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Theorem gsumvalx 15502
Description: Expand out the substitutions in df-gsum 14381. (Contributed by Mario Carneiro, 18-Sep-2015.)
Hypotheses
Ref Expression
gsumval.b  |-  B  =  ( Base `  G
)
gsumval.z  |-  .0.  =  ( 0g `  G )
gsumval.p  |-  .+  =  ( +g  `  G )
gsumval.o  |-  O  =  { s  e.  B  |  A. t  e.  B  ( ( s  .+  t )  =  t  /\  ( t  .+  s )  =  t ) }
gsumval.w  |-  ( ph  ->  W  =  ( `' F " ( _V 
\  O ) ) )
gsumval.g  |-  ( ph  ->  G  e.  V )
gsumvalx.f  |-  ( ph  ->  F  e.  X )
gsumvalx.a  |-  ( ph  ->  dom  F  =  A )
Assertion
Ref Expression
gsumvalx  |-  ( ph  ->  ( G  gsumg  F )  =  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) ) )
Distinct variable groups:    t, s, x, B    f, m, n, x, ph    f, F, m, n, x    f, G, m, n, x    .+ , s,
t, x    f, O, m, n, x
Allowed substitution hints:    ph( t, s)    A( x, t, f, m, n, s)    B( f, m, n)    .+ ( f, m, n)    F( t, s)    G( t, s)    O( t, s)    V( x, t, f, m, n, s)    W( x, t, f, m, n, s)    X( x, t, f, m, n, s)    .0. ( x, t, f, m, n, s)

Proof of Theorem gsumvalx
Dummy variables  g 
o  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-gsum 14381 . . 3  |-  gsumg  =  ( w  e. 
_V ,  g  e. 
_V  |->  [_ { x  e.  ( Base `  w
)  |  A. y  e.  ( Base `  w
) ( ( x ( +g  `  w
) y )  =  y  /\  ( y ( +g  `  w
) x )  =  y ) }  / 
o ]_ if ( ran  g  C_  o , 
( 0g `  w
) ,  if ( dom  g  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( dom  g  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) ) ) ,  ( iota x E. f [. ( `' g " ( _V 
\  o ) )  /  y ]. (
f : ( 1 ... ( # `  y
) ) -1-1-onto-> y  /\  x  =  (  seq 1
( ( +g  `  w
) ,  ( g  o.  f ) ) `
 ( # `  y
) ) ) ) ) ) )
21a1i 11 . 2  |-  ( ph  -> 
gsumg  =  ( w  e. 
_V ,  g  e. 
_V  |->  [_ { x  e.  ( Base `  w
)  |  A. y  e.  ( Base `  w
) ( ( x ( +g  `  w
) y )  =  y  /\  ( y ( +g  `  w
) x )  =  y ) }  / 
o ]_ if ( ran  g  C_  o , 
( 0g `  w
) ,  if ( dom  g  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( dom  g  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) ) ) ,  ( iota x E. f [. ( `' g " ( _V 
\  o ) )  /  y ]. (
f : ( 1 ... ( # `  y
) ) -1-1-onto-> y  /\  x  =  (  seq 1
( ( +g  `  w
) ,  ( g  o.  f ) ) `
 ( # `  y
) ) ) ) ) ) ) )
3 simprl 755 . . . . . . . 8  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  ->  w  =  G )
43fveq2d 5695 . . . . . . 7  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( Base `  w )  =  ( Base `  G
) )
5 gsumval.b . . . . . . 7  |-  B  =  ( Base `  G
)
64, 5syl6eqr 2493 . . . . . 6  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( Base `  w )  =  B )
73fveq2d 5695 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( +g  `  w )  =  ( +g  `  G
) )
8 gsumval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2493 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( +g  `  w )  =  .+  )
109oveqd 6108 . . . . . . . . 9  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( x ( +g  `  w ) y )  =  ( x  .+  y ) )
1110eqeq1d 2451 . . . . . . . 8  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( ( x ( +g  `  w ) y )  =  y  <-> 
( x  .+  y
)  =  y ) )
129oveqd 6108 . . . . . . . . 9  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( y ( +g  `  w ) x )  =  ( y  .+  x ) )
1312eqeq1d 2451 . . . . . . . 8  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( ( y ( +g  `  w ) x )  =  y  <-> 
( y  .+  x
)  =  y ) )
1411, 13anbi12d 710 . . . . . . 7  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( ( ( x ( +g  `  w
) y )  =  y  /\  ( y ( +g  `  w
) x )  =  y )  <->  ( (
x  .+  y )  =  y  /\  (
y  .+  x )  =  y ) ) )
156, 14raleqbidv 2931 . . . . . 6  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( A. y  e.  ( Base `  w
