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Theorem gsumvalx 16456
Description: Expand out the substitutions in df-gsum 15284. (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 15284 . . 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 762 . . . . . . . 8  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  ->  w  =  G )
43fveq2d 5829 . . . . . . 7  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( Base `  w )  =  ( Base `  G
) )
5 gsumval.b . . . . . . 7  |-  B  =  ( Base `  G
)
64, 5syl6eqr 2480 . . . . . 6  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( Base `  w )  =  B )
73fveq2d 5829 . . . . . . . . . . 11  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( +g  `  w )  =  ( +g  `  G
) )
8 gsumval.p . . . . . . . . . . 11  |-  .+  =  ( +g  `  G )
97, 8syl6eqr 2480 . . . . . . . . . 10  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( +g  `  w )  =  .+  )
109oveqd 6266 . . . . . . . . 9  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( x ( +g  `  w ) y )  =  ( x  .+  y ) )
1110eqeq1d 2430 . . . . . . . 8  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( ( x ( +g  `  w ) y )  =  y  <-> 
( x  .+  y
)  =  y ) )
129oveqd 6266 . . . . . . . . 9  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( y ( +g  `  w ) x )  =  ( y  .+  x ) )
1312eqeq1d 2430 . . . . . . . 8  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( ( y ( +g  `  w ) x )  =  y  <-> 
( y  .+  x
)  =  y ) )
1411, 13anbi12d 715 . . . . . . 7  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  -> 
( ( ( x ( +g  `  w
) y )  =  y  /\  ( y ( +g  `  w
) x )  =  y )  <->  ( (
x  .+  y )  =  y  /\  (
y  .+  x )  =  y ) ) )
156, 14raleqbidv 2978 . . . . . 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 3017 . . . . 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 6257 . . . . . . . . . . 11  |-  ( t  =  y  ->  (
s  .+  t )  =  ( s  .+  y ) )
19 id 22 . . . . . . . . . . 11  |-  ( t  =  y  ->  t  =  y )
2018, 19eqeq12d 2443 . . . . . . . . . 10  |-  ( t  =  y  ->  (
( s  .+  t
)  =  t  <->  ( s  .+  y )  =  y ) )
21 oveq1 6256 . . . . . . . . . . 11  |-  ( t  =  y  ->  (
t  .+  s )  =  ( y  .+  s ) )
2221, 19eqeq12d 2443 . . . . . . . . . 10  |-  ( t  =  y  ->  (
( t  .+  s
)  =  t  <->  ( y  .+  s )  =  y ) )
2320, 22anbi12d 715 . . . . . . . . 9  |-  ( t  =  y  ->  (
( ( s  .+  t )  =  t  /\  ( t  .+  s )  =  t )  <->  ( ( s 
.+  y )  =  y  /\  ( y 
.+  s )  =  y ) ) )
2423cbvralv 2996 . . . . . . . 8  |-  ( A. t  e.  B  (
( s  .+  t
)  =  t  /\  ( t  .+  s
)  =  t )  <->  A. y  e.  B  ( ( s  .+  y )  =  y  /\  ( y  .+  s )  =  y ) )
25 oveq1 6256 . . . . . . . . . . 11  |-  ( s  =  x  ->  (
s  .+  y )  =  ( x  .+  y ) )
2625eqeq1d 2430 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( s  .+  y
)  =  y  <->  ( x  .+  y )  =  y ) )
27 oveq2 6257 . . . . . . . . . . 11  |-  ( s  =  x  ->  (
y  .+  s )  =  ( y  .+  x ) )
2827eqeq1d 2430 . . . . . . . . . 10  |-  ( s  =  x  ->  (
( y  .+  s
)  =  y  <->  ( y  .+  x )  =  y ) )
2926, 28anbi12d 715 . . . . . . . . 9  |-  ( s  =  x  ->  (
( ( s  .+  y )  =  y  /\  ( y  .+  s )  =  y )  <->  ( ( x 
.+  y )  =  y  /\  ( y 
.+  x )  =  y ) ) )
3029ralbidv 2804 . . . . . . . 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 260 . . . . . . 7  |-  ( s  =  x  ->  ( A. t  e.  B  ( ( s  .+  t )  =  t  /\  ( t  .+  s )  =  t )  <->  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) ) )
3231cbvrabv 3021 . . . . . 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 2450 . . . . 5  |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) }
3416, 33syl6eqr 2480 . . . 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 3345 . . 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 5835 . . . . . . 7  |-  ( Base `  G )  e.  _V
375, 36eqeltri 2502 . . . . . 6  |-  B  e. 
_V
3817, 37rabex2 4520 . . . . 5  |-  O  e. 
