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Theorem gsumzmhm 15210
Description: Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.)
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 "
( _V  \  {  .0.  } ) )  e. 
Fin )
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 2283 . . . . . . . 8  |-  ( 0g
`  H )  =  ( 0g `  H
)
43gsumz 14458 . . . . . . 7  |-  ( ( H  e.  Mnd  /\  A  e.  V )  ->  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) )  =  ( 0g `  H
) )
51, 2, 4syl2anc 642 . . . . . 6  |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) )  =  ( 0g `  H
) )
65adantr 451 . . . . 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 14423 . . . . . . 7  |-  ( K  e.  ( G MndHom  H
)  ->  ( K `  .0.  )  =  ( 0g `  H ) )
107, 9syl 15 . . . . . 6  |-  ( ph  ->  ( K `  .0.  )  =  ( 0g `  H ) )
1110adantr 451 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K `  .0.  )  =  ( 0g `  H
) )
126, 11eqtr4d 2318 . . . 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 14391 . . . . . . . . 9  |-  ( G  e.  Mnd  ->  .0.  e.  B )
1613, 15syl 15 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
1716ad2antrr 706 . . . . . . 7  |-  ( ( ( ph  /\  ( `' F " ( _V 
\  {  .0.  }
) )  =  (/) )  /\  k  e.  A
)  ->  .0.  e.  B )
18 gsumzmhm.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
19 ssid 3197 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2019a1i 10 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
2118, 20gsumcllem 15193 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
22 eqid 2283 . . . . . . . . . . 11  |-  ( Base `  H )  =  (
Base `  H )
2314, 22mhmf 14420 . . . . . . . . . 10  |-  ( K  e.  ( G MndHom  H
)  ->  K : B
--> ( Base `  H
) )
247, 23syl 15 . . . . . . . . 9  |-  ( ph  ->  K : B --> ( Base `  H ) )
2524feqmptd 5575 . . . . . . . 8  |-  ( ph  ->  K  =  ( x  e.  B  |->  ( K `
 x ) ) )
2625adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  K  =  ( x  e.  B  |->  ( K `  x ) ) )
27 fveq2 5525 . . . . . . 7  |-  ( x  =  .0.  ->  ( K `  x )  =  ( K `  .0.  ) )
2817, 21, 26, 27fmptco 5691 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K  o.  F )  =  ( k  e.  A  |->  ( K `  .0.  ) ) )
2910mpteq2dv 4107 . . . . . . 7  |-  ( ph  ->  ( k  e.  A  |->  ( K `  .0.  ) )  =  ( k  e.  A  |->  ( 0g `  H ) ) )
3029adantr 451 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  (
k  e.  A  |->  ( K `  .0.  )
)  =  ( k  e.  A  |->  ( 0g
`  H ) ) )
3128, 30eqtrd 2315 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K  o.  F )  =  ( k  e.  A  |->  ( 0g `  H ) ) )
3231oveq2d 5874 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( K  o.  F
) )  =  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) ) )
3321oveq2d 5874 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
348gsumz 14458 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
3513, 2, 34syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
3635adantr 451 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
3733, 36eqtrd 2315 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  .0.  )
3837fveq2d 5529 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K `  ( G  gsumg  F ) )  =  ( K `  .0.  )
)
3912, 32, 383eqtr4d 2325 . . 3  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( K  o.  F
) )  =  ( K `  ( G 
gsumg  F ) ) )
4039ex 423 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  ->  ( H  gsumg  ( K  o.  F ) )  =  ( K `  ( G  gsumg  F ) ) ) )
4113adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  G  e.  Mnd )
42 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
4314, 42mndcl 14372 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
44433expb 1152 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  G
) y )  e.  B )
4541, 44sylan 457 . . . . . . 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 )
4618adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> B )
47 f1of1 5471 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) ) )
4847ad2antll 709 . . . . . . . . . . 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.  } ) ) )
49 cnvimass 5033 . . . . . . . . . . . 12  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
50 fdm 5393 . . . . . . . . . . . . 13  |-  ( F : A --> B  ->  dom  F  =  A )
5146, 50syl 15 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  dom  F  =  A )
5249, 51syl5sseq 3226 . . . . . . . . . . 11  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  A )
53 f1ss 5442 . . . . . . . . . . 11  |-  ( ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) )  /\  ( `' F " ( _V 
\  {  .0.  }
) )  C_  A
)  ->  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-> A )
5448, 52, 53syl2anc 642 . . . . . . . . . 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 )
55 f1f 5437 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> A  ->  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) --> A )
5654, 55syl 15 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) --> A )
57 fco 5398 . . . . . . . . 9  |-  ( ( F : A --> B  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) --> A )  ->  ( F  o.  f ) : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) --> B )
5846, 56, 57syl2anc 642 . . . . . . . 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 )
59 ffvelrn 5663 . . . . . . . 8  |-  ( ( ( F  o.  f
) : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) --> B  /\  x  e.  ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) )  ->  (
( F  o.  f
) `  x )  e.  B )
6058, 59sylan 457 . . . . . . 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 )
61 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )
62 nnuz 10263 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
6361, 62syl6eleq 2373 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  ( ZZ>= `  1 )
)
647adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  K  e.  ( G MndHom  H ) )
65 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  H )  =  ( +g  `  H )
