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Theorem gsumress 15829
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 3505 . . . . . . . 8  |-  ( ph  ->  .0.  e.  B )
4 gsumress.c . . . . . . . . 9  |-  ( (
ph  /\  x  e.  B )  ->  (
(  .0.  .+  x
)  =  x  /\  ( x  .+  .0.  )  =  x ) )
54ralrimiva 2878 . . . . . . . 8  |-  ( ph  ->  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )
6 oveq1 6291 . . . . . . . . . . . 12  |-  ( y  =  .0.  ->  (
y  .+  x )  =  (  .0.  .+  x
) )
76eqeq1d 2469 . . . . . . . . . . 11  |-  ( y  =  .0.  ->  (
( y  .+  x
)  =  x  <->  (  .0.  .+  x )  =  x ) )
8 oveq2 6292 . . . . . . . . . . . 12  |-  ( y  =  .0.  ->  (
x  .+  y )  =  ( x  .+  .0.  ) )
98eqeq1d 2469 . . . . . . . . . . 11  |-  ( y  =  .0.  ->  (
( x  .+  y
)  =  x  <->  ( x  .+  .0.  )  =  x ) )
107, 9anbi12d 710 . . . . . . . . . 10  |-  ( y  =  .0.  ->  (
( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x )  <->  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
1110ralbidv 2903 . . . . . . . . 9  |-  ( y  =  .0.  ->  ( A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x )  <->  A. x  e.  B  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
1211elrab 3261 . . . . . . . 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 664 . . . . . . 7  |-  ( ph  ->  .0.  e.  { y  e.  B  |  A. x  e.  B  (
( y  .+  x
)  =  x  /\  ( x  .+  y )  =  x ) } )
1413snssd 4172 . . . . . 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 2467 . . . . . . . . 9  |-  ( 0g
`  G )  =  ( 0g `  G
)
18 gsumress.o . . . . . . . . 9  |-  .+  =  ( +g  `  G )
19 eqid 2467 . . . . . . . . 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 15822 . . . . . . . 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 3505 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  { ( 0g `  G ) } )
23 elsni 4052 . . . . . . . . 9  |-  (  .0. 
e.  { ( 0g
`  G ) }  ->  .0.  =  ( 0g `  G ) )
2422, 23syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  G ) )
2524sneqd 4039 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  G ) } )
2621, 25sseqtr4d 3541 . . . . . 6  |-  ( ph  ->  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) }  C_  {  .0.  } )
2714, 26eqssd 3521 . . . . 5  |-  ( ph  ->  {  .0.  }  =  { y  e.  B  |  A. x  e.  B  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x ) } )
281sselda 3504 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  S )  ->  x  e.  B )
2928, 4syldan 470 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  S )  ->  (
(  .0.  .+  x
)  =  x  /\  ( x  .+  .0.  )  =  x ) )
3029ralrimiva 2878 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  S  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) )
3110ralbidv 2903 . . . . . . . . . 10  |-  ( y  =  .0.  ->  ( A. x  e.  S  ( ( y  .+  x )  =  x  /\  ( x  .+  y )  =  x )  <->  A. x  e.  S  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x
) ) )
3231elrab 3261 . . . . . . . . 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 664 . . . . . . . 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 14546 . . . . . . . . . 10  |-  ( S 
C_  B  ->  S  =  ( Base `  H
) )
361, 35syl 16 . . . . . . . . 9  |-  ( ph  ->  S  =  ( Base `  H ) )
37 fvex 5876 . . . . . . . . . . . . . . 15  |-  ( Base `  H )  e.  _V
3836, 37syl6eqel 2563 . . . . . . . . . . . . . 14  |-  ( ph  ->  S  e.  _V )
3934, 18ressplusg 14597 . . . . . . . . . . . . . 14  |-  ( S  e.  _V  ->  .+  =  ( +g  `  H ) )
4038, 39syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  .+  =  ( +g  `  H ) )
4140oveqd 6301 . . . . . . . . . . . 12  |-  ( ph  ->  ( y  .+  x
)  =  ( y ( +g  `  H
) x ) )
4241eqeq1d 2469 . . . . . . . . . . 11  |-  ( ph  ->  ( ( y  .+  x )  =  x  <-> 
( y ( +g  `  H ) x )  =  x ) )
4340oveqd 6301 . . . . . . . . . . . 12  |-  ( ph  ->  ( x  .+  y
)  =  ( x ( +g  `  H
) y ) )
4443eqeq1d 2469 . . . . . . . . . . 11  |-  ( ph  ->  ( ( x  .+  y )  =  x  <-> 
( x ( +g  `  H ) y )  =  x ) )
4542, 44anbi12d 710 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( y 
.+  x )  =  x  /\  ( x 
.+  y )  =  x )  <->  ( (
y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) ) )
4636, 45raleqbidv 3072 . . . . . . . . 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 3108 . . . . . . . 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 2557 . . . . . . 7  |-  ( ph  ->  .0.  e.  { y  e.  ( Base `  H
)  |  A. x  e.  ( Base `  H
) ( ( y ( +g  `  H
) x )  =  x  /\  ( x ( +g  `  H
) y )  =  x ) } )
4948snssd 4172 . . . . . 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 6309 . . . . . . . . . 10  |-  ( Gs  S )  e.  _V
5134, 50eqeltri 2551 . . . . . . . . 9  |-  H  e. 
