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Theorem gsumpropd 16459
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 16506 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 6313 . . . . . . . 8  |-  ( ph  ->  ( s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )
43eqeq1d 2422 . . . . . . 7  |-  ( ph  ->  ( ( s ( +g  `  G ) t )  =  t  <-> 
( s ( +g  `  H ) t )  =  t ) )
52oveqd 6313 . . . . . . . 8  |-  ( ph  ->  ( t ( +g  `  G ) s )  =  ( t ( +g  `  H ) s ) )
65eqeq1d 2422 . . . . . . 7  |-  ( ph  ->  ( ( t ( +g  `  G ) s )  =  t  <-> 
( t ( +g  `  H ) s )  =  t ) )
74, 6anbi12d 715 . . . . . 6  |-  ( ph  ->  ( ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t )  <->  ( (
s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) ) )
81, 7raleqbidv 3037 . . . . 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 3073 . . . 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 3489 . . 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 2421 . . . 4  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
122oveqdr 6320 . . . 4  |-  ( (
ph  /\  ( a  e.  ( Base `  G
)  /\  b  e.  ( Base `  G )
) )  ->  (
a ( +g  `  G
) b )  =  ( a ( +g  `  H ) b ) )
1311, 1, 12grpidpropd 16448 . . 3  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
142seqeq2d 12206 . . . . . . . . . 10  |-  ( ph  ->  seq m ( ( +g  `  G ) ,  F )  =  seq m ( ( +g  `  H ) ,  F ) )
1514fveq1d 5874 . . . . . . . . 9  |-  ( ph  ->  (  seq m ( ( +g  `  G
) ,  F ) `
 n )  =  (  seq m ( ( +g  `  H
) ,  F ) `
 n ) )
1615eqeq2d 2434 . . . . . . . 8  |-  ( ph  ->  ( x  =  (  seq m ( ( +g  `  G ) ,  F ) `  n )  <->  x  =  (  seq m ( ( +g  `  H ) ,  F ) `  n ) ) )
1716anbi2d 708 . . . . . . 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 2937 . . . . . 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 1758 . . . . 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 5577 . . . 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 3580 . . . . . . . . . . . 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 5179 . . . . . . . . . . 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 5876 . . . . . . . . . 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 6312 . . . . . . . . 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 5814 . . . . . . . . 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 17 . . . . . . . 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 5815 . . . . . . . . 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 17 . . . . . . . 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 256 . . . . . . 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 12206 . . . . . . . . 9  |-  ( ph  ->  seq 1 ( ( +g  `  G ) ,  ( F  o.  f ) )  =  seq 1 ( ( +g  `  H ) ,  ( F  o.  f ) ) )
3130, 23fveq12d 5878 . . . . . . . 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 2434 . . . . . . 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 715 . . . . . 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 1758 . . . . 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 5577 . . . 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 3926 . . 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 3933 . 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 2420 . . 3  |-  ( Base `  G )  =  (
Base `  G )
39 eqid 2420 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
40 eqid 2420 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
41 eqid 2420 . . 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 2421 . . 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 2421 . . 3  |-  ( ph  ->  dom  F  =  dom  F )
4638, 39, 40, 41, 42, 43, 44, 45gsumvalx 16457 . 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 2420 . . 3  |-  ( Base `  H )  =  (
Base `  H )
48 eqid 2420 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
49 eqid 2420 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
50 eqid 2420 . . 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 2421 . . 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 16457 . 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 2471 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867   A.wral 2773   E.wrex 2774   {crab 2777   _Vcvv 3078    \ cdif 3430    C_ wss 3433   ifcif 3906   `'ccnv 4844   dom cdm 4845   ran crn 4846   "cima 4848    o. ccom 4849   iotacio 5554   -1-1-onto->wf1o 5591   ` cfv 5592  (class class class)co 6296   1c1 9529   ZZ>=cuz 11148   ...cfz 11771    seqcseq 12199   #chash 12501   Basecbs 15073   +g cplusg 15142   0gc0g 15290    gsumg cgsu 15291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-seq 12200  df-0g 15292  df-gsum 15293
This theorem is referenced by:  psropprmul  18759  ply1coe  18817  ply1coeOLD  18818  frlmgsum  19254  matgsum  19386  tsmspropd  21070
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