzmhm 17563
Description: Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) (Revised by AV, 6-Jun-2019.)
Hypotheses
Ref Expression
gsumzmhm.b  |-  B  =  ( Base `  G
)
gsumzmhm.z  |-  Z  =  (Cntz `  G )
gsumzmhm.g  |-  ( ph  ->  G  e.  Mnd )
gsumzmhm.h  |-  ( ph  ->  H  e.  Mnd )
gsumzmhm.a  |-  ( ph  ->  A  e.  V )
gsumzmhm.k  |-  ( ph  ->  K  e.  ( G MndHom  H ) )
gsumzmhm.f  |-  ( ph  ->  F : A --> B )
gsumzmhm.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzmhm.0  |-  .0.  =  ( 0g `  G )
gsumzmhm.w  |-  ( ph  ->  F finSupp  .0.  )
Assertion
Ref Expression
gsumzmhm  |-  ( ph  ->  ( H  gsumg  ( K  o.  F
) )  =  ( K `  ( G 
gsumg  F ) ) )

Proof of Theorem gsumzmhm
Dummy variables  k  x  y  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzmhm.h . . . . . . 7  |-  ( ph  ->  H  e.  Mnd )
2 gsumzmhm.a . . . . . . 7  |-  ( ph  ->  A  e.  V )
3 eqid 2450 . . . . . . . 8  |-  ( 0g
`  H )  =  ( 0g `  H
)
43gsumz 16614 . . . . . . 7  |-  ( ( H  e.  Mnd  /\  A  e.  V )  ->  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) )  =  ( 0g `  H
) )
51, 2, 4syl2anc 666 . . . . . 6  |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) )  =  ( 0g `  H
) )
65adantr 467 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) )  =  ( 0g `  H
) )
7 gsumzmhm.k . . . . . . 7  |-  ( ph  ->  K  e.  ( G MndHom  H ) )
8 gsumzmhm.0 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
98, 3mhm0 16583 . . . . . . 7  |-  ( K  e.  ( G MndHom  H
)  ->  ( K `  .0.  )  =  ( 0g `  H ) )
107, 9syl 17 . . . . . 6  |-  ( ph  ->  ( K `  .0.  )  =  ( 0g `  H ) )
1110adantr 467 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K `  .0.  )  =  ( 0g `  H
) )
126, 11eqtr4d 2487 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) )  =  ( K `  .0.  ) )
13 gsumzmhm.g . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
14 gsumzmhm.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
1514, 8mndidcl 16547 . . . . . . . . 9  |-  ( G  e.  Mnd  ->  .0.  e.  B )
1613, 15syl 17 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
1716ad2antrr 731 . . . . . . 7  |-  ( ( ( ph  /\  ( `' F " ( _V 
\  {  .0.  }
) )  =  (/) )  /\  k  e.  A
)  ->  .0.  e.  B )
18 gsumzmhm.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
19 fvex 5873 . . . . . . . . . 10  |-  ( 0g
`  G )  e. 
_V
208, 19eqeltri 2524 . . . . . . . . 9  |-  .0.  e.  _V
2120a1i 11 . . . . . . . 8  |-  ( ph  ->  .0.  e.  _V )
22 fex 6136 . . . . . . . . . . 11  |-  ( ( F : A --> B  /\  A  e.  V )  ->  F  e.  _V )
2318, 2, 22syl2anc 666 . . . . . . . . . 10  |-  ( ph  ->  F  e.  _V )
24 suppimacnv 6922 . . . . . . . . . 10  |-  ( ( F  e.  _V  /\  .0.  e.  _V )  -> 
( F supp  .0.  )  =  ( `' F " ( _V  \  {  .0.  } ) ) )
2523, 21, 24syl2anc 666 . . . . . . . . 9  |-  ( ph  ->  ( F supp  .0.  )  =  ( `' F " ( _V  \  {  .0.  } ) ) )
26 ssid 3450 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2725, 26syl6eqss 3481 . . . . . . . 8  |-  ( ph  ->  ( F supp  .0.  )  C_  ( `' F "
( _V  \  {  .0.  } ) ) )
2818, 2, 21, 27gsumcllem 17535 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
29 eqid 2450 . . . . . . . . . . 11  |-  ( Base `  H )  =  (
Base `  H )
3014, 29mhmf 16580 . . . . . . . . . 10  |-  ( K  e.  ( G MndHom  H
)  ->  K : B
--> ( Base `  H
) )
317, 30syl 17 . . . . . . . . 9  |-  ( ph  ->  K : B --> ( Base `  H ) )
3231feqmptd 5916 . . . . . . . 8  |-  ( ph  ->  K  =  ( x  e.  B  |->  ( K `
 x ) ) )
3332adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  K  =  ( x  e.  B  |->  ( K `  x ) ) )
34 fveq2 5863 . . . . . . 7  |-  ( x  =  .0.  ->  ( K `  x )  =  ( K `  .0.  ) )
3517, 28, 33, 34fmptco 6054 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K  o.  F )  =  ( k  e.  A  |->  ( K `  .0.  ) ) )
3610mpteq2dv 4489 . . . . . . 7  |-  ( ph  ->  ( k  e.  A  |->  ( K `  .0.  ) )  =  ( k  e.  A  |->  ( 0g `  H ) ) )
3736adantr 467 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  (