) ( ( x ( +g  `  w
) y )  =  y  /\  ( y ( +g  `  w
) x )  =  y )  <->  A. y  e.  B  ( (
x  .+  y )  =  y  /\  (
y  .+  x )  =  y ) ) )
166, 15rabeqbidv 2967 . . . . 5  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  ->  { x  e.  ( Base `  w )  | 
A. y  e.  (
Base `  w )
( ( x ( +g  `  w ) y )  =  y  /\  ( y ( +g  `  w ) x )  =  y ) }  =  {
x  e.  B  |  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) } )
17 gsumval.o . . . . . 6  |-  O  =  { s  e.  B  |  A. t  e.  B  ( ( s  .+  t )  =  t  /\  ( t  .+  s )  =  t ) }
18 oveq2 6099 . . . . . . . . . . 11  |-  ( t  =  y  ->  (
s  .+  t )  =  ( s  .+  y ) )
19 id 22 . . . . . . . . . . 11  |-  ( t  =  y  ->  t  =  y )
2018, 19eqeq12d 2457 . . . . . . . . . 10  |-  ( t  =  y  ->  (
( s  .+  t
)  =  t  <->  ( s  .+  y )  =  y ) )
21 oveq1 6098 . . . . . . . . . . 11  |-  ( t  =  y  ->  (
t  .+  s )  =  ( y  .+  s ) )
2221, 19eqeq12d 2457 . . . . . . . . . 10  |-  ( t  =  y  ->  (
( t  .+  s
)  =  t  <->  ( y  .+  s )  =  y ) )
2320, 22anbi12d 710 . . . . . . . . 9  |-  ( t  =  y  ->  (
( ( s  .+  t )  =  t  /\  ( t  .+  s )  =  t )  <->  ( ( s 
.+  y )  =  y  /\  ( y 
.+  s )  =  y ) ) )
2423cbvralv 2947 . . . . . . . 8  |-  ( A. t  e.  B  (
( s  .+  t
)  =  t  /\  ( t  .+  s
)  =  t )  <->  A. y  e.  B  ( ( s  .+  y )  =  y  /\  ( y  .+  s )  =  y ) )
25 oveq1 6098 . . . . . . . . . . 11  |-  ( s  =  x  ->  (
s  .+  y )  =  ( x  .+  y ) )
2625eqeq1d 2451 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( s  .+  y
)  =  y  <->  ( x  .+  y )  =  y ) )
27 oveq2 6099 . . . . . . . . . . 11  |-  ( s  =  x  ->  (
y  .+  s )  =  ( y  .+  x ) )
2827eqeq1d 2451 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( y  .+  s
)  =  y  <->  ( y  .+  x )  =  y ) )
2926, 28anbi12d 710 . . . . . . . . 9  |-  ( s  =  x  ->  (
( ( s  .+  y )  =  y  /\  ( y  .+  s )  =  y )  <->  ( ( x 
.+  y )  =  y  /\  ( y 
.+  x )  =  y ) ) )
3029ralbidv 2735 . . . . . . . 8  |-  ( s  =  x  ->  ( A. y  e.  B  ( ( s  .+  y )  =  y  /\  ( y  .+  s )  =  y )  <->  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) ) )
3124, 30syl5bb 257 . . . . . . 7  |-  ( s  =  x  ->  ( A. t  e.  B  ( ( s  .+  t )  =  t  /\  ( t  .+  s )  =  t )  <->  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) ) )
3231cbvrabv 2971 . . . . . 6  |-  { s  e.  B  |  A. t  e.  B  (
( s  .+  t
)  =  t  /\  ( t  .+  s
)  =  t ) }  =  { x  e.  B  |  A. y  e.  B  (
( x  .+  y
)  =  y  /\  ( y  .+  x
)  =  y ) }
3317, 32eqtri 2463 . . . . 5  |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) }
3416, 33syl6eqr 2493 . . . 4  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  ->  { x  e.  ( Base `  w )  | 
A. y  e.  (
Base `  w )
( ( x ( +g  `  w ) y )  =  y  /\  ( y ( +g  `  w ) x )  =  y ) }  =  O )
3534csbeq1d 3295 . . 3  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  ->  [_ { x  e.  (
Base `  w )  |  A. y  e.  (
Base `  w )
( ( x ( +g  `  w ) y )  =  y  /\  ( y ( +g  `  w ) x )  =  y ) }  /  o ]_ if ( ran  g  C_  o ,  ( 0g
`  w ) ,  if ( dom  g  e.  ran  ... ,  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( dom  g  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) ) ) ,  ( iota x E. f [. ( `' g "
( _V  \  o
) )  /  y ]. ( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) ) ) ) )  = 
[_ O  /  o ]_ if ( ran  g  C_  o ,  ( 0g
`  w ) ,  if ( dom  g  e.  ran  ... ,  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( dom  g  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) ) ) ,  ( iota x E. f [. ( `' g "
( _V  \  o
) )  /  y ]. ( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) ) ) ) ) )
36 fvex 5701 . . . . . . . 8  |-  ( Base `  G )  e.  _V
375, 36eqeltri 2513 . . . . . . 7  |-  B  e. 
