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Theorem gsumpropd 14453
Description: The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 14398 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
gsumpropd.f  |-  ( ph  ->  F  e.  V )
gsumpropd.g  |-  ( ph  ->  G  e.  W )
gsumpropd.h  |-  ( ph  ->  H  e.  X )
gsumpropd.b  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
gsumpropd.p  |-  ( ph  ->  ( +g  `  G
)  =  ( +g  `  H ) )
Assertion
Ref Expression
gsumpropd  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )

Proof of Theorem gsumpropd
Dummy variables  a 
b  f  m  n  s  t  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumpropd.b . . . . 5  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  H ) )
2 gsumpropd.p . . . . . . . . 9  |-  ( ph  ->  ( +g  `  G
)  =  ( +g  `  H ) )
32oveqd 5875 . . . . . . . 8  |-  ( ph  ->  ( s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )
43eqeq1d 2291 . . . . . . 7  |-  ( ph  ->  ( ( s ( +g  `  G ) t )  =  t  <-> 
( s ( +g  `  H ) t )  =  t ) )
52oveqd 5875 . . . . . . . 8  |-  ( ph  ->  ( t ( +g  `  G ) s )  =  ( t ( +g  `  H ) s ) )
65eqeq1d 2291 . . . . . . 7  |-  ( ph  ->  ( ( t ( +g  `  G ) s )  =  t  <-> 
( t ( +g  `  H ) s )  =  t ) )
74, 6anbi12d 691 . . . . . 6  |-  ( ph  ->  ( ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t )  <->  ( (
s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) ) )
81, 7raleqbidv 2748 . . . . 5  |-  ( ph  ->  ( A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t )  <->  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) ) )
91, 8rabeqbidv 2783 . . . 4  |-  ( ph  ->  { s  e.  (
Base `  G )  |  A. t  e.  (
Base `  G )
( ( s ( +g  `  G ) t )  =  t  /\  ( t ( +g  `  G ) s )  =  t ) }  =  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } )
109sseq2d 3206 . . 3  |-  ( ph  ->  ( ran  F  C_  { s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) }  <->  ran  F  C_  { s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )
11 eqidd 2284 . . . 4  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
122proplem3 13593 . . . 4  |-  ( (
ph  /\  ( a  e.  ( Base `  G
)  /\  b  e.  ( Base `  G )
) )  ->  (
a ( +g  `  G
) b )  =  ( a ( +g  `  H ) b ) )
1311, 1, 12grpidpropd 14399 . . 3  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
142seqeq2d 11053 . . . . . . . . . 10  |-  ( ph  ->  seq  m ( ( +g  `  G ) ,  F )  =  seq  m ( ( +g  `  H ) ,  F ) )
1514fveq1d 5527 . . . . . . . . 9  |-  ( ph  ->  (  seq  m ( ( +g  `  G
) ,  F ) `
 n )  =  (  seq  m ( ( +g  `  H
) ,  F ) `
 n ) )
1615eqeq2d 2294 . . . . . . . 8  |-  ( ph  ->  ( x  =  (  seq  m ( ( +g  `  G ) ,  F ) `  n )  <->  x  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) )
1716anbi2d 684 . . . . . . 7  |-  ( ph  ->  ( ( dom  F  =  ( m ... n )  /\  x  =  (  seq  m ( ( +g  `  G
) ,  F ) `
 n ) )  <-> 
( dom  F  =  ( m ... n
)  /\  x  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) )
1817rexbidv 2564 . . . . . 6  |-  ( ph  ->  ( E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq  m ( ( +g  `  G ) ,  F ) `  n ) )  <->  E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) )
1918exbidv 1612 . . . . 5  |-  ( ph  ->  ( E. m E. n  e.  ( ZZ>= `  m ) ( dom 
F  =  ( m ... n )  /\  x  =  (  seq  m ( ( +g  `  G ) ,  F
) `  n )
)  <->  E. m E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) )
2019iotabidv 5240 . . . 4  |-  ( ph  ->  ( iota x E. m E. n  e.  (
ZZ>= `  m ) ( dom  F  =  ( m ... n )  /\  x  =  (  seq  m ( ( +g  `  G ) ,  F ) `  n ) ) )  =  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) )
219difeq2d 3294 . . . . . . . . . . . 12  |-  ( ph  ->  ( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } )  =  ( _V  \  { s  e.  (
Base `  H )  |  A. t  e.  (
Base `  H )
( ( s ( +g  `  H ) t )  =  t  /\  ( t ( +g  `  H ) s )  =  t ) } ) )
2221imaeq2d 5012 . . . . . . . . . . 11  |-  ( ph  ->  ( `' F "
( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  =  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) )
2322fveq2d 5529 . . . . . . . . . 10  |-  ( ph  ->  ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) )  =  (
# `  ( `' F " ( _V  \  { s  e.  (
Base `  H )  |  A. t  e.  (
Base `  H )
( ( s ( +g  `  H ) t )  =  t  /\  ( t ( +g  `  H ) s )  =  t ) } ) ) ) )
2423oveq2d 5874 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... ( # `
 ( `' F " ( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) )  =  ( 1 ... ( # `
 ( `' F " ( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) )
25 f1oeq2 5464 . . . . . . . . 9  |-  ( ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) )  =  ( 1 ... ( # `
 ( `' F " ( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) )  -> 
( f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  <->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) )
2624, 25syl 15 . . . . . . . 8  |-  ( ph  ->  ( f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  <->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) )
27 f1oeq3 5465 . . . . . . . . 9  |-  ( ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  =  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )  ->  ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  <->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) )
2822, 27syl 15 . . . . . . . 8  |-  ( ph  ->  ( f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  <->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) )
2926, 28bitrd 244 . . . . . . 7  |-  ( ph  ->  ( f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  <->  f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) )
302seqeq2d 11053 . . . . . . . . 9  |-  ( ph  ->  seq  1 ( ( +g  `  G ) ,  ( F  o.  f ) )  =  seq  1 ( ( +g  `  H ) ,  ( F  o.  f ) ) )
3130, 23fveq12d 5531 . . . . . . . 8  |-  ( ph  ->  (  seq  1 ( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) )  =  (  seq  1 ( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) )
3231eqeq2d 2294 . . . . . . 7  |-  ( ph  ->  ( x  =  (  seq  1 ( ( +g  `  G ) ,  ( F  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) )  <->  x  =  (  seq  1 ( ( +g  `  H ) ,  ( F  o.  f ) ) `  ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) ) )
3329, 32anbi12d 691 . . . . . 6  |-  ( ph  ->  ( ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) )  <-> 
( f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) ) ) )
3433exbidv 1612 . . . . 5  |-  ( ph  ->  ( E. f ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) )  <->  E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) ) ) )
3534iotabidv 5240 . . . 4  |-  ( ph  ->  ( iota x E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) ) )  =  ( iota
x E. f ( f : ( 1 ... ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) ) ) )
3620, 35ifeq12d 3581 . . 3  |-  ( ph  ->  if ( dom  F  e.  ran  ... ,  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( dom 
F  =  ( m ... n )  /\  x  =  (  seq  m ( ( +g  `  G ) ,  F
) `  n )
) ) ,  ( iota x E. f
( f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) ) ) )  =  if ( dom  F  e. 
