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Theorem gsumress 14454
Description: The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither  G nor 
H need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
gsumress.b  |-  B  =  ( Base `  G
)
gsumress.o  |-  .+  =  ( +g  `  G )
gsumress.h  |-  H  =  ( Gs  S )
gsumress.g  |-  ( ph  ->  G  e.  V )
gsumress.a  |-  ( ph  ->  A  e.  X )
gsumress.s  |-  ( ph  ->  S  C_  B )
gsumress.f  |-  ( ph  ->  F : A --> S )
gsumress.z  |-  ( ph  ->  .0.  e.  S )
gsumress.c  |-  ( (
ph  /\  x  e.  B )  ->  (
(  .0.  .+  x
)  =  x  /\  ( x  .+  .0.  )  =  x ) )
Assertion
Ref Expression
gsumress  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
Distinct variable groups:    x, B    x, G    ph, x    x, S    x, H    x,  .+    x,  .0.
Allowed substitution hints:    A( x)    F( x)    V( x)    X( x)

Proof of Theorem gsumress
Dummy variables  f  m  n  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumress.s . . . . . . . . 9  |-  ( ph  ->  S  C_  B )
2 gsumress.z . . . . . . . . 9  |-  ( ph  ->  .0.  e.  S )
31, 2sseldd 3181 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
4 gsumress.c . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  (
(  .0.  .+  x
)  =  x  /\  ( x  .+  .0.  )  =  x ) )
54ralrimiva 2626 . . . . . . . 8  |-  ( ph  ->  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )
6 oveq1 5865 . . . . . . . . . . . 12  |-  ( y  =  .0.  ->  (
y  .+  x )  =  (  .0.  .+  x
) )
76eqeq1d 2291 . . . . . . . . . . 11  |-  ( y  =  .0.  ->  (
( y  .+  x
)  =  x  <->  (  .0.  .+  x )  =  x ) )
8 oveq2 5866 . . . . . . . . . . . 12  |-  ( y  =  .0.  ->  (
x  .+  y )  =  ( x  .+  .0.  ) )
98eqeq1d 2291 . . . . . . . . . . 11  |-  ( y  =  .0.  ->  (
( x  .+  y
)  =  x  <->  ( x  .+  .0.  )  =  x ) )
107, 9anbi12d 691 . . . . . . . . . 10  |-  ( y  =  .0.  ->  (
( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x )  <->  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
1110ralbidv 2563 . . . . . . . . 9  |-  ( y  =  .0.  ->  ( A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x )  <->  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
1211elrab 2923 . . . . . . . 8  |-  (  .0. 
e.  { y  e.  B  |  A. x  e.  B  ( (
y  .+  x )  =  x  /\  (
x  .+  y )  =  x ) }  <->  (  .0.  e.  B  /\  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x 
.+  .0.  )  =  x ) ) )
133, 5, 12sylanbrc 645 . . . . . . 7  |-  ( ph  ->  .0.  e.  { y  e.  B  |  A. x  e.  B  (
( y  .+  x
)  =  x  /\  ( x  .+  y )  =  x ) } )
1413snssd 3760 . . . . . 6  |-  ( ph  ->  {  .0.  }  C_  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) } )
15 gsumress.g . . . . . . . 8  |-  ( ph  ->  G  e.  V )
16 gsumress.b . . . . . . . . 9  |-  B  =  ( Base `  G
)
17 eqid 2283 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
18 gsumress.o . . . . . . . . 9  |-  .+  =  ( +g  `  G )
19 eqid 2283 . . . . . . . . 9  |-  { y  e.  B  |  A. x  e.  B  (
( y  .+  x
)  =  x  /\  ( x  .+  y )  =  x ) }  =  { y  e.  B  |  A. x  e.  B  ( (
y  .+  x )  =  x  /\  (
x  .+  y )  =  x ) }
2016, 17, 18, 19gsumvallem1 14448 . . . . . . . 8  |-  ( G  e.  V  ->  { y  e.  B  |  A. x  e.  B  (
( y  .+  x
)  =  x  /\  ( x  .+  y )  =  x ) } 
C_  { ( 0g
`  G ) } )
2115, 20syl 15 . . . . . . 7  |-  ( ph  ->  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) }  C_  { ( 0g `  G ) } )
2221, 13sseldd 3181 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  { ( 0g `  G ) } )
23 elsni 3664 . . . . . . . . 9  |-  (  .0. 
