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Theorem gsumress 15487
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 3345 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
4 gsumress.c . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  (
(  .0.  .+  x
)  =  x  /\  ( x  .+  .0.  )  =  x ) )
54ralrimiva 2789 . . . . . . . 8  |-  ( ph  ->  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )
6 oveq1 6087 . . . . . . . . . . . 12  |-  ( y  =  .0.  ->  (
y  .+  x )  =  (  .0.  .+  x
) )
76eqeq1d 2441 . . . . . . . . . . 11  |-  ( y  =  .0.  ->  (
( y  .+  x
)  =  x  <->  (  .0.  .+  x )  =  x ) )
8 oveq2 6088 . . . . . . . . . . . 12  |-  ( y  =  .0.  ->  (
x  .+  y )  =  ( x  .+  .0.  ) )
98eqeq1d 2441 . . . . . . . . . . 11  |-  ( y  =  .0.  ->  (
( x  .+  y
)  =  x  <->  ( x  .+  .0.  )  =  x ) )
107, 9anbi12d 703 . . . . . . . . . 10  |-  ( y  =  .0.  ->  (
( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x )  <->  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
1110ralbidv 2725 . . . . . . . . 9  |-  ( y  =  .0.  ->  ( A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x )  <->  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
1211elrab 3106 . . . . . . . 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 657 . . . . . . 7  |-  ( ph  ->  .0.  e.  { y  e.  B  |  A. x  e.  B  (
( y  .+  x
)  =  x  /\  ( x  .+  y )  =  x ) } )
1413snssd 4006 . . . . . 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 2433 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
18 gsumress.o . . . . . . . . 9  |-  .+  =  ( +g  `  G )
19 eqid 2433 . . . . . . . . 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 15482 . . . . . . . 8  |-  ( G  e.  V  ->  { y  e.  B  |  A. x  e.  B  (
( y  .+  x
)  =  x  /\  ( x  .+  y )  =  x ) } 
C_  { ( 0g
`  G ) } )
2115, 20syl 16 . . . . . . 7  |-  ( ph  ->  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) }  C_  { ( 0g `  G ) } )
2221, 13sseldd 3345 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  { ( 0g `  G ) } )
23 elsni 3890 . . . . . . . . 9  |-  (  .0. 
e.  { ( 0g
`  G ) }  ->  .0.  =  ( 0g `  G ) )
2422, 23syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
2524sneqd 3877 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  G ) } )
2621, 25sseqtr4d 3381 . . . . . 6  |-  ( ph  ->  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) }  C_  {  .0.  } )
2714, 26eqssd 3361 . . . . 5  |-  ( ph  ->  {  .0.  }  =  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) } )
281sselda 3344 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  B )
2928, 4syldan 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  (
(  .0.  .+  x
)  =  x  /\  ( x  .+  .0.  )  =  x ) )
3029ralrimiva 2789 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )
3110ralbidv 2725 . . . . . . . . . 10  |-  ( y  =  .0.  ->  ( A. x  e.  S  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x )  <->  A. x  e.  S  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
3231elrab 3106 . . . . . . . . 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 657 . . . . . . . 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 14212 . . . . . . . . . 10  |-  ( S 
C_  B  ->  S  =  ( Base `  H
) )
361, 35syl 16 . . . . . . . . 9  |-  ( ph  ->  S  =  ( Base `  H ) )
37 fvex 5689 . . . . . . . . . . . . . . 15  |-  ( Base `  H )  e.  _V
3836, 37syl6eqel 2521 . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  _V )
3934, 18ressplusg 14263 . . . . . . . . . . . . . 14  |-  ( S  e.  _V  ->  .+  =  ( +g  `  H ) )
4038, 39syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  .+  =  ( +g  `  H ) )
4140oveqd 6097 . . . . . . . . . . . 12  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  H
) x ) )
4241eqeq1d 2441 . . . . . . . . . . 11  |-  ( ph  ->  ( ( y  .+  x )  =  x  <-> 
( y ( +g  `  H ) x )  =  x ) )
4340oveqd 6097 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  .+  y
)  =  ( x ( +g  `  H
) y ) )
4443eqeq1d 2441 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  .+  y )  =  x  <-> 
( x ( +g  `  H ) y )  =  x ) )
4542, 44anbi12d 703 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( y 
.+  x )  =  x  /\  ( x 
.+  y )  =  x )  <->  ( (
y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) ) )
4636, 45raleqbidv 2921 . . . . . . . . 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 2957 . . . . . . . 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 2509 . . . . . . 7  |-  ( ph  ->  .0.  e.  { y  e.  ( Base `  H
)  |  A. x  e.  ( Base `  H
) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } )
4948snssd 4006 . . . . . 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 6105 . . . . . . . . . 10  |-  ( Gs  S )  e.  _V
5134, 50eqeltri 2503 . . . . . . . . 9  |-  H  e. 