_V
3938a1i 11 . . . 4  |-  ( (
ph  /\  ( w  =  G  /\  g  =  F ) )  ->  O  e.  _V )
40 simplrr 769 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  g  =  F )
4140rneqd 5024 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ran  g  =  ran  F )
42 simpr 462 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  o  =  O )
4341, 42sseq12d 3436 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( ran  g  C_  o  <->  ran  F  C_  O ) )
443adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  w  =  G )
4544fveq2d 5829 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( 0g `  w )  =  ( 0g `  G
) )
46 gsumval.z . . . . . 6  |-  .0.  =  ( 0g `  G )
4745, 46syl6eqr 2480 . . . . 5  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( 0g `  w )  =  .0.  )
4840dmeqd 4999 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  dom  g  =  dom  F )
49 gsumvalx.a . . . . . . . . 9  |-  ( ph  ->  dom  F  =  A )
5049ad2antrr 730 . . . . . . . 8  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  dom  F  =  A )
5148, 50eqtrd 2462 . . . . . . 7  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  dom  g  =  A )
5251eleq1d 2490 . . . . . 6  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( dom  g  e.  ran  ...  <->  A  e.  ran  ... )
)
5351eqeq1d 2430 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( dom  g  =  (
m ... n )  <->  A  =  ( m ... n
) ) )
549adantr 466 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( +g  `  w )  = 
.+  )
5554seqeq2d 12170 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq m ( ( +g  `  w ) ,  g )  =  seq m
(  .+  ,  g
) )
5640seqeq3d 12171 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq m (  .+  , 
g )  =  seq m (  .+  ,  F ) )
5755, 56eqtrd 2462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq m ( ( +g  `  w ) ,  g )  =  seq m
(  .+  ,  F
) )
5857fveq1d 5827 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  (  seq m ( ( +g  `  w ) ,  g ) `  n )  =  (  seq m
(  .+  ,  F
) `  n )
)
5958eqeq2d 2438 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  (
x  =  (  seq m ( ( +g  `  w ) ,  g ) `  n )  <-> 
x  =  (  seq m (  .+  ,  F ) `  n
) ) )
6053, 59anbi12d 715 . . . . . . . . 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 )
) ) )
6160rexbidv 2878 . . . . . . . 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
) ) ) )
6261exbidv 1762 . . . . . . 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
) ) ) )
6362iotabidv 5529 . . . . . 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
) ) ) )
6442difeq2d 3526 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( _V  \  o )  =  ( _V  \  O
) )
6564imaeq2d 5130 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( `' F " ( _V 
\  o ) )  =  ( `' F " ( _V  \  O
) ) )
6640cnveqd 4972 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  `' g  =  `' F
)
6766imaeq1d 5129 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( `' g " ( _V  \  o ) )  =  ( `' F " ( _V  \  o
) ) )
68 gsumval.w . . . . . . . . . . . 12  |-  ( ph  ->  W  =  ( `' F " ( _V 
\  O ) ) )
6968ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  W  =  ( `' F " ( _V  \  O
) ) )
7065, 67, 693eqtr4d 2472 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  ( `' g " ( _V  \  o ) )  =  W )
7170sbceq1d 3247 . . . . . . . . 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 ) ) ) ) )
72 gsumvalx.f . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  X )
73 cnvexg 6697 . . . . . . . . . . . . 13  |-  ( F  e.  X  ->  `' F  e.  _V )
74 imaexg 6688 . . . . . . . . . . . . 13  |-  ( `' F  e.  _V  ->  ( `' F " ( _V 
\  O ) )  e.  _V )
7572, 73, 743syl 18 . . . . . . . . . . . 12  |-  ( ph  ->  ( `' F "
( _V  \  O
) )  e.  _V )
7668, 75eqeltrd 2506 . . . . . . . . . . 11  |-  ( ph  ->  W  e.  _V )
7776ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  W  e.  _V )
78 fveq2 5825 . . . . . . . . . . . . . . 15  |-  ( y  =  W  ->  ( # `
 y )  =  ( # `  W
) )
7978adantl 467 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  ( # `
 y )  =  ( # `  W
) )
8079oveq2d 6265 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (
1 ... ( # `  y
) )  =  ( 1 ... ( # `  W ) ) )
81 f1oeq2 5766 . . . . . . . . . . . . 13  |-  ( ( 1 ... ( # `  y ) )  =  ( 1 ... ( # `
 W ) )  ->  ( f : ( 1 ... ( # `
