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Theorem psrval 16434
Description: Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
psrval.s  |-  S  =  ( I mPwSer  R )
psrval.k  |-  K  =  ( Base `  R
)
psrval.a  |-  .+  =  ( +g  `  R )
psrval.m  |-  .x.  =  ( .r `  R )
psrval.o  |-  O  =  ( TopOpen `  R )
psrval.d  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
psrval.b  |-  ( ph  ->  B  =  ( K  ^m  D ) )
psrval.p  |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )
psrval.t  |-  .X.  =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) 
.x.  ( g `  ( k  o F  -  x ) ) ) ) ) ) )
psrval.v  |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F  .x.  f
) )
psrval.j  |-  ( ph  ->  J  =  ( Xt_ `  ( D  X.  { O } ) ) )
psrval.i  |-  ( ph  ->  I  e.  W )
psrval.r  |-  ( ph  ->  R  e.  X )
Assertion
Ref Expression
psrval  |-  ( ph  ->  S  =  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } ) )
Distinct variable groups:    y, h    f, g, k, x, ph    B, f, g, k, x   
f, h, I, g, k, x    R, f, g, k, x    y,
f, D, g, k, x
Allowed substitution hints:    ph( y, h)    B( y, h)    D( h)    .+ ( x, y, f, g, h, k)    .+b ( x, y, f, g, h, k)    R( y, h)    S( x, y, f, g, h, k)    .xb (
x, y, f, g, h, k)    .x. ( x, y, f, g, h, k)    .X. ( x, y, f, g, h, k)    I( y)    J( x, y, f, g, h, k)    K( x, y, f, g, h, k)    O( x, y, f, g, h, k)    W( x, y, f, g, h, k)    X( x, y, f, g, h, k)

Proof of Theorem psrval
Dummy variables  i 
r  b  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrval.s . 2  |-  S  =  ( I mPwSer  R )
2 df-psr 16422 . . . 4  |- mPwSer  =  ( i  e.  _V , 
r  e.  _V  |->  [_ { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  /  d ]_ [_ (
( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  o F  -  x
) ) ) ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } ) )
32a1i 11 . . 3  |-  ( ph  -> mPwSer 
=  ( i  e. 
_V ,  r  e. 
_V  |->  [_ { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  / 
d ]_ [_ ( (
Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  o F  -  x
) ) ) ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } ) ) )
4 simprl 734 . . . . . . . 8  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  -> 
i  =  I )
54oveq2d 6100 . . . . . . 7  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  -> 
( NN0  ^m  i
)  =  ( NN0 
^m  I ) )
6 rabeq 2952 . . . . . . 7  |-  ( ( NN0  ^m  i )  =  ( NN0  ^m  I )  ->  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } )
75, 6syl 16 . . . . . 6  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  ->  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin } )
8 psrval.d . . . . . 6  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
97, 8syl6eqr 2488 . . . . 5  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  ->  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  =  D )
109csbeq1d 3259 . . . 4  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  ->  [_ { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  /  d ]_ [_ (
( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  o F  -  x
) ) ) ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } )  =  [_ D  /  d ]_ [_ (
( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  o F  -  x
) ) ) ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } ) )
11 ovex 6109 . . . . . . 7  |-  ( NN0 
^m  i )  e. 
_V
1211rabex 4357 . . . . . 6  |-  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  e.  _V
139, 12syl6eqelr 2527 . . . . 5  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  ->  D  e.  _V )
14 simplrr 739 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  r  =  R )
1514fveq2d 5735 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  ( Base `  r )  =  ( Base `  R
) )
16 psrval.k . . . . . . . . . 10  |-  K  =  ( Base `  R
)
1715, 16syl6eqr 2488 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  ( Base `  r )  =  K )
18 simpr 449 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  d  =  D )
1917, 18oveq12d 6102 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  (
( Base `  r )  ^m  d )  =  ( K  ^m  D ) )
20 psrval.b . . . . . . . . 9  |-  ( ph  ->  B  =  ( K  ^m  D ) )
2120ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  B  =  ( K  ^m  D ) )
2219, 21eqtr4d 2473 . . . . . . 7  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  (
( Base `  r )  ^m  d )  =  B )
2322csbeq1d 3259 . . . . . 6  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  [_ (
( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  o F  -  x
) ) ) ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } )  =  [_ B  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  (  o F ( +g  `  r
)  |`  ( b  X.  b ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r 
gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `
 x ) ( .r `  r ) ( g `  (
k  o F  -  x ) ) ) ) ) ) )
>. }  u.  { <. (Scalar `  ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } ) )
24 ovex 6109 . . . . . . . 8  |-  ( (
Base `  r )  ^m  d )  e.  _V
2522, 24syl6eqelr 2527 . . . . . . 7  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  B  e.  _V )
26 simpr 449 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  b  =  B )
2726opeq2d 3993 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  <. ( Base `  ndx ) ,  b >.  =  <. (
Base `  ndx ) ,  B >. )
2814adantr 453 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  r  =  R )
