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Theorem gsumzmhmOLD 16937
Description: Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) Obsolete version of gsumzmhm 16936 as of 6-Jun-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
gsumzmhmOLD.b  |-  B  =  ( Base `  G
)
gsumzmhmOLD.z  |-  Z  =  (Cntz `  G )
gsumzmhmOLD.g  |-  ( ph  ->  G  e.  Mnd )
gsumzmhmOLD.h  |-  ( ph  ->  H  e.  Mnd )
gsumzmhmOLD.a  |-  ( ph  ->  A  e.  V )
gsumzmhmOLD.k  |-  ( ph  ->  K  e.  ( G MndHom  H ) )
gsumzmhmOLD.f  |-  ( ph  ->  F : A --> B )
gsumzmhmOLD.c  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
gsumzmhmOLD.0  |-  .0.  =  ( 0g `  G )
gsumzmhmOLD.w  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
Assertion
Ref Expression
gsumzmhmOLD  |-  ( ph  ->  ( H  gsumg  ( K  o.  F
) )  =  ( K `  ( G 
gsumg  F ) ) )

Proof of Theorem gsumzmhmOLD
Dummy variables  k  x  y  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumzmhmOLD.h . . . . . . 7  |-  ( ph  ->  H  e.  Mnd )
2 gsumzmhmOLD.a . . . . . . 7  |-  ( ph  ->  A  e.  V )
3 eqid 2443 . . . . . . . 8  |-  ( 0g
`  H )  =  ( 0g `  H
)
43gsumz 15984 . . . . . . 7  |-  ( ( H  e.  Mnd  /\  A  e.  V )  ->  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) )  =  ( 0g `  H
) )
51, 2, 4syl2anc 661 . . . . . 6  |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) )  =  ( 0g `  H
) )
65adantr 465 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) )  =  ( 0g `  H
) )
7 gsumzmhmOLD.k . . . . . . 7  |-  ( ph  ->  K  e.  ( G MndHom  H ) )
8 gsumzmhmOLD.0 . . . . . . . 8  |-  .0.  =  ( 0g `  G )
98, 3mhm0 15953 . . . . . . 7  |-  ( K  e.  ( G MndHom  H
)  ->  ( K `  .0.  )  =  ( 0g `  H ) )
107, 9syl 16 . . . . . 6  |-  ( ph  ->  ( K `  .0.  )  =  ( 0g `  H ) )
1110adantr 465 . . . . 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 gsumzmhmOLD.g . . . . . . . . 9  |-  ( ph  ->  G  e.  Mnd )
14 gsumzmhmOLD.b . . . . . . . . . 10  |-  B  =  ( Base `  G
)
1514, 8mndidcl 15917 . . . . . . . . 9  |-  ( G  e.  Mnd  ->  .0.  e.  B )
1613, 15syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
1716ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  ( `' F " ( _V 
\  {  .0.  }
) )  =  (/) )  /\  k  e.  A
)  ->  .0.  e.  B )
18 gsumzmhmOLD.f . . . . . . . 8  |-  ( ph  ->  F : A --> B )
19 ssid 3508 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2019a1i 11 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
2118, 20gsumcllemOLD 16892 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
22 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  H )  =  (
Base `  H )
2314, 22mhmf 15950 . . . . . . . . . 10  |-  ( K  e.  ( G MndHom  H
)  ->  K : B
--> ( Base `  H
) )
247, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  K : B --> ( Base `  H ) )
2524feqmptd 5911 . . . . . . . 8  |-  ( ph  ->  K  =  ( x  e.  B  |->  ( K `
 x ) ) )
2625adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  K  =  ( x  e.  B  |->  ( K `  x ) ) )
27 fveq2 5856 . . . . . . 7  |-  ( x  =  .0.  ->  ( K `  x )  =  ( K `  .0.  ) )
2817, 21, 26, 27fmptco 6049 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K  o.  F )  =  ( k  e.  A  |->  ( K `  .0.  ) ) )
2910mpteq2dv 4524 . . . . . . 7  |-  ( ph  ->  ( k  e.  A  |->  ( K `  .0.  ) )  =  ( k  e.  A  |->  ( 0g `  H ) ) )
3029adantr 465 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  (
k  e.  A  |->  ( K `  .0.  )
)  =  ( k  e.  A  |->  ( 0g
`  H ) ) )
3128, 30eqtrd 2484 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K  o.  F )  =  ( k  e.  A  |->  ( 0g `  H ) ) )
3231oveq2d 6297 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( K  o.  F
) )  =  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) ) )
3321oveq2d 6297 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
348gsumz 15984 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  A  e.  V )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
3513, 2, 34syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
3635adantr 465 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
3733, 36eqtrd 2484 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  .0.  )
3837fveq2d 5860 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K `  ( G  gsumg  F ) )  =  ( K `  .0.  )
)
3912, 32, 383eqtr4d 2494 . . 3  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( K  o.  F
) )  =  ( K `  ( G 