_V
5251a1i 11 . . . . . . . 8  |-  ( ph  ->  H  e.  _V )
53 eqid 2467 . . . . . . . . 9  |-  ( Base `  H )  =  (
Base `  H )
54 eqid 2467 . . . . . . . . 9  |-  ( 0g
`  H )  =  ( 0g `  H
)
55 eqid 2467 . . . . . . . . 9  |-  ( +g  `  H )  =  ( +g  `  H )
56 eqid 2467 . . . . . . . . 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 15822 . . . . . . . 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 3505 . . . . . . . . 9  |-  ( ph  ->  .0.  e.  { ( 0g `  H ) } )
60 elsni 4052 . . . . . . . . 9  |-  (  .0. 
e.  { ( 0g
`  H ) }  ->  .0.  =  ( 0g `  H ) )
6159, 60syl 16 . . . . . . . 8  |-  ( ph  ->  .0.  =  ( 0g
`  H ) )
6261sneqd 4039 . . . . . . 7  |-  ( ph  ->  {  .0.  }  =  { ( 0g `  H ) } )
6358, 62sseqtr4d 3541 . . . . . 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 3521 . . . . 5  |-  ( ph  ->  {  .0.  }  =  { y  e.  (
Base `  H )  |  A. x  e.  (
Base `  H )
( ( y ( +g  `  H ) x )  =  x  /\  ( x ( +g  `  H ) y )  =  x ) } )
6527, 64eqtr3d 2510 . . . 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 3532 . . 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 2510 . . 3  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
6840seqeq2d 12082 . . . . . . . . . 10  |-  ( ph  ->  seq m (  .+  ,  F )  =  seq m ( ( +g  `  H ) ,  F
) )
6968fveq1d 5868 . . . . . . . . 9  |-  ( ph  ->  (  seq m ( 
.+  ,  F ) `
 n )  =  (  seq m ( ( +g  `  H
) ,  F ) `
 n ) )
7069eqeq2d 2481 . . . . . . . 8  |-  ( ph  ->  ( z  =  (  seq m (  .+  ,  F ) `  n
)  <->  z  =  (  seq m ( ( +g  `  H ) ,  F ) `  n ) ) )
7170anbi2d 703 . . . . . . 7  |-  ( ph  ->  ( ( A  =  ( m ... n
)  /\  z  =  (  seq m (  .+  ,  F ) `  n
) )  <->  ( A  =  ( m ... n )  /\  z  =  (  seq m
( ( +g  `  H
) ,  F ) `
 n ) ) ) )
7271rexbidv 2973 . . . . . 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 1690 . . . . 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 5572 . . . 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 12082 . . . . . . . . 9  |-  ( ph  ->  seq 1 (  .+  ,  ( F  o.  f ) )  =  seq 1 ( ( +g  `  H ) ,  ( F  o.  f ) ) )
7675fveq1d 5868 . . . . . . . 8  |-  ( ph  ->  (  seq 1 ( 
.+  ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  =  (  seq 1
( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) ) )
7776eqeq2d 2481 . . . . . . 7  |-  ( ph  ->  ( z  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  {  .0.  }
) ) ) )  <-> 
z  =  (  seq 1 ( ( +g  `  H ) ,  ( F  o.  f ) ) `  ( # `  ( `' F "
( _V  \  {  .0.  } ) ) ) ) ) )
7877anbi2d 703 . . . . . 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 1690 . . . . 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 5572 . . . 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 3959 . . 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 3966 . 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 3622 . . . 4  |-  ( ph  ->  ( _V  \  {  .0.  } )  =  ( _V  \  { y  e.  B  |  A. x  e.  B  (
( y  .+  x
)  =  x  /\  ( x  .+  y )  =  x ) } ) )
8483imaeq2d 5337 . . 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 5739 . . . 4  |-  ( ( F : A --> S  /\  S  C_  B )  ->  F : A --> B )
8886, 1, 87syl2anc 661 . . 3  |-  ( ph  ->  F : A --> B )
8916, 17, 18, 19, 84, 15, 85, 88gsumval 15825 . 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 3622 . . . 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 5337 . . 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 5715 . . . . 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 15825 . 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 2518 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2814   E.wrex 2815   {crab 2818   _Vcvv 3113    \ cdif 3473    C_ wss 3476   ifcif 3939   {csn 4027   `'ccnv 4998   ran crn 5000   "cima 5002    o. ccom 5003   iotacio 5549   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588  (class class class)co 6284   1c1 9493   ZZ>=cuz 11082   ...cfz 11672    seqcseq 12075   #chash 12373   Basecbs 14490   ↾s cress 14491   +g cplusg 14555   0gc0g 14695    gsumg cgsu 14696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-seq 12076  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-0g 14697  df-gsum 14698
This theorem is referenced by:  gsumsubm  15830  regsumsupp  18453  frlmgsumOLD  18596  frlmgsum  18597  imasdsf1olem  20639  regsumfsum  27463  esumpfinvallem  27748
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