e.  { ( 0g
`  G ) }  ->  .0.  =  ( 0g `  G ) )
2422, 23syl 15 . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
2524sneqd 3653 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  G ) } )
2621, 25sseqtr4d 3215 . . . . . 6  |-  ( ph  ->  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) }  C_  {  .0.  } )
2714, 26eqssd 3196 . . . . 5  |-  ( ph  ->  {  .0.  }  =  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) } )
281sselda 3180 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  B )
2928, 4syldan 456 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  (
(  .0.  .+  x
)  =  x  /\  ( x  .+  .0.  )  =  x ) )
3029ralrimiva 2626 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )
3110ralbidv 2563 . . . . . . . . . 10  |-  ( y  =  .0.  ->  ( A. x  e.  S  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x )  <->  A. x  e.  S  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
3231elrab 2923 . . . . . . . . 9  |-  (  .0. 
e.  { y  e.  S  |  A. x  e.  S  ( (
y  .+  x )  =  x  /\  (
x  .+  y )  =  x ) }  <->  (  .0.  e.  S  /\  A. x  e.  S  ( (  .0.  .+  x )  =  x  /\  ( x 
.+  .0.  )  =  x ) ) )
332, 30, 32sylanbrc 645 . . . . . . . 8  |-  ( ph  ->  .0.  e.  { y  e.  S  |  A. x  e.  S  (
( y  .+  x
)  =  x  /\  ( x  .+  y )  =  x ) } )
34 gsumress.h . . . . . . . . . . 11  |-  H  =  ( Gs  S )
3534, 16ressbas2 13199 . . . . . . . . . 10  |-  ( S 
C_  B  ->  S  =  ( Base `  H
) )
361, 35syl 15 . . . . . . . . 9  |-  ( ph  ->  S  =  ( Base `  H ) )
37 fvex 5539 . . . . . . . . . . . . . . 15  |-  ( Base `  H )  e.  _V
3836, 37syl6eqel 2371 . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  _V )
3934, 18ressplusg 13250 . . . . . . . . . . . . . 14  |-  ( S  e.  _V  ->  .+  =  ( +g  `  H ) )
4038, 39syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  .+  =  ( +g  `  H ) )
4140oveqd 5875 . . . . . . . . . . . 12  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  H
) x ) )
4241eqeq1d 2291 . . . . . . . . . . 11  |-  ( ph  ->  ( ( y  .+  x )  =  x  <-> 
( y ( +g  `  H ) x )  =  x ) )
4340oveqd 5875 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  .+  y
)  =  ( x ( +g  `  H
) y ) )
4443eqeq1d 2291 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  .+  y )  =  x  <-> 
( x ( +g  `  H ) y )  =  x ) )
4542, 44anbi12d 691 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( y 
.+  x )  =  x  /\  ( x 
.+  y )  =  x )  <->  ( (
y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) ) )
4636, 45raleqbidv 2748 . . . . . . . . 9  |-  ( ph  ->  ( A. x  e.  S  ( ( y 
.+  x )  =  x  /\  ( x 
.+  y )  =  x )  <->  A. x  e.  ( Base `  H
) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) ) )
4736, 46rabeqbidv 2783 . . . . . . . 8  |-  ( ph  ->  { y  e.  S  |  A. x  e.  S  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) }  =  {
y  e.  ( Base `  H )  |  A. x  e.  ( Base `  H ) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } )
4833, 47eleqtrd 2359 . . . . . . 7  |-  ( ph  ->  .0.  e.  { y  e.  ( Base `  H
)  |  A. x  e.  ( Base `  H
) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } )
4948snssd 3760 . . . . . 6  |-  ( ph  ->  {  .0.  }  C_  { y  e.  ( Base `  H )  |  A. x  e.  ( Base `  H ) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } )
50 ovex 5883 . . . . . . . . . 10  |-  ( Gs  S )  e.  _V
5134, 50eqeltri 2353 . . . . . . . . 9  |-  H  e. 
_V
5251a1i 10 . . . . . . . 8  |-  ( ph  ->  H  e.  _V )
53 eqid 2283 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
54 eqid 2283 . . . . . . . . 9  |-  ( 0g
`  H )  =  ( 0g `  H
)
55 eqid 2283 . . . . . . . . 9  |-  ( +g  `  H )  =  ( +g  `  H )
56 eqid 2283 . . . . . . . . 9  |-  { y  e.  ( Base `  H
)  |  A. x  e.  ( Base `  H
) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) }  =  { y  e.  (
Base `  H )  |  A. x  e.  (
Base `  H )
( ( y ( +g  `  H ) x )  =  x  /\  ( x ( +g  `  H ) y )  =  x ) }
5753, 54, 55, 56gsumvallem1 14448 . . . . . . . 8  |-  ( H  e.  _V  ->  { y  e.  ( Base `  H
)  |  A. x  e.  ( Base `  H
) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) }  C_  { ( 0g `  H
) } )
5852, 57syl 15 . . . . . . 7  |-  ( ph  ->  { y  e.  (
Base `  H )  |  A. x  e.  (
Base `  H )
( ( y ( +g  `  H ) x )  =  x  /\  ( x ( +g  `  H ) y )  =  x ) }  C_  { ( 0g `  H ) } )
5958, 48sseldd 3181 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  { ( 0g `  H ) } )
60 elsni 3664 . . . . . . . . 9  |-  (  .0. 