_V
5251a1i 11 . . . . . . . 8  |-  ( ph  ->  H  e.  _V )
53 eqid 2433 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
54 eqid 2433 . . . . . . . . 9  |-  ( 0g
`  H )  =  ( 0g `  H
)
55 eqid 2433 . . . . . . . . 9  |-  ( +g  `  H )  =  ( +g  `  H )
56 eqid 2433 . . . . . . . . 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 15482 . . . . . . . 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 16 . . . . . . 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 3345 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  { ( 0g `  H ) } )
60 elsni 3890 . . . . . . . . 9  |-  (  .0. 
e.  { ( 0g
`  H ) }  ->  .0.  =  ( 0g `  H ) )
6159, 60syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  H ) )
6261sneqd 3877 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  H ) } )
6358, 62sseqtr4d 3381 . . . . . 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 3361 . . . . 5  |-  ( ph  ->  {  .0.  }  =  { y  e.  (
Base `  H )  |  A. x  e.  (
Base `  H )
( ( y ( +g  `  H ) x )  =  x  /\  ( x ( +g  `  H ) y )  =  x ) } )
6527, 64eqtr3d 2467 . . . 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 3372 . . 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 2467 . . 3  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
6840seqeq2d 11797 . . . . . . . . . 10  |-  ( ph  ->  seq m (  .+  ,  F )  =  seq m ( ( +g  `  H ) ,  F
) )
6968fveq1d 5681 . . . . . . . . 9  |-  ( ph  ->  (  seq m ( 
.+  ,  F ) `
 n )  =  (  seq m ( ( +g  `  H
) ,  F ) `
 n ) )
7069eqeq2d 2444 . . . . . . . 8  |-  ( ph  ->  ( z  =  (  seq m (  .+  ,  F ) `  n
)  <->  z  =  (  seq m ( ( +g  `  H ) ,  F ) `  n ) ) )
7170anbi2d 696 . . . . . . 7  |-  ( ph  ->  ( ( A  =  ( m ... n
)  /\  z  =  (  seq m (  .+  ,  F ) `  n
) )  <->  ( A  =  ( m ... n )  /\  z  =  (  seq m
( ( +g  `  H
) ,  F ) `
 n ) ) ) )
7271rexbidv 2726 . . . . . 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 1679 . . . . 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 5390 . . . 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 11797 . . . . . . . . 9  |-  ( ph  ->  seq 1 (  .+  ,  ( F  o.  f ) )  =  seq 1 ( ( +g  `  H ) ,  ( F  o.  f ) ) )
7675fveq1d 5681 . . . . . . . 8  |-  ( ph  ->  (  seq 1 ( 
.+  ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  =  (  seq 1
( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
7776eqeq2d 2444 . . . . . . 7  |-  ( ph  ->  ( z  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  <-> 
z  =  (  seq 1 ( ( +g  `  H ) ,  ( F  o.  f ) ) `  ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) ) )
7877anbi2d 696 . . . . . 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 1679 . . . . 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 5390 . . . 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 3797 . . 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 3804 . 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 3462 . . . 4  |-  ( ph  ->  ( _V  \  {  .0.  } )  =  ( _V  \  { y  e.  B  |  A. x  e.  B  (
( y  .+  x
)  =  x  /\  ( x  .+  y )  =  x ) } ) )
8483imaeq2d 5157 . . 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 5555 . . . 4  |-  ( ( F : A --> S  /\  S  C_  B )  ->  F : A --> B )
8886, 1, 87syl2anc 654 . . 3  |-  ( ph  ->  F : A --> B )
8916, 17, 18, 19, 84, 15, 85, 88gsumval 15485 . 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 3462 . . . 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 5157 . . 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 5532 . . . . 5  |-  ( S  =  ( Base `  H
)  ->  ( F : A --> S  <->  F : A
--> ( Base `  H
) ) )
9336, 92syl 16 . . . 4  |-  ( ph  ->  ( F : A --> S 
<->  F : A --> ( Base `  H ) ) )
9486, 93mpbid 210 . . 3  |-  ( ph  ->  F : A --> ( Base `  H ) )
9553, 54, 55, 56, 91, 52, 85, 94gsumval 15485 . 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 2475 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1362   E.wex 1589    e. wcel 1755   A.wral 2705   E.wrex 2706   {crab 2709   _Vcvv 2962    \ cdif 3313    C_ wss 3316   ifcif 3779   {csn 3865   `'ccnv 4826   ran crn 4828   "cima 4830    o. ccom 4831   iotacio 5367   -->wf 5402   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080   1c1 9271   ZZ>=cuz 10849   ...cfz 11424    seqcseq 11790   #chash 12087   Basecbs 14157   ↾s cress 14158   +g cplusg 14221   0gc0g 14361    gsumg cgsu 14362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-seq 11791  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-0g 14363  df-gsum 14364
This theorem is referenced by:  gsumsubm  15488  regsumsupp  17894  frlmgsumOLD  18037  frlmgsum  18038  imasdsf1olem  19790  regsumfsum  26103  esumpfinvallem  26377
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