k  e.  A  |->  ( K `  .0.  )
)  =  ( k  e.  A  |->  ( 0g
`  H ) ) )
3835, 37eqtrd 2484 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K  o.  F )  =  ( k  e.  A  |->  ( 0g `  H ) ) )
3938oveq2d 6304 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( K  o.  F
) )  =  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) ) )
4028oveq2d 6304 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
418gsumz 16614 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
4213, 2, 41syl2anc 666 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
4342adantr 467 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
4440, 43eqtrd 2484 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  .0.  )
4544fveq2d 5867 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K `  ( G  gsumg  F ) )  =  ( K `  .0.  )
)
4612, 39, 453eqtr4d 2494 . . 3  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( K  o.  F
) )  =  ( K `  ( G 
gsumg  F ) ) )
4746ex 436 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  ->  ( H  gsumg  ( K  o.  F ) )  =  ( K `  ( G  gsumg  F ) ) ) )
4813adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  G  e.  Mnd )
49 eqid 2450 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
5014, 49mndcl 16538 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
51503expb 1208 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  G
) y )  e.  B )
5248, 51sylan 474 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  G ) y )  e.  B )
5318adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> B )
54 f1of1 5811 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) ) )
5554ad2antll 734 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-> ( `' F " ( _V  \  {  .0.  } ) ) )
56 cnvimass 5187 . . . . . . . . . . . 12  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
57 fdm 5731 . . . . . . . . . . . . 13  |-  ( F : A --> B  ->  dom  F  =  A )
5853, 57syl 17 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  dom  F  =  A )
5956, 58syl5sseq 3479 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  A )
60 f1ss 5782 . . . . . . . . . . 11  |-  ( ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)  ->  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-> A )
6155, 59, 60syl2anc 666 . . . . . . . . . 10  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-> A )
62 f1f 5777 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> A  ->  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) --> A )
6361, 62syl 17 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) --> A )
64 fco 5737 . . . . . . . . 9  |-  ( ( F : A --> B  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) --> A )  ->  ( F  o.  f ) : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) --> B )
6553, 63, 64syl2anc 666 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F  o.  f ) : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) --> B )
6665ffvelrnda 6020 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F  o.  f
) `  x )  e.  B )
67 simprl 763 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )
68 nnuz 11191 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
6967, 68syl6eleq 2538 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  ( ZZ>= `  1 )
)
707adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  K  e.  ( G MndHom  H ) )
71 eqid 2450 . . . . . . . . . 10  |-  ( +g  `  H )  =  ( +g  `  H )
7214, 49, 71mhmlin 16582 . . . . . . . . 9  |-  ( ( K  e.  ( G MndHom  H )  /\  x  e.  B  /\  y  e.  B )  ->  ( K `  ( x
( +g  `  G ) y ) )  =  ( ( K `  x ) ( +g  `  H ) ( K `
 y ) ) )
73723expb 1208 . . . . . . . 8  |-  ( ( K  e.  ( G MndHom  H )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( K `  ( x ( +g  `  G ) y ) )  =  ( ( K `  x ) ( +g  `  H
) ( K `  y ) ) )
7470, 73sylan 474 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( K `  (
x ( +g  `  G
) y ) )  =  ( ( K `
 x ) ( +g  `  H ) ( K `  y
) ) )