 y ) ) -1-1-onto-> y  <-> 
f : ( 1 ... ( # `  W
) ) -1-1-onto-> y ) )
8280, 81syl 17 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (
f : ( 1 ... ( # `  y
) ) -1-1-onto-> y  <->  f : ( 1 ... ( # `  W ) ) -1-1-onto-> y ) )
83 f1oeq3 5767 . . . . . . . . . . . . 13  |-  ( y  =  W  ->  (
f : ( 1 ... ( # `  W
) ) -1-1-onto-> y  <->  f : ( 1 ... ( # `  W ) ) -1-1-onto-> W ) )
8483adantl 467 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (
f : ( 1 ... ( # `  W
) ) -1-1-onto-> y  <->  f : ( 1 ... ( # `  W ) ) -1-1-onto-> W ) )
8582, 84bitrd 256 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (
f : ( 1 ... ( # `  y
) ) -1-1-onto-> y  <->  f : ( 1 ... ( # `  W ) ) -1-1-onto-> W ) )
8654seqeq2d 12170 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) )  =  seq 1
(  .+  ,  (
g  o.  f ) ) )
8740coeq1d 4958 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  (
g  o.  f )  =  ( F  o.  f ) )
8887seqeq3d 12171 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq 1 (  .+  , 
( g  o.  f
) )  =  seq 1 (  .+  , 
( F  o.  f
) ) )
8986, 88eqtrd 2462 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
w  =  G  /\  g  =  F )
)  /\  o  =  O )  ->  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) )  =  seq 1
(  .+  ,  ( F  o.  f )
) )
9089adantr 466 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) )  =  seq 1
(  .+  ,  ( F  o.  f )
) )
9190, 79fveq12d 5831 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( w  =  G  /\  g  =  F
) )  /\  o  =  O )  /\  y  =  W )  ->  (  seq 1 ( ( +g  `  w ) ,  ( g  o.  f ) ) `  ( # `  y ) )  =  (  seq 1 ( 
.+  ,  ( F  o.  f ) ) `
 ( # `  W
) ) )
9291eqeq2d 2438 . . . . . . . . . . 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 ) ) ) )
9385, 92anbi12d 715 . . . . . . . . . 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 ) ) ) ) )
9477, 93sbcied 3279 . . . . . . . . 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 ) ) ) ) )
9571, 94bitrd 256 . . . . . . . 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 ) ) ) ) )
9695exbidv 1762 . . . . . . 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 ) ) ) ) )
9796iotabidv 5529 . . . . . 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 ) ) ) ) )
9852, 63, 97ifbieq12d 3881 . . . . 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 ) ) ) ) ) )
9943, 47, 98ifbieq12d 3881 . . . 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 ) ) ) ) ) ) )
10039, 99csbied 3365 . . 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 ) ) ) ) ) ) )
10135, 100eqtrd 2462 . 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 ) ) ) ) ) ) )
102 gsumval.g . . 3  |-  ( ph  ->  G  e.  V )
103 elex 3031 . . 3  |-  ( G  e.  V  ->  G  e.  _V )
104102, 103syl 17 . 2  |-  ( ph  ->  G  e.  _V )
105 elex 3031 . . 3  |-  ( F  e.  X  ->  F  e.  _V )
10672, 105syl 17 . 2  |-  ( ph  ->  F  e.  _V )
107 fvex 5835 . . . . 5  |-  ( 0g
`  G )  e. 
_V
10846, 107eqeltri 2502 . . . 4  |-  .0.  e.  _V
109 iotaex 5525 . . . . 5  |-  ( iota
x E. m E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  =  (  seq m
(  .+  ,  F
) `  n )
) )  e.  _V
110 iotaex 5525 . . . . 5  |-  ( iota
x E. f ( f : ( 1 ... ( # `  W
) ) -1-1-onto-> W  /\  x  =  (  seq 1 ( 
.+  ,  ( F  o.  f ) ) `
 ( # `  W
) ) ) )  e.  _V
111109, 110ifex 3922 . . . 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
112108, 111ifex 3922 . . 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
113112a1i 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 )
1142, 101, 104, 106, 113ovmpt2d 6382 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 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   A.wral 2714   E.wrex 2715   {crab 2718   _Vcvv 3022   [.wsbc 3242   [_csb 3338    \ cdif 3376    C_ wss 3379   ifcif 3854   `'ccnv 4795   dom cdm 4796   ran crn 4797   "cima 4799    o. ccom 4800   iotacio 5506   -1-1-onto->wf1o 5543   ` cfv 5544  (class class class)co 6249    |-> cmpt2 6251   1c1 9491   ZZ>=cuz 11110   ...cfz 11735    seqcseq 12163   #chash 12465   Basecbs 15064   +g cplusg 15133   0gc0g 15281    gsumg cgsu 15282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-seq 12164  df-gsum 15284
This theorem is referenced by:  gsumval  16457  gsumpropd  16458  gsumpropd2lem  16459
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