_V
3837rabex 4443 . . . . . 6  |-  { x  e.  B  |  A. y  e.  B  (
( x  .+  y
)  =  y  /\  ( y  .+  x
)  =  y ) }  e.  _V
3933, 38eqeltri 2513 . . . . 5  |-  O  e. 
_V
4039a1i 11 . . . 4  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  ->  O  e.  _V )
41 simplrr 760 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  g  =  F )
4241rneqd 5067 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ran  g  =  ran  F )
43 simpr 461 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  o  =  O )
4442, 43sseq12d 3385 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( ran  g  C_  o  <->  ran  F  C_  O ) )
453adantr 465 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  w  =  G )
4645fveq2d 5695 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( 0g `  w )  =  ( 0g `  G
) )
47 gsumval.z . . . . . 6  |-  .0.  =  ( 0g `  G )
4846, 47syl6eqr 2493 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( 0g `  w )  =  .0.  )
4941dmeqd 5042 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  dom  g  =  dom  F )
50 gsumvalx.a . . . . . . . . 9  |-  ( ph  ->  dom  F  =  A )
5150ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  dom  F  =  A )
5249, 51eqtrd 2475 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  dom  g  =  A )
5352eleq1d 2509 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( dom  g  e.  ran  ...  <->  A  e.  ran  ... )
)
5452eqeq1d 2451 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( dom  g  =  (
m ... n )  <->  A  =  ( m ... n
) ) )
559adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( +g  `  w )  = 
.+  )
5655seqeq2d 11813 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq m ( ( +g  `  w ) ,  g )  =  seq m
(  .+  ,  g
) )
5741seqeq3d 11814 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq m (  .+  , 
g )  =  seq m (  .+  ,  F ) )
5856, 57eqtrd 2475 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq m ( ( +g  `  w ) ,  g )  =  seq m
(  .+  ,  F
) )
5958fveq1d 5693 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  (  seq m ( ( +g  `  w ) ,  g ) `  n )  =  (  seq m
(  .+  ,  F
) `  n )
)
6059eqeq2d 2454 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  (
x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n )  <-> 
x  =  (  seq m (  .+  ,  F ) `  n
) ) )
6154, 60anbi12d 710 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  (
( dom  g  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) )  <->  ( A  =  ( m ... n )  /\  x  =  (  seq m
(  .+  ,  F
) `  n )
) ) )
6261rexbidv 2736 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( E. n  e.  ( ZZ>=
`  m ) ( dom  g  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) )  <->  E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) )
6362exbidv 1680 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( E. m E. n  e.  ( ZZ>= `  m )
( dom  g  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) )  <->  E. m E. n  e.  ( ZZ>=
`  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) )
6463iotabidv 5402 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( iota x E. m E. n  e.  ( ZZ>= `  m ) ( dom  g  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) ) )  =  ( iota x E. m E. n  e.  ( ZZ>=
`  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) )
6543difeq2d 3474 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( _V  \  o )  =  ( _V  \  O
) )
6665imaeq2d 5169 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( `' F " ( _V 
\  o ) )  =  ( `' F " ( _V  \  O
) ) )
6741cnveqd 5015 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  `' g  =  `' F
)
6867imaeq1d 5168 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( `' g " ( _V  \  o ) )  =  ( `' F " ( _V  \  o
) ) )
69 gsumval.w . . . . . . . . . . . 12  |-  ( ph  ->  W  =  ( `' F " ( _V 
\  O ) ) )
7069ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  W  =  ( `' F " ( _V  \  O
) ) )
7166, 68, 703eqtr4d 2485 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( `' g " ( _V  \  o ) )  =  W )
72 dfsbcq 3188 . . . . . . . . . 10  |-  ( ( `' g " ( _V  \  o ) )  =  W  ->  ( [. ( `' g "
( _V  \  o
) )  /  y ]. ( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) )  <->  [. W  /  y ]. ( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) ) ) )