75 coass 5353 . . . . . . . . 9  |-  ( ( K  o.  F )  o.  f )  =  ( K  o.  ( F  o.  f )
)
7675fveq1i 5864 . . . . . . . 8  |-  ( ( ( K  o.  F
)  o.  f ) `
 x )  =  ( ( K  o.  ( F  o.  f
) ) `  x
)
77 fvco3 5940 . . . . . . . . 9  |-  ( ( ( F  o.  f
) : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) --> B  /\  x  e.  ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) )  ->  (
( K  o.  ( F  o.  f )
) `  x )  =  ( K `  ( ( F  o.  f ) `  x
) ) )
7865, 77sylan 474 . . . . . . . 8  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) )  ->  (
( K  o.  ( F  o.  f )
) `  x )  =  ( K `  ( ( F  o.  f ) `  x
) ) )
7976, 78syl5req 2497 . . . . . . 7  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) )  ->  ( K `  ( ( F  o.  f ) `  x ) )  =  ( ( ( K  o.  F )  o.  f ) `  x
) )
8052, 66, 69, 74, 79seqhomo 12257 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( K `  (  seq 1 ( ( +g  `  G ) ,  ( F  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )  =  (  seq 1 ( ( +g  `  H ) ,  ( ( K  o.  F
)  o.  f ) ) `  ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) )
81 gsumzmhm.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
822adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  A  e.  V
)
83 gsumzmhm.c . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
8483adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
8527adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F supp  .0.  )  C_  ( `' F " ( _V  \  {  .0.  } ) ) )
86 f1ofo 5819 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
87 forn 5794 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
8886, 87syl 17 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ran  f  =  ( `' F " ( _V  \  {  .0.  } ) ) )
8988ad2antll 734 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
9085, 89sseqtr4d 3468 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F supp  .0.  )  C_  ran  f )
91 eqid 2450 . . . . . . . 8  |-  ( ( F  o.  f ) supp 
.0.  )  =  ( ( F  o.  f
) supp  .0.  )
9214, 8, 49, 81, 48, 82, 53, 84, 67, 61, 90, 91gsumval3 17534 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( G  gsumg  F )  =  (  seq 1
( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
9392fveq2d 5867 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( K `  ( G  gsumg  F ) )  =  ( K `  (  seq 1 ( ( +g  `  G ) ,  ( F  o.  f ) ) `  ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) ) )
94 eqid 2450 . . . . . . 7  |-  (Cntz `  H )  =  (Cntz `  H )
951adantr 467 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  H  e.  Mnd )
9631adantr 467 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  K : B --> ( Base `  H )
)
97 fco 5737 . . . . . . . 8  |-  ( ( K : B --> ( Base `  H )  /\  F : A --> B )  -> 
( K  o.  F
) : A --> ( Base `  H ) )
9896, 53, 97syl2anc 666 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( K  o.  F ) : A --> ( Base `  H )
)
9981, 94cntzmhm2 16986 . . . . . . . . 9  |-  ( ( K  e.  ( G MndHom  H )  /\  ran  F 
C_  ( Z `  ran  F ) )  -> 
( K " ran  F )  C_  ( (Cntz `  H ) `  ( K " ran  F ) ) )
10070, 84, 99syl2anc 666 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( K " ran  F )  C_  (
(Cntz `  H ) `  ( K " ran  F ) ) )
101 rnco2 5341 . . . . . . . 8  |-  ran  ( K  o.  F )  =  ( K " ran  F )
102101fveq2i 5866 . . . . . . . 8  |-  ( (Cntz `  H ) `  ran  ( K  o.  F
) )  =  ( (Cntz `  H ) `  ( K " ran  F ) )
103100, 101, 1023sstr4g 3472 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  ( K  o.  F )  C_  (
(Cntz `  H ) `  ran  ( K  o.  F ) ) )
104 eldifi 3554 . . . . . . . . . . 11  |-  ( x  e.  ( A  \ 
( `' F "
( _V  \  {  .0.  } ) ) )  ->  x  e.  A
)
105 fvco3 5940 . . . . . . . . . . 11  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( K  o.  F ) `  x
)  =  ( K `
 ( F `  x ) ) )
10653, 104, 105syl2an 480 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( K  o.  F ) `  x )  =  ( K `  ( F `
 x ) ) )
10720a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  .0.  e.  _V )
