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Theorem gsumzmhmOLD 16430
Description: Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.) Obsolete version of gsumzmhm 16429 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 2442 . . . . . . . 8  |-  ( 0g
`  H )  =  ( 0g `  H
)
43gsumz 15510 . . . . . . 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 15471 . . . . . . 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 2477 . . . 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 15438 . . . . . . . . 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 3374 . . . . . . . . 9  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  ( `' F " ( _V 
\  {  .0.  }
) )
2019a1i 11 . . . . . . . 8  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  C_  ( `' F " ( _V 
\  {  .0.  }
) ) )
2118, 20gsumcllemOLD 16386 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  F  =  ( k  e.  A  |->  .0.  ) )
22 eqid 2442 . . . . . . . . . . 11  |-  ( Base `  H )  =  (
Base `  H )
2314, 22mhmf 15468 . . . . . . . . . 10  |-  ( K  e.  ( G MndHom  H
)  ->  K : B
--> ( Base `  H
) )
247, 23syl 16 . . . . . . . . 9  |-  ( ph  ->  K : B --> ( Base `  H ) )
2524feqmptd 5743 . . . . . . . 8  |-  ( ph  ->  K  =  ( x  e.  B  |->  ( K `
 x ) ) )
2625adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  K  =  ( x  e.  B  |->  ( K `  x ) ) )
27 fveq2 5690 . . . . . . 7  |-  ( x  =  .0.  ->  ( K `  x )  =  ( K `  .0.  ) )
2817, 21, 26, 27fmptco 5875 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K  o.  F )  =  ( k  e.  A  |->  ( K `  .0.  ) ) )
2910mpteq2dv 4378 . . . . . . 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 2474 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K  o.  F )  =  ( k  e.  A  |->  ( 0g `  H ) ) )
3231oveq2d 6106 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( H  gsumg  ( K  o.  F
) )  =  ( H  gsumg  ( k  e.  A  |->  ( 0g `  H
) ) ) )
3321oveq2d 6106 . . . . . 6  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  ( G  gsumg  ( k  e.  A  |->  .0.  ) ) )
348gsumz 15510 . . . . . . . 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 2474 . . . . 5  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( G  gsumg  F )  =  .0.  )
3837fveq2d 5694 . . . 4  |-  ( (
ph  /\  ( `' F " ( _V  \  {  .0.  } ) )  =  (/) )  ->  ( K `  ( G  gsumg  F ) )  =  ( K `  .0.  )
)
3912, 32, 383eqtr4d 2484 . . 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 2442 . . . . . . . . . 10  |-  ( +g  `  G )  =  ( +g  `  G )
4314, 42mndcl 15419 . . . . . . . . 9  |-  ( ( G  e.  Mnd  /\  x  e.  B  /\  y  e.  B )  ->  ( x ( +g  `  G ) y )  e.  B )
44433expb 1188 . . . . . . . 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 5639 . . . . . . . . . . . 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 5188 . . . . . . . . . . . 12  |-  ( `' F " ( _V 
\  {  .0.  }
) )  C_  dom  F
50 fdm 5562 . . . . . . . . . . . . 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 3403 . . . . . . . . . . 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 5610 . . . . . . . . . . 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 5605 . . . . . . . . . 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 5567 . . . . . . . . 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 5842 . . . . . . 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 755 . . . . . . . 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 10895 . . . . . . . 8  |-  NN  =  ( ZZ>= `  1 )
6260, 61syl6eleq 2532 . . . . . . 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 2442 . . . . . . . . . 10  |-  ( +g  `  H )  =  ( +g  `  H )
6514, 42, 64mhmlin 15470 . . . . . . . . 9  |-  ( ( K  e.  ( G MndHom  H )  /\  x  e.  B  /\  y  e.  B )  ->  ( K `  ( x
( +g  `  G ) y ) )  =  ( ( K `  x ) ( +g  `  H ) ( K `
 y ) ) )
66653expb 1188 . . . . . . . 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 5355 . . . . . . . . 9  |-  ( ( K  o.  F )  o.  f )  =  ( K  o.  ( F  o.  f )
)
6968fveq1i 5691 . . . . . . . 8  |-  ( ( ( K  o.  F
)  o.  f ) `
 x )  =  ( ( K  o.  ( F  o.  f
) ) `  x
)
70 fvco3 5767 . . . . . . . . 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 2487 . . . . . . 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 11852 . . . . . 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 5647 . . . . . . . . . . 11  |-  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  ->  f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )
-onto-> ( `' F "
( _V  \  {  .0.  } ) ) )
80 forn 5622 . . . . . . . . . . 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 3392 . . . . . . . 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 2442 . . . . . . . 8  |-  ( `' ( F  o.  f
) " ( _V 
\  {  .0.  }
) )  =  ( `' ( F  o.  f ) " ( _V  \  {  .0.  }
) )
8514, 8, 42, 74, 41, 75, 46, 77, 60, 54, 83, 84gsumval3OLD 16381 . . . . . . 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 5694 . . . . . 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 2442 . . . . . . 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 5567 . . . . . . . 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 15856 . . . . . . . . 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 5344 . . . . . . . 8  |-  ran  ( K  o.  F )  =  ( K " ran  F )
9594fveq2i 5693 . . . . . . . 8  |-  ( (Cntz `  H ) `  ran  ( K  o.  F
) )  =  ( (Cntz `  H ) `  ( K " ran  F ) )
9693, 94, 953sstr4g 3396 . . . . . . 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 3477 . . . . . . . . . . 11  |-  ( x  e.  ( A  \ 
( `' F "
( _V  \  {  .0.  } ) ) )  ->  x  e.  A
)
98 fvco3 5767 . . . . . . . . . . 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 5836 . . . . . . . . . . 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 5694 . . . . . . . . . 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 2478 . . . . . . . . 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 5835 . . . . . . . 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 3392 . . . . . . 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 2442 . . . . . . 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 16381 . . . . . 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 2485 . . . . 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 1690 . . 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 13186 . . 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 1369   E.wex 1586    e. wcel 1756   _Vcvv 2971    \ cdif 3324    C_ wss 3327   (/)c0 3636   {csn 3876    e. cmpt 4349   `'ccnv 4838   dom cdm 4839   ran crn 4840   "cima 4842    o. ccom 4843   -->wf 5413   -1-1->wf1 5414   -onto->wfo 5415   -1-1-onto->wf1o 5416   ` cfv 5417  (class class class)co 6090   Fincfn 7309   1c1 9282   NNcn 10321   ZZ>=cuz 10860   ...cfz 11436    seqcseq 11805   #chash 12102   Basecbs 14173   +g cplusg 14237   0gc0g 14377    gsumg cgsu 14378   Mndcmnd 15408   MndHom cmhm 15461  Cntzccntz 15832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-pre-mulgt0 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-int 4128  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-tr 4385  df-eprel 4631  df-id 4635  df-po 4640  df-so 4641  df-fr 4678  df-se 4679  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6831  df-rdg 6865  df-1o 6919  df-oadd 6923  df-er 7100  df-map 7215  df-en 7310  df-dom 7311  df-sdom 7312  df-fin 7313  df-oi 7723  df-card 8108  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-sub 9596  df-neg 9597  df-nn 10322  df-n0 10579  df-z 10646  df-uz 10861  df-fz 11437  df-fzo 11548  df-seq 11806  df-hash 12103  df-0g 14379  df-gsum 14380  df-mnd 15414  df-mhm 15463  df-cntz 15834
This theorem is referenced by:  gsummhmOLD  16432  gsumzinvOLD  16442
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