gsumg  F ) ) )
4039ex 434 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  ->  ( H  gsumg  ( K  o.  F ) )  =  ( K `  ( G  gsumg  F ) ) ) )
4113adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  G  e.  Mnd )
42 eqid 2443 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
4314, 42mndcl 15908 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
44433expb 1198 . . . . . . . 8  |-  ( ( G  e.  Mnd  /\  ( x  e.  B  /\  y  e.  B
) )  ->  (
x ( +g  `  G
) y )  e.  B )
4541, 44sylan 471 . . . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  F : A --> B )
47 f1of1 5805 . . . . . . . . . . . 12  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> ( `' F "
( _V  \  {  .0.  } ) ) )
4847ad2antll 728 . . . . . . . . . . 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 5347 . . . . . . . . . . . 12  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
50 fdm 5725 . . . . . . . . . . . . 13  |-  ( F : A --> B  ->  dom  F  =  A )
5146, 50syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  dom  F  =  A )
5249, 51syl5sseq 3537 . . . . . . . . . . 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 5776 . . . . . . . . . . 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 661 . . . . . . . . . 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 5771 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-1-1-> A  ->  f :
( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) --> A )
5654, 55syl 16 . . . . . . . . 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 5731 . . . . . . . . 9  |-  ( ( F : A --> B  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) --> A )  ->  ( F  o.  f ) : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) --> B )
5846, 56, 57syl2anc 661 . . . . . . . 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 )
5958ffvelrnda 6016 . . . . . . 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 )
60 simprl 756 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  NN )
61 nnuz 11127 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
6260, 61syl6eleq 2541 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) )  e.  ( ZZ>= `  1 )
)
637adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  K  e.  ( G MndHom  H ) )
64 eqid 2443 . . . . . . . . . 10  |-  ( +g  `  H )  =  ( +g  `  H )
6514, 42, 64mhmlin 15952 . . . . . . . . 9  |-  ( ( K  e.  ( G MndHom  H )  /\  x  e.  B  /\  y  e.  B )  ->  ( K `  ( x
( +g  `  G ) y ) )  =  ( ( K `  x ) ( +g  `  H ) ( K `
 y ) ) )
66653expb 1198 . . . . . . . 8  |-  ( ( K  e.  ( G MndHom  H )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( K `  ( x ( +g  `  G ) y ) )  =  ( ( K `  x ) ( +g  `  H
) ( K `  y ) ) )
6763, 66sylan 471 . . . . . . 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
) ) )
68 coass 5516 . . . . . . . . 9  |-  ( ( K  o.  F )  o.  f )  =  ( K  o.  ( F  o.  f )
)
6968fveq1i 5857 . . . . . . . 8  |-  ( ( ( K  o.  F
)  o.  f ) `
 x )  =  ( ( K  o.  ( F  o.  f
) ) `  x
)
70 fvco3 5935 . . . . . . . . 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
) ) )
7158, 70sylan 471 . . . . . . . 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
) ) )
7269, 71syl5req 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
) )
7345, 59, 62, 67, 72seqhomo 12136 . . . . . 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.  } ) ) ) ) )
74 gsumzmhmOLD.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
752adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  A  e.  V
)
76 gsumzmhmOLD.c . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
7776adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  F  C_  ( Z `  ran  F ) )
7819a1i 11 . . . . . . . . 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.  }
) ) )
79 f1ofo 5813 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
80 forn 5788 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
8179, 80syl 16 . . . . . . . . . 10  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  ran  f  =  ( `' F " ( _V  \  {  .0.  } ) ) )
8281ad2antll 728 . . . . . . . . 9  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ran  f  =  ( `' F " ( _V 
\  {  .0.  }
) ) )
8378, 82sseqtr4d 3526 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  ran  f )
84 eqid 2443 . . . . . . . 8  |-  ( `' ( F  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  o.  f ) " ( _V  \  {  .0.  }
) )