10853, 85, 82, 107suppssr 6943 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( F `  x )  =  .0.  )
109108fveq2d 5867 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( K `  ( F `  x ) )  =  ( K `
 .0.  ) )
11010ad2antrr 731 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( K `  .0.  )  =  ( 0g `  H ) )
111106, 109, 1103eqtrd 2488 . . . . . . . . 9  |-  ( ( ( ph  /\  (
( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN  /\  f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) )  /\  x  e.  ( A  \  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( K  o.  F ) `  x )  =  ( 0g `  H ) )
11298, 111suppss 6942 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( K  o.  F ) supp  ( 0g `  H ) ) 
C_  ( `' F " ( _V  \  {  .0.  } ) ) )
113112, 89sseqtr4d 3468 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( ( K  o.  F ) supp  ( 0g `  H ) ) 
C_  ran  f )
114 eqid 2450 . . . . . . 7  |-  ( ( ( K  o.  F
)  o.  f ) supp  ( 0g `  H
) )  =  ( ( ( K  o.  F )  o.  f
) supp  ( 0g `  H ) )
11529, 3, 71, 94, 95, 82, 98, 103, 67, 61, 113, 114gsumval3 17534 . . . . . 6  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( H  gsumg  ( K  o.  F ) )  =  (  seq 1
( ( +g  `  H
) ,  ( ( K  o.  F )  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
11680, 93, 1153eqtr4rd 2495 . . . . 5  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( H  gsumg  ( K  o.  F ) )  =  ( K `  ( G  gsumg  F ) ) )
117116expr 619 . . . 4  |-  ( (
ph  /\  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )  ->  (
f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ( H  gsumg  ( K  o.  F
) )  =  ( K `  ( G 
gsumg  F ) ) ) )
118117exlimdv 1778 . . 3  |-  ( (
ph  /\  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )  ->  ( E. f  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  -> 
( H  gsumg  ( K  o.  F
) )  =  ( K `  ( G 
gsumg  F ) ) ) )
119118expimpd 607 . 2  |-  ( ph  ->  ( ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )  ->  ( H  gsumg  ( K  o.  F ) )  =  ( K `  ( G  gsumg  F ) ) ) )
120 gsumzmhm.w . . . . 5  |-  ( ph  ->  F finSupp  .0.  )
121120fsuppimpd 7887 . . . 4  |-  ( ph  ->  ( F supp  .0.  )  e.  Fin )
12225, 121eqeltrrd 2529 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
123 fz1f1o 13769 . . 3  |-  ( ( `' F " ( _V 
\  {  .0.  }
) )  e.  Fin  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  \/  ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) ) )
124122, 123syl 17 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  \/  ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) ) )
12547, 119, 124mpjaod 383 1  |-  ( ph  ->  ( H  gsumg  ( K  o.  F
) )  =  ( K `  ( G 
gsumg  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1443   E.wex 1662    e. wcel 1886   _Vcvv 3044    \ cdif 3400    C_ wss 3403   (/)c0 3730   {csn 3967   class class class wbr 4401    |-> cmpt 4460   `'ccnv 4832   dom cdm 4833   ran crn 4834   "cima 4836    o. ccom 4837   -->wf 5577   -1-1->wf1 5578   -onto->wfo 5579   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6288   supp csupp 6911   Fincfn 7566   finSupp cfsupp 7880   1c1 9537   NNcn 10606   ZZ>=cuz 11156   ...cfz 11781    seqcseq 12210   #chash 12512   Basecbs 15114   +g cplusg 15183   0gc0g 15331    gsumg cgsu 15332   Mndcmnd 16528   MndHom cmhm 16573  Cntzccntz 16962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-se 4793  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-supp 6912  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-map 7471  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-fsupp 7881  df-oi 8022  df-card 8370  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-fzo 11913  df-seq 12211  df-hash 12513  df-0g 15333  df-gsum 15334  df-mgm 16481  df-sgrp 16520  df-mnd 16530  df-mhm 16575  df-cntz 16964
This theorem is referenced by:  gsummhm  17564  gsumzinv  17571
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