6614, 42, 65mhmlin 14422 . . . . . . . . 9  |-  ( ( K  e.  ( G MndHom  H )  /\  x  e.  B  /\  y  e.  B )  ->  ( K `  ( x
( +g  `  G ) y ) )  =  ( ( K `  x ) ( +g  `  H ) ( K `
 y ) ) )
67663expb 1152 . . . . . . . 8  |-  ( ( K  e.  ( G MndHom  H )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( K `  ( x ( +g  `  G ) y ) )  =  ( ( K `  x ) ( +g  `  H
) ( K `  y ) ) )
6864, 67sylan 457 . . . . . . 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
) ) )
69 coass 5191 . . . . . . . . 9  |-  ( ( K  o.  F )  o.  f )  =  ( K  o.  ( F  o.  f )
)
7069fveq1i 5526 . . . . . . . 8  |-  ( ( ( K  o.  F
)  o.  f ) `
 x )  =  ( ( K  o.  ( F  o.  f
) ) `  x
)
71 fvco3 5596 . . . . . . . . 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
) ) )
7258, 71sylan 457 . . . . . . . 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
) ) )
7370, 72syl5req 2328 . . . . . . 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
) )
7445, 60, 63, 68, 73seqhomo 11093 . . . . . 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.  } ) ) ) ) )
75 gsumzmhm.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
762adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  A  e.  V
)
77 gsumzmhm.c . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
7877adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
7919a1i 10 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
80 f1ofo 5479 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
81 forn 5454 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
8280, 81syl 15 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ran  f  =  ( `' F " ( _V  \  {  .0.  } ) ) )
8382ad2antll 709 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
8479, 83sseqtr4d 3215 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  ran  f )
85 eqid 2283 . . . . . . . 8  |-  ( `' ( F  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  o.  f ) " ( _V  \  {  .0.  }
) )
8614, 8, 42, 75, 41, 76, 46, 78, 61, 54, 84, 85gsumval3 15191 . . . . . . 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.  }
) ) ) ) )
8786fveq2d 5529 . . . . . 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.  } ) ) ) ) ) )
88 eqid 2283 . . . . . . 7  |-  (Cntz `  H )  =  (Cntz `  H )
891adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  H  e.  Mnd )
9024adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  K : B --> ( Base `  H )
)
91 fco 5398 . . . . . . . 8  |-  ( ( K : B --> ( Base `  H )  /\  F : A --> B )  -> 
( K  o.  F
) : A --> ( Base `  H ) )
9290, 46, 91syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( K  o.  F ) : A --> ( Base `  H )
)
9375, 88cntzmhm2 14815 . . . . . . . . 9  |-  ( ( K  e.  ( G MndHom  H )  /\  ran  F 
C_  ( Z `  ran  F ) )  -> 
( K " ran  F )  C_  ( (Cntz `  H ) `  ( K " ran  F ) ) )
9464, 78, 93syl2anc 642 . . . . . . . 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 ) ) )
95 rnco2 5180 . . . . . . . 8  |-  ran  ( K  o.  F )  =  ( K " ran  F )
9695fveq2i 5528 . . . . . . . 8  |-  ( (Cntz `  H ) `  ran  ( K  o.  F
) )  =  ( (Cntz `  H ) `  ( K " ran  F ) )
9794, 95, 963sstr4g 3219 . . . . . . 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 ) ) )
98 eldifi 3298 . . . . . . . . . . 11  |-  ( x  e.  ( A  \ 
( `' F "
( _V  \  {  .0.  } ) ) )  ->  x  e.  A
)
99 fvco3 5596 . . . . . . . . . . 11  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( K  o.  F ) `  x
)  =  ( K `
 ( F `  x ) ) )
10046, 98, 99syl2an 463 . . . . . . . . . 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 ) ) )
10146, 79suppssr 5659 . . . . . . . . . . 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.  )
102101fveq2d 5529 . . . . . . . . . 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.  ) )
10310ad2antrr 706 . . . . . . . . . 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 ) )
104100, 102, 1033eqtrd 2319 . . . . . . . . 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 ) )
10592, 104suppss 5658 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' ( K  o.  F )
" ( _V  \  { ( 0g `  H ) } ) )  C_  ( `' F " ( _V  \  {  .0.  } ) ) )
106105, 83sseqtr4d 3215 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' ( K  o.  F )
" ( _V  \  { ( 0g `  H ) } ) )  C_  ran  f )
107 eqid 2283 . . . . . . 7  |-  ( `' ( ( K  o.  F )  o.  f
) " ( _V 
\  { ( 0g
`  H ) } ) )  =  ( `' ( ( K  o.  F )  o.  f ) " ( _V  \  { ( 0g
`  H ) } ) )
10822, 3, 65, 88, 89, 76, 92, 97, 61, 54, 106, 107gsumval3 15191 . . . . . 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.  }
) ) ) ) )
10974, 87, 1083eqtr4rd 2326 . . . . 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 ) ) )
110109expr 598 . . . 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 ) ) ) )
111110exlimdv 1664 . . 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 ) ) ) )
112111expimpd 586 . 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 ) ) ) )
113 gsumzmhm.w . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
114 fz1f1o 12183 . . 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.  } ) ) ) ) )
115113, 114syl 15 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  \/  ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) ) )
11640, 112, 115mpjaod 370 1  |-  ( ph  ->  ( H  gsumg  ( K  o.  F
) )  =  ( K `  ( G 
gsumg  F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640    e. cmpt 4077   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    o. ccom 4693   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Fincfn 6863   1c1 8738   NNcn 9746   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   #chash 11337   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361   MndHom cmhm 14413  Cntzccntz 14791
This theorem is referenced by:  gsummhm  15211  gsumzinv  15217
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-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  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-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-0g 13404  df-gsum 13405  df-mnd 14367  df-mhm 14415  df-cntz 14793
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