e.  { ( 0g
`  H ) }  ->  .0.  =  ( 0g `  H ) )
6159, 60syl 15 . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  H ) )
6261sneqd 3653 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  H ) } )
6358, 62sseqtr4d 3215 . . . . . 6  |-  ( ph  ->  { y  e.  (
Base `  H )  |  A. x  e.  (
Base `  H )
( ( y ( +g  `  H ) x )  =  x  /\  ( x ( +g  `  H ) y )  =  x ) }  C_  {  .0.  } )
6449, 63eqssd 3196 . . . . 5  |-  ( ph  ->  {  .0.  }  =  { y  e.  (
Base `  H )  |  A. x  e.  (
Base `  H )
( ( y ( +g  `  H ) x )  =  x  /\  ( x ( +g  `  H ) y )  =  x ) } )
6527, 64eqtr3d 2317 . . . 4  |-  ( ph  ->  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) }  =  {
y  e.  ( Base `  H )  |  A. x  e.  ( Base `  H ) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } )
6665sseq2d 3206 . . 3  |-  ( ph  ->  ( ran  F  C_  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) }  <->  ran  F  C_  { y  e.  ( Base `  H )  |  A. x  e.  ( Base `  H ) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } ) )
6724, 61eqtr3d 2317 . . 3  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
6840seqeq2d 11053 . . . . . . . . . 10  |-  ( ph  ->  seq  m (  .+  ,  F )  =  seq  m ( ( +g  `  H ) ,  F
) )
6968fveq1d 5527 . . . . . . . . 9  |-  ( ph  ->  (  seq  m ( 
.+  ,  F ) `
 n )  =  (  seq  m ( ( +g  `  H
) ,  F ) `
 n ) )
7069eqeq2d 2294 . . . . . . . 8  |-  ( ph  ->  ( z  =  (  seq  m (  .+  ,  F ) `  n
)  <->  z  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) )
7170anbi2d 684 . . . . . . 7  |-  ( ph  ->  ( ( A  =  ( m ... n
)  /\  z  =  (  seq  m (  .+  ,  F ) `  n
) )  <->  ( A  =  ( m ... n )  /\  z  =  (  seq  m ( ( +g  `  H
) ,  F ) `
 n ) ) ) )
7271rexbidv 2564 . . . . . 6  |-  ( ph  ->  ( E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  =  (  seq  m (  .+  ,  F ) `  n
) )  <->  E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) )
7372exbidv 1612 . . . . 5  |-  ( ph  ->  ( E. m E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  z  =  (  seq  m ( 
.+  ,  F ) `
 n ) )  <->  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) )
7473iotabidv 5240 . . . 4  |-  ( ph  ->  ( iota z E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  =  (  seq  m (  .+  ,  F ) `  n
) ) )  =  ( iota z E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) )
7540seqeq2d 11053 . . . . . . . . 9  |-  ( ph  ->  seq  1 (  .+  ,  ( F  o.  f ) )  =  seq  1 ( ( +g  `  H ) ,  ( F  o.  f ) ) )
7675fveq1d 5527 . . . . . . . 8  |-  ( ph  ->  (  seq  1 ( 
.+  ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  =  (  seq  1
( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
7776eqeq2d 2294 . . . . . . 7  |-  ( ph  ->  ( z  =  (  seq  1 (  .+  ,  ( F  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  <-> 
z  =  (  seq  1 ( ( +g  `  H ) ,  ( F  o.  f ) ) `  ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) ) )
7877anbi2d 684 . . . . . 6  |-  ( ph  ->  ( ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  /\  z  =  (  seq  1 (  .+  , 
( F  o.  f
) ) `  ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) )  <->  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  /\  z  =  (  seq  1
( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) ) ) )
7978exbidv 1612 . . . . 5  |-  ( ph  ->  ( E. f ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  /\  z  =  (  seq  1
(  .+  ,  ( F  o.  f )
) `  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )  <->  E. f ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  /\  z  =  (  seq  1
( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) ) ) )
8079iotabidv 5240 . . . 4  |-  ( ph  ->  ( iota z E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  /\  z  =  (  seq  1 (  .+  , 
( F  o.  f
) ) `  ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) ) )  =  ( iota z E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  /\  z  =  (  seq  1 ( ( +g  `  H ) ,  ( F  o.  f ) ) `  ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) ) ) )
8174, 80ifeq12d 3581 . . 3  |-  ( ph  ->  if ( A  e. 