8514, 8, 42, 74, 41, 75, 46, 77, 60, 54, 83, 84gsumval3OLD 16887 . . . . . . 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.  }
) ) ) ) )
8685fveq2d 5860 . . . . . 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.  } ) ) ) ) ) )
87 eqid 2443 . . . . . . 7  |-  (Cntz `  H )  =  (Cntz `  H )
881adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  H  e.  Mnd )
8924adantr 465 . . . . . . . 8  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  K : B --> ( Base `  H )
)
90 fco 5731 . . . . . . . 8  |-  ( ( K : B --> ( Base `  H )  /\  F : A --> B )  -> 
( K  o.  F
) : A --> ( Base `  H ) )
9189, 46, 90syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  ( ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) )  e.  NN  /\  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) ) ) )  ->  ( K  o.  F ) : A --> ( Base `  H )
)
9274, 87cntzmhm2 16356 . . . . . . . . 9  |-  ( ( K  e.  ( G MndHom  H )  /\  ran  F 
C_  ( Z `  ran  F ) )  -> 
( K " ran  F )  C_  ( (Cntz `  H ) `  ( K " ran  F ) ) )
9363, 77, 92syl2anc 661 . . . . . . . 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 ) ) )
94 rnco2 5504 . . . . . . . 8  |-  ran  ( K  o.  F )  =  ( K " ran  F )
9594fveq2i 5859 . . . . . . . 8  |-  ( (Cntz `  H ) `  ran  ( K  o.  F
) )  =  ( (Cntz `  H ) `  ( K " ran  F ) )
9693, 94, 953sstr4g 3530 . . . . . . 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 ) ) )
97 eldifi 3611 . . . . . . . . . . 11  |-  ( x  e.  ( A  \ 
( `' F "
( _V  \  {  .0.  } ) ) )  ->  x  e.  A
)
98 fvco3 5935 . . . . . . . . . . 11  |-  ( ( F : A --> B  /\  x  e.  A )  ->  ( ( K  o.  F ) `  x
)  =  ( K `
 ( F `  x ) ) )
9946, 97, 98syl2an 477 . . . . . . . . . 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 ) ) )
10046, 78suppssrOLD 6006 . . . . . . . . . . 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.  )
101100fveq2d 5860 . . . . . . . . . 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.  ) )
10210ad2antrr 725 . . . . . . . . . 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 ) )
10399, 101, 1023eqtrd 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 ) )
10491, 103suppssOLD 6005 . . . . . . . 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.  } ) ) )
105104, 82sseqtr4d 3526 . . . . . . 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 )
106 eqid 2443 . . . . . . 7  |-  ( `' ( ( K  o.  F )  o.  f
) " ( _V 
\  { ( 0g
`  H ) } ) )  =  ( `' ( ( K  o.  F )  o.  f ) " ( _V  \  { ( 0g
`  H ) } ) )
10722, 3, 64, 87, 88, 75, 91, 96, 60, 54, 105, 106gsumval3OLD 16887 . . . . . 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.  }
) ) ) ) )
10873, 86, 1073eqtr4rd 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 ) ) )
109108expr 615 . . . 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 ) ) ) )
110109exlimdv 1711 . . 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 ) ) ) )
111110expimpd 603 . 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 ) ) ) )
112 gsumzmhmOLD.w . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  e. 
Fin )
113 fz1f1o 13514 . . 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.  } ) ) ) ) )
114112, 113syl 16 . 2  |-  ( ph  ->  ( ( `' F " ( _V  \  {  .0.  } ) )  =  (/)  \/  ( ( # `  ( `' F "
( _V  \  {  .0.  } ) ) )  e.  NN  /\  E. f  f : ( 1 ... ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) ) ) ) )
11540, 111, 114mpjaod 381 1  |-  ( ph  ->  ( H  gsumg  ( K  o.  F
) )  =  ( K `  ( G 
gsumg  F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1383   E.wex 1599    e. wcel 1804   _Vcvv 3095    \ cdif 3458    C_ wss 3461   (/)c0 3770   {csn 4014    |-> cmpt 4495   `'ccnv 4988   dom cdm 4989   ran crn 4990   "cima 4992    o. ccom 4993   -->wf 5574   -1-1->wf1 5575   -onto->wfo 5576   -1-1-onto->wf1o 5577   ` cfv 5578  (class class class)co 6281   Fincfn 7518   1c1 9496   NNcn 10543   ZZ>=cuz 11092   ...cfz 11683    seqcseq 12089   #chash 12387   Basecbs 14614   +g cplusg 14679   0gc0g 14819    gsumg cgsu 14820   Mndcmnd 15898   MndHom cmhm 15943  Cntzccntz 16332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-fzo 11807  df-seq 12090  df-hash 12388  df-0g 14821  df-gsum 14822  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-mhm 15945  df-cntz 16334
This theorem is referenced by:  gsummhmOLD  16939  gsumzinvOLD  16949
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