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Theorem gsumzmhm 15226
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 2296 . . . . . . . 8  |-  ( 0g
`  H )  =  ( 0g `  H
)
43gsumz 14474 . . . . . . 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 14439 . . . . . . 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 2331 . . . 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 14407 . . . . . . . . 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 3210 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2019a1i 10 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
2118, 20gsumcllem 15209 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
22 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  H )  =  (
Base `  H )
2314, 22mhmf 14436 . . . . . . . . . 10  |-  ( K  e.  ( G MndHom  H
)  ->  K : B
--> ( Base `  H
) )
247, 23syl 15 . . . . . . . . 9  |-  ( ph  ->  K : B --> ( Base `  H ) )
2524feqmptd 5591 . . . . . . . 8  |-  ( ph  ->  K  =  ( x  e.  B  |->  ( K `
 x ) ) )
2625adantr 451 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  K  =  ( x  e.  B  |->  ( K `  x ) ) )
27 fveq2 5541 . . . . . . 7  |-  ( x  =  .0.  ->  ( K `  x )  =  ( K `  .0.  ) )
2817, 21, 26, 27fmptco 5707 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K  o.  F )  =  ( k  e.  A  |->  ( K `  .0.  ) ) )
2910mpteq2dv 4123 . . . . . . 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 2328 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K  o.  F )  =  ( k  e.  A  |->  ( 0g `  H ) ) )
3231oveq2d 5890 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( K  o.  F
) )  =  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) ) )
3321oveq2d 5890 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
348gsumz 14474 . . . . . . . 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 2328 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  .0.  )
3837fveq2d 5545 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K `  ( G  gsumg  F ) )  =  ( K `  .0.  )
)
3912, 32, 383eqtr4d 2338 . . 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 2296 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
4314, 42mndcl 14388 . . . . . . . . 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 5487 . . . . . . . . . . . 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 5049 . . . . . . . . . . . 12  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
50 fdm 5409 . . . . . . . . . . . . 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 3239 . . . . . . . . . . 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 5458 . . . . . . . . . . 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 5453 . . . . . . . . . 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 5414 . . . . . . . . 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 5679 . . . . . . . 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 10279 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
6361, 62syl6eleq 2386 . . . . . . 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 2296 . . . . . . . . . 10  |-  ( +g  `  H )  =  ( +g  `  H )
6614, 42, 65mhmlin 14438 . . . . . . . . 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 5207 . . . . . . . . 9  |-  ( ( K  o.  F )  o.  f )  =  ( K  o.  ( F  o.  f )
)
7069fveq1i 5542 . . . . . . . 8  |-  ( ( ( K  o.  F
)  o.  f ) `
 x )  =  ( ( K  o.  ( F  o.  f
) ) `  x
)
71 fvco3 5612 . . . . . . . . 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 2341 . . . . . . 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 11109 . . . . . 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 5495 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
81 forn 5470 . . . . . . . . . . 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 3228 . . . . . . . 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 2296 . . . . . . . 8  |-  ( `' ( F  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  o.  f ) " ( _V  \  {  .0.  }
) )
8614, 8, 42, 75, 41, 76, 46, 78, 61, 54, 84, 85gsumval3 15207 . . . . . . 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 5545 . . . . . 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 2296 . . . . . . 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 5414 . . . . . . . 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 14831 . . . . . . . . 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 5196 . . . . . . . 8  |-  ran  ( K  o.  F )  =  ( K " ran  F )
9695fveq2i 5544 . . . . . . . 8  |-  ( (Cntz `  H ) `  ran  ( K  o.  F
) )  =  ( (Cntz `  H ) `  ( K " ran  F ) )
9794, 95, 963sstr4g 3232 . . . . . . 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 3311 . . . . . . . . . . 11  |-  ( x  e.  ( A  \ 
( `' F "
( _V  \  {  .0.  } ) ) )  ->  x  e.  A
)
99 fvco3 5612 . . . . . . . . . . 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 5675 . . . . . . . . . . 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 5545 . . . . . . . . . 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 2332 . . . . . . . . 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 5674 . . . . . . . 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 3228 . . . . . . 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 2296 . . . . . . 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 15207 . . . . . 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 2339 . . . . 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 1626 . . 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 12199 . . 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 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653    e. cmpt 4093   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    o. ccom 4709   -->wf 5267   -1-1->wf1 5268   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   Fincfn 6879   1c1 8754   NNcn 9762   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062   #chash 11353   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417   Mndcmnd 14377   MndHom cmhm 14429  Cntzccntz 14807
This theorem is referenced by:  gsummhm  15227  gsumzinv  15233
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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  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-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-seq 11063  df-hash 11354  df-0g 13420  df-gsum 13421  df-mnd 14383  df-mhm 14431  df-cntz 14809
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