ran  ... ,  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( dom 
F  =  ( m ... n )  /\  x  =  (  seq  m ( ( +g  `  H ) ,  F
) `  n )
) ) ,  ( iota x E. f
( f : ( 1 ... ( # `  ( `' F "
( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) ) ) ) )
3710, 13, 36ifbieq12d 3587 . 2  |-  ( ph  ->  if ( ran  F  C_ 
{ s  e.  (
Base `  G )  |  A. t  e.  (
Base `  G )
( ( s ( +g  `  G ) t )  =  t  /\  ( t ( +g  `  G ) s )  =  t ) } ,  ( 0g `  G ) ,  if ( dom 
F  e.  ran  ... ,  ( iota x E. m E. n  e.  (
ZZ>= `  m ) ( dom  F  =  ( m ... n )  /\  x  =  (  seq  m ( ( +g  `  G ) ,  F ) `  n ) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) ) ) ) )  =  if ( ran  F  C_ 
{ s  e.  (
Base `  H )  |  A. t  e.  (
Base `  H )
( ( s ( +g  `  H ) t )  =  t  /\  ( t ( +g  `  H ) s )  =  t ) } ,  ( 0g `  H ) ,  if ( dom 
F  e.  ran  ... ,  ( iota x E. m E. n  e.  (
ZZ>= `  m ) ( dom  F  =  ( m ... n )  /\  x  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) ) ) ) ) )
38 eqid 2283 . . 3  |-  ( Base `  G )  =  (
Base `  G )
39 eqid 2283 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
40 eqid 2283 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
41 eqid 2283 . . 3  |-  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) }  =  { s  e.  (
Base `  G )  |  A. t  e.  (
Base `  G )
( ( s ( +g  `  G ) t )  =  t  /\  ( t ( +g  `  G ) s )  =  t ) }
42 eqidd 2284 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  =  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) )
43 gsumpropd.g . . 3  |-  ( ph  ->  G  e.  W )
44 gsumpropd.f . . 3  |-  ( ph  ->  F  e.  V )
45 eqidd 2284 . . 3  |-  ( ph  ->  dom  F  =  dom  F )
4638, 39, 40, 41, 42, 43, 44, 45gsumvalx 14451 . 2  |-  ( ph  ->  ( G  gsumg  F )  =  if ( ran  F  C_  { s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } , 
( 0g `  G
) ,  if ( dom  F  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq  m ( ( +g  `  G ) ,  F ) `  n ) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {
s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  G
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  G
)  |  A. t  e.  ( Base `  G
) ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t ) } ) ) ) ) ) ) ) ) )
47 eqid 2283 . . 3  |-  ( Base `  H )  =  (
Base `  H )
48 eqid 2283 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
49 eqid 2283 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
50 eqid 2283 . . 3  |-  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) }  =  { s  e.  (
Base `  H )  |  A. t  e.  (
Base `  H )
( ( s ( +g  `  H ) t )  =  t  /\  ( t ( +g  `  H ) s )  =  t ) }
51 eqidd 2284 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )  =  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) )
52 gsumpropd.h . . 3  |-  ( ph  ->  H  e.  X )
5347, 48, 49, 50, 51, 52, 44, 45gsumvalx 14451 . 2  |-  ( ph  ->  ( H  gsumg  F )  =  if ( ran  F  C_  { s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } , 
( 0g `  H
) ,  if ( dom  F  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( dom  F  =  ( m ... n
)  /\  x  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 ( `' F " ( _V  \  {
s  e.  ( Base `  H )  |  A. t  e.  ( Base `  H ) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) -1-1-onto-> ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) )  /\  x  =  (  seq  1 ( ( +g  `  H
) ,  ( F  o.  f ) ) `
 ( # `  ( `' F " ( _V 
\  { s  e.  ( Base `  H
)  |  A. t  e.  ( Base `  H
) ( ( s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) } ) ) ) ) ) ) ) ) )
5437, 46, 533eqtr4d 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   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692    o. ccom 4693   iotacio 5217   -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   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401
This theorem is referenced by:  psropprmul  16316  ply1coe  16368  tsmspropd  17814  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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-recs 6388  df-rdg 6423  df-seq 11047  df-0g 13404  df-gsum 13405
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