7371, 72syl 16 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( [. ( `' g "
( _V  \  o
) )  /  y ]. ( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) )  <->  [. W  /  y ]. ( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) ) ) )
74 gsumvalx.f . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  X )
75 cnvexg 6524 . . . . . . . . . . . . 13  |-  ( F  e.  X  ->  `' F  e.  _V )
76 imaexg 6515 . . . . . . . . . . . . 13  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  O ) )  e.  _V )
7774, 75, 763syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F "
( _V  \  O
) )  e.  _V )
7869, 77eqeltrd 2517 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  _V )
7978ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  W  e.  _V )
80 fveq2 5691 . . . . . . . . . . . . . . 15  |-  ( y  =  W  ->  ( # `
 y )  =  ( # `  W
) )
8180adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  ( # `
 y )  =  ( # `  W
) )
8281oveq2d 6107 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (
1 ... ( # `  y
) )  =  ( 1 ... ( # `  W ) ) )
83 f1oeq2 5633 . . . . . . . . . . . . 13  |-  ( ( 1 ... ( # `  y ) )  =  ( 1 ... ( # `
 W ) )  ->  ( f : ( 1 ... ( # `
 y ) ) -1-1-onto-> y  <-> 
f : ( 1 ... ( # `  W
) ) -1-1-onto-> y ) )
8482, 83syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (
f : ( 1 ... ( # `  y
) ) -1-1-onto-> y  <->  f : ( 1 ... ( # `  W ) ) -1-1-onto-> y ) )
85 f1oeq3 5634 . . . . . . . . . . . . 13  |-  ( y  =  W  ->  (
f : ( 1 ... ( # `  W
) ) -1-1-onto-> y  <->  f : ( 1 ... ( # `  W ) ) -1-1-onto-> W ) )
8685adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (
f : ( 1 ... ( # `  W
) ) -1-1-onto-> y  <->  f : ( 1 ... ( # `  W ) ) -1-1-onto-> W ) )
8784, 86bitrd 253 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (
f : ( 1 ... ( # `  y
) ) -1-1-onto-> y  <->  f : ( 1 ... ( # `  W ) ) -1-1-onto-> W ) )
8855seqeq2d 11813 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) )  =  seq 1
(  .+  ,  (
g  o.  f ) ) )
8941coeq1d 5001 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  (
g  o.  f )  =  ( F  o.  f ) )
9089seqeq3d 11814 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq 1 (  .+  , 
( g  o.  f
) )  =  seq 1 (  .+  , 
( F  o.  f
) ) )
9188, 90eqtrd 2475 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) )  =  seq 1
(  .+  ,  ( F  o.  f )
) )
9291adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) )  =  seq 1
(  .+  ,  ( F  o.  f )
) )
9392, 81fveq12d 5697 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) )  =  (  seq 1 ( 
.+  ,  ( F  o.  f ) ) `
 ( # `  W
) ) )
9493eqeq2d 2454 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (
x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) )  <->  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) )
9587, 94anbi12d 710 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (
( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) )  <-> 
( f : ( 1 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 W ) ) ) ) )
9679, 95sbcied 3223 . . . . . . . . 9  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( [. W  /  y ]. ( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) )  <-> 
( f : ( 1 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 W ) ) ) ) )
9773, 96bitrd 253 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( [. ( `' g "
( _V  \  o
) )  /  y ]. ( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) )  <-> 
( f : ( 1 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 W ) ) ) ) )
9897exbidv 1680 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( E. f [. ( `' g " ( _V 
\  o ) )  /  y ]. (
f : ( 1 ... ( # `  y
) ) -1-1-onto-> y  /\  x  =  (  seq 1
( ( +g  `  w
) ,  ( g  o.  f ) ) `
 ( # `  y
) ) )  <->  E. f
( f : ( 1 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  , 
( F  o.  f
) ) `  ( # `
 W ) ) ) ) )
9998iotabidv 5402 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( iota x E. f [. ( `' g " ( _V  \  o ) )  /  y ]. (
f : ( 1 ... ( # `  y
) ) -1-1-onto-> y  /\  x  =  (  seq 1
( ( +g  `  w
) ,  ( g  o.  f ) ) `
 ( # `  y
) ) ) )  =  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) )
