MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumpropd Structured version   Unicode version

Theorem gsumpropd 14768
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 14713 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 6090 . . . . . . . 8  |-  ( ph  ->  ( s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )
43eqeq1d 2443 . . . . . . 7  |-  ( ph  ->  ( ( s ( +g  `  G ) t )  =  t  <-> 
( s ( +g  `  H ) t )  =  t ) )
52oveqd 6090 . . . . . . . 8  |-  ( ph  ->  ( t ( +g  `  G ) s )  =  ( t ( +g  `  H ) s ) )
65eqeq1d 2443 . . . . . . 7  |-  ( ph  ->  ( ( t ( +g  `  G ) s )  =  t  <-> 
( t ( +g  `  H ) s )  =  t ) )
74, 6anbi12d 692 . . . . . 6  |-  ( ph  ->  ( ( ( s ( +g  `  G
) t )  =  t  /\  ( t ( +g  `  G
) s )  =  t )  <->  ( (
s ( +g  `  H
) t )  =  t  /\  ( t ( +g  `  H
) s )  =  t ) ) )
81, 7raleqbidv 2908 . . . . 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 2943 . . . 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 3368 . . 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 2436 . . . 4  |-  ( ph  ->  ( Base `  G
)  =  ( Base `  G ) )
122proplem3 13908 . . . 4  |-  ( (
ph  /\  ( a  e.  ( Base `  G
)  /\  b  e.  ( Base `  G )
) )  ->  (
a ( +g  `  G
) b )  =  ( a ( +g  `  H ) b ) )
1311, 1, 12grpidpropd 14714 . . 3  |-  ( ph  ->  ( 0g `  G
)  =  ( 0g
`  H ) )
142seqeq2d 11322 . . . . . . . . . 10  |-  ( ph  ->  seq  m ( ( +g  `  G ) ,  F )  =  seq  m ( ( +g  `  H ) ,  F ) )
1514fveq1d 5722 . . . . . . . . 9  |-  ( ph  ->  (  seq  m ( ( +g  `  G
) ,  F ) `
 n )  =  (  seq  m ( ( +g  `  H
) ,  F ) `
 n ) )
1615eqeq2d 2446 . . . . . . . 8  |-  ( ph  ->  ( x  =  (  seq  m ( ( +g  `  G ) ,  F ) `  n )  <->  x  =  (  seq  m ( ( +g  `  H ) ,  F ) `  n ) ) )
1716anbi2d 685 . . . . . . 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 2718 . . . . . 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 1636 . . . . 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 5431 . . . 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 3457 . . . . . . . . . . . 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 5195 . . . . . . . . . . 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 5724 . . . . . . . . . 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 6089 . . . . . . . . 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 5658 . . . . . . . . 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 16 . . . . . . . 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 5659 . . . . . . . . 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 16 . . . . . . . 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 245 . . . . . . 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 11322 . . . . . . . . 9  |-  ( ph  ->  seq  1 ( ( +g  `  G ) ,  ( F  o.  f ) )  =  seq  1 ( ( +g  `  H ) ,  ( F  o.  f ) ) )
3130, 23fveq12d 5726 . . . . . . . 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 2446 . . . . . . 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 692 . . . . . 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 1636 . . . . 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 5431 . . . 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 3747 . . 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 3753 . 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 2435 . . 3  |-  ( Base `  G )  =  (
Base `  G )
39 eqid 2435 . . 3  |-  ( 0g
`  G )  =  ( 0g `  G
)
40 eqid 2435 . . 3  |-  ( +g  `  G )  =  ( +g  `  G )
41 eqid 2435 . . 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 2436 . . 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 2436 . . 3  |-  ( ph  ->  dom  F  =  dom  F )
4638, 39, 40, 41, 42, 43, 44, 45gsumvalx 14766 . 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 2435 . . 3  |-  ( Base `  H )  =  (
Base `  H )
48 eqid 2435 . . 3  |-  ( 0g
`  H )  =  ( 0g `  H
)
49 eqid 2435 . . 3  |-  ( +g  `  H )  =  ( +g  `  H )
50 eqid 2435 . . 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 2436 . . 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 14766 . 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 2477 1  |-  ( ph  ->  ( G  gsumg  F )  =  ( H  gsumg  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    \ cdif 3309    C_ wss 3312   ifcif 3731   `'ccnv 4869   dom cdm 4870   ran crn 4871   "cima 4873    o. ccom 4874   iotacio 5408   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073   1c1 8983   ZZ>=cuz 10480   ...cfz 11035    seq cseq 11315   #chash 11610   Basecbs 13461   +g cplusg 13521   0gc0g 13715    gsumg cgsu 13716
This theorem is referenced by:  psropprmul  16624  ply1coe  16676  tsmspropd  18153  frlmgsum  27200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-seq 11316  df-0g 13719  df-gsum 13720
  Copyright terms: Public domain W3C validator