ran  ... ,  ( iota z E. m E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  z  =  (  seq  m ( 
.+  ,  F ) `
 n ) ) ) ,  ( iota z E. f ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  /\  z  =  (  seq  1
(  .+  ,  ( F  o.  f )
) `  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) ) ) )  =  if ( A  e. 
ran  ... ,  ( iota z E. m E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  z  =  (  seq  m ( ( +g  `  H
) ,  F ) `
 n ) ) ) ,  ( iota z E. f ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  {  .0.  }
) )  /\  z  =  (  seq  1
( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) ) ) ) )
8266, 67, 81ifbieq12d 3587 . 2  |-  ( ph  ->  if ( ran  F  C_ 
{ y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) } ,  ( 0g `  G ) ,  if ( A  e.  ran  ... , 
( iota z E. m E. n  e.  ( ZZ>=
`  m ) ( A  =  ( m ... n )  /\  z  =  (  seq  m (  .+  ,  F ) `  n
) ) ) ,  ( iota z E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  /\  z  =  (  seq  1 (  .+  , 
( F  o.  f
) ) `  ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) ) ) ) )  =  if ( ran  F  C_  { y  e.  ( Base `  H
)  |  A. x  e.  ( Base `  H
) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } , 
( 0g `  H
) ,  if ( A  e.  ran  ... ,  ( iota z E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) ,  ( iota z E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  /\  z  =  (  seq  1 ( ( +g  `  H ) ,  ( F  o.  f ) ) `  ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) ) ) ) ) )
8327difeq2d 3294 . . . 4  |-  ( ph  ->  ( _V  \  {  .0.  } )  =  ( _V  \  { y  e.  B  |  A. x  e.  B  (
( y  .+  x
)  =  x  /\  ( x  .+  y )  =  x ) } ) )
8483imaeq2d 5012 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  =  ( `' F "
( _V  \  {
y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) } ) ) )
85 gsumress.a . . 3  |-  ( ph  ->  A  e.  X )
86 gsumress.f . . . 4  |-  ( ph  ->  F : A --> S )
87 fss 5397 . . . 4  |-  ( ( F : A --> S  /\  S  C_  B )  ->  F : A --> B )
8886, 1, 87syl2anc 642 . . 3  |-  ( ph  ->  F : A --> B )
8916, 17, 18, 19, 84, 15, 85, 88gsumval 14452 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  if ( ran  F  C_  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) } ,  ( 0g `  G ) ,  if ( A  e.  ran  ... , 
( iota z E. m E. n  e.  ( ZZ>=
`  m ) ( A  =  ( m ... n )  /\  z  =  (  seq  m (  .+  ,  F ) `  n
) ) ) ,  ( iota z E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  /\  z  =  (  seq  1 (  .+  , 
( F  o.  f
) ) `  ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) ) ) ) ) )
9064difeq2d 3294 . . . 4  |-  ( ph  ->  ( _V  \  {  .0.  } )  =  ( _V  \  { y  e.  ( Base `  H
)  |  A. x  e.  ( Base `  H
) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } ) )
9190imaeq2d 5012 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {  .0.  } ) )  =  ( `' F "
( _V  \  {
y  e.  ( Base `  H )  |  A. x  e.  ( Base `  H ) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } ) ) )
92 feq3 5377 . . . . 5  |-  ( S  =  ( Base `  H
)  ->  ( F : A --> S  <->  F : A
--> ( Base `  H
) ) )
9336, 92syl 15 . . . 4  |-  ( ph  ->  ( F : A --> S 
<->  F : A --> ( Base `  H ) ) )
9486, 93mpbid 201 . . 3  |-  ( ph  ->  F : A --> ( Base `  H ) )
9553, 54, 55, 56, 91, 52, 85, 94gsumval 14452 . 2  |-  ( ph  ->  ( H  gsumg  F )  =  if ( ran  F  C_  { y  e.  ( Base `  H )  |  A. x  e.  ( Base `  H ) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } , 
( 0g `  H
) ,  if ( A  e.  ran  ... ,  ( iota z E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  z  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) ,  ( iota z E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {  .0.  } ) ) ) ) -1-1-onto-> ( `' F "
( _V  \  {  .0.  } ) )  /\  z  =  (  seq  1 ( ( +g  `  H ) ,  ( F  o.  f ) ) `  ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) ) ) ) ) )
9682, 89, 953eqtr4d 2325 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149    C_ wss 3152   ifcif 3565   {csn 3640   `'ccnv 4688   ran crn 4690   "cima 4692    o. ccom 4693   iotacio 5217   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   1c1 8738   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046   #chash 11337   Basecbs 13148   ↾s cress 13149   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401
This theorem is referenced by:  gsumsubm  14455  imasdsf1olem  17937  esumpfinvallem  23442  frlmgsum  27232
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-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-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-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-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-2 9804  df-seq 11047  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405
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