10053, 64, 99ifbieq12d 3816 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  if ( dom  g  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( dom  g  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) ) ) ,  ( iota x E. f [. ( `' g " ( _V 
\  o ) )  /  y ]. (
f : ( 1 ... ( # `  y
) ) -1-1-onto-> y  /\  x  =  (  seq 1
( ( +g  `  w
) ,  ( g  o.  f ) ) `
 ( # `  y
) ) ) ) )  =  if ( A  e.  ran  ... ,  ( iota x E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) )
10144, 48, 100ifbieq12d 3816 . . . 4  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  if ( ran  g  C_  o ,  ( 0g `  w ) ,  if ( dom  g  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( dom  g  =  ( m ... n
)  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) ) ) ,  ( iota x E. f [. ( `' g " ( _V 
\  o ) )  /  y ]. (
f : ( 1 ... ( # `  y
) ) -1-1-onto-> y  /\  x  =  (  seq 1
( ( +g  `  w
) ,  ( g  o.  f ) ) `
 ( # `  y
) ) ) ) ) )  =  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) ) )
10240, 101csbied 3314 . . 3  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  ->  [_ O  /  o ]_ if ( ran  g  C_  o ,  ( 0g
`  w ) ,  if ( dom  g  e.  ran  ... ,  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( dom  g  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) ) ) ,  ( iota x E. f [. ( `' g "
( _V  \  o
) )  /  y ]. ( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) ) ) ) )  =  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) ) )
10335, 102eqtrd 2475 . 2  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  ->  [_ { x  e.  (
Base `  w )  |  A. y  e.  (
Base `  w )
( ( x ( +g  `  w ) y )  =  y  /\  ( y ( +g  `  w ) x )  =  y ) }  /  o ]_ if ( ran  g  C_  o ,  ( 0g
`  w ) ,  if ( dom  g  e.  ran  ... ,  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( dom  g  =  ( m ... n )  /\  x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n ) ) ) ,  ( iota x E. f [. ( `' g "
( _V  \  o
) )  /  y ]. ( f : ( 1 ... ( # `  y ) ) -1-1-onto-> y  /\  x  =  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) ) ) ) ) )  =  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) ) )
104 gsumval.g . . 3  |-  ( ph  ->  G  e.  V )
105 elex 2981 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
106104, 105syl 16 . 2  |-  ( ph  ->  G  e.  _V )
107 elex 2981 . . 3  |-  ( F  e.  X  ->  F  e.  _V )
10874, 107syl 16 . 2  |-  ( ph  ->  F  e.  _V )
109 fvex 5701 . . . . 5  |-  ( 0g
`  G )  e. 
_V
11047, 109eqeltri 2513 . . . 4  |-  .0.  e.  _V
111 iotaex 5398 . . . . 5  |-  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m
(  .+  ,  F
) `  n )
) )  e.  _V
112 iotaex 5398 . . . . 5  |-  ( iota
x E. f ( f : ( 1 ... ( # `  W
) ) -1-1-onto-> W  /\  x  =  (  seq 1 ( 
.+  ,  ( F  o.  f ) ) `
 ( # `  W
) ) ) )  e.  _V
113111, 112ifex 3858 . . . 4  |-  if ( A  e.  ran  ... ,  ( iota x E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) )  e.  _V
114110, 113ifex 3858 . . 3  |-  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ... ,  ( iota x E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) )  e.  _V
115114a1i 11 . 2  |-  ( ph  ->  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) )  e.  _V )
1162, 103, 106, 108, 115ovmpt2d 6218 1  |-  ( ph  ->  ( G  gsumg  F )  =  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ...
,  ( iota x E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  =  (  seq m (  .+  ,  F ) `  n
) ) ) ,  ( iota x E. f ( f : ( 1 ... ( # `
 W ) ) -1-1-onto-> W  /\  x  =  (  seq 1 (  .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   A.wral 2715   E.wrex 2716   {crab 2719   _Vcvv 2972   [.wsbc 3186   [_csb 3288    \ cdif 3325    C_ wss 3328   ifcif 3791   `'ccnv 4839   dom cdm 4840   ran crn 4841   "cima 4843    o. ccom 4844   iotacio 5379   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1c1 9283   ZZ>=cuz 10861   ...cfz 11437    seqcseq 11806   #chash 12103   Basecbs 14174   +g cplusg 14238   0gc0g 14378    gsumg cgsu 14379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-recs 6832  df-rdg 6866  df-seq 11807  df-gsum 14381
This theorem is referenced by:  gsumval  15503  gsumpropd  15504  gsumpropd2lem  15505
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