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Theorem gsumpropd 14469
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 14414 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 5891 . . . . . . . 8  |-  ( ph  ->  ( s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )
43eqeq1d 2304 . . . . . . 7  |-  ( ph  ->  ( ( s ( +g  `  G ) t )  =  t  <-> 
( s ( +g  `  H ) t )  =  t ) )
52oveqd 5891 . . . . . . . 8  |-  ( ph  ->  ( t ( +g  `  G ) s )  =  ( t ( +g  `  H ) s ) )
65eqeq1d 2304 . . . . . . 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 2761 . . . . 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 2796 . . . 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 3219 . . 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 2297 . . . 4  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
122proplem3 13609 . . . 4  |-  ( (
ph  /\  ( a  e.  ( Base `  G
)  /\  b  e.  ( Base `  G )
) )  ->  (
a ( +g  `  G
) b )  =  ( a ( +g  `  H ) b ) )
1311, 1, 12grpidpropd 14415 . . 3  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
142seqeq2d 11069 . . . . . . . . . 10  |-  ( ph  ->  seq  m ( ( +g  `  G ) ,  F )  =  seq  m ( ( +g  `  H ) ,  F ) )
1514fveq1d 5543 . . . . . . . . 9  |-  ( ph  ->  (  seq  m ( ( +g  `  G
) ,  F ) `
 n )  =  (  seq  m ( ( +g  `  H
) ,  F ) `
 n ) )
1615eqeq2d 2307 . . . . . . . 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 2577 . . . . . 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 1616 . . . . 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 5256 . . . 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 3307 . . . . . . . . . . . 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 5028 . . . . . . . . . . 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 5545 . . . . . . . . . 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 5890 . . . . . . . . 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 5480 . . . . . . . . 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 5481 . . . . . . . . 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 11069 . . . . . . . . 9  |-  ( ph  ->  seq  1 ( ( +g  `  G ) ,  ( F  o.  f ) )  =  seq  1 ( ( +g  `  H ) ,  ( F  o.  f ) ) )
3130, 23fveq12d 5547 . . . . . . . 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 2307 . . . . . . 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 1616 . . . . 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 5256 . . . 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 3594 . . 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 3600 . 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 2296 . . 3  |-  ( Base `  G )  =  (
Base `  G )
39 eqid 2296 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
40 eqid 2296 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
41 eqid 2296 . . 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 2297 . . 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 2297 . . 3  |-  ( ph  ->  dom  F  =  dom  F )
4638, 39, 40, 41, 42, 43, 44, 45gsumvalx 14467 . 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 2296 . . 3  |-  ( Base `  H )  =  (
Base `  H )
48 eqid 2296 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
49 eqid 2296 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
50 eqid 2296 . . 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 2297 . . 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 14467 . 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 2338 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162    C_ wss 3165   ifcif 3578   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708    o. ccom 4709   iotacio 5233   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874   1c1 8754   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062   #chash 11353   Basecbs 13164   +g cplusg 13224   0gc0g 13416    gsumg cgsu 13417
This theorem is referenced by:  psropprmul  16332  ply1coe  16384  tsmspropd  17830  frlmgsum  27335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063  df-0g 13420  df-gsum 13421
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