2928fveq2d 5735 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( +g  `  r )  =  ( +g  `  R
) )
30 psrval.a . . . . . . . . . . . . . 14  |-  .+  =  ( +g  `  R )
3129, 30syl6eqr 2488 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( +g  `  r )  = 
.+  )
32 ofeq 6310 . . . . . . . . . . . . 13  |-  ( ( +g  `  r )  =  .+  ->  o F ( +g  `  r
)  =  o F 
.+  )
3331, 32syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  o F ( +g  `  r
)  =  o F 
.+  )
3426, 26xpeq12d 4906 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
b  X.  b )  =  ( B  X.  B ) )
3533, 34reseq12d 5150 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (  o F ( +g  `  r
)  |`  ( b  X.  b ) )  =  (  o F  .+  |`  ( B  X.  B
) ) )
36 psrval.p . . . . . . . . . . 11  |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )
3735, 36syl6eqr 2488 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (  o F ( +g  `  r
)  |`  ( b  X.  b ) )  = 
.+b  )
3837opeq2d 3993 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r )  |`  ( b  X.  b
) ) >.  =  <. ( +g  `  ndx ) ,  .+b  >. )
3918adantr 453 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  d  =  D )
40 rabeq 2952 . . . . . . . . . . . . . . . 16  |-  ( d  =  D  ->  { y  e.  d  |  y  o R  <_  k }  =  { y  e.  D  |  y  o R  <_  k } )
4139, 40syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  { y  e.  d  |  y  o R  <_  k }  =  { y  e.  D  |  y  o R  <_  k } )
4228fveq2d 5735 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( .r `  r )  =  ( .r `  R
) )
43 psrval.m . . . . . . . . . . . . . . . . 17  |-  .x.  =  ( .r `  R )
4442, 43syl6eqr 2488 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( .r `  r )  = 
.x.  )
4544oveqd 6101 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
( f `  x
) ( .r `  r ) ( g `
 ( k  o F  -  x ) ) )  =  ( ( f `  x
)  .x.  ( g `  ( k  o F  -  x ) ) ) )
4641, 45mpteq12dv 4290 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `
 x ) ( .r `  r ) ( g `  (
k  o F  -  x ) ) ) )  =  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( f `  x )  .x.  (
g `  ( k  o F  -  x
) ) ) ) )
4728, 46oveq12d 6102 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
r  gsumg  ( x  e.  {
y  e.  d  |  y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  r
) ( g `  ( k  o F  -  x ) ) ) ) )  =  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) 
.x.  ( g `  ( k  o F  -  x ) ) ) ) ) )
4839, 47mpteq12dv 4290 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
k  e.  d  |->  ( r  gsumg  ( x  e.  {
y  e.  d  |  y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  r
) ( g `  ( k  o F  -  x ) ) ) ) ) )  =  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( f `  x )  .x.  (
g `  ( k  o F  -  x
) ) ) ) ) ) )
4926, 26, 48mpt2eq123dv 6139 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  {
y  e.  d  |  y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  r
) ( g `  ( k  o F  -  x ) ) ) ) ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R 
gsumg  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( f `
 x )  .x.  ( g `  (
k  o F  -  x ) ) ) ) ) ) ) )
50 psrval.t . . . . . . . . . . 11  |-  .X.  =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) 
.x.  ( g `  ( k  o F  -  x ) ) ) ) ) ) )
5149, 50syl6eqr 2488 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  {
y  e.  d  |  y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  r
) ( g `  ( k  o F  -  x ) ) ) ) ) ) )  =  .X.  )
5251opeq2d 3993 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b 
|->  ( k  e.  d 
|->  ( r  gsumg  ( x  e.  {
y  e.  d  |  y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  r
) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >.  =  <. ( .r `  ndx ) ,  .X.  >. )
5327, 38, 52tpeq123d 3900 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  { <. (
Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  o F  -  x
) ) ) ) ) ) ) >. }  =  { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.X.  >. } )
5428opeq2d 3993 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  <. (Scalar ` 
ndx ) ,  r
>.  =  <. (Scalar `  ndx ) ,  R >. )
5517adantr 453 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( Base `  r )  =  K )
56 ofeq 6310 . . . . . . . . . . . . . 14  |-  ( ( .r `  r )  =  .x.  ->  o F ( .r `  r )  =  o F  .x.  )
5744, 56syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  o F ( .r `  r )  =  o F  .x.  )
5839xpeq1d 4904 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
d  X.  { x } )  =  ( D  X.  { x } ) )
59 eqidd 2439 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  f  =  f )
6057, 58, 59oveq123d 6105 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
( d  X.  {
x } )  o F ( .r `  r ) f )  =  ( ( D  X.  { x }
)  o F  .x.  f ) )
6155, 26, 60mpt2eq123dv 6139 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
)  o F  .x.  f ) ) )
62 psrval.v . . . . . . . . . . 11  |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F  .x.  f
) )
6361, 62syl6eqr 2488 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )  =  .xb  )
6463opeq2d 3993 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  r ) ,  f  e.  b  |->  ( ( d  X. 
{ x } )  o F ( .r
`  r ) f ) ) >.  =  <. ( .s `  ndx ) ,  .xb  >. )
6528fveq2d 5735 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( TopOpen
`  r )  =  ( TopOpen `  R )
)
66 psrval.o . . . . . . . . . . . . . . 15  |-  O  =  ( TopOpen `  R )
6765, 66syl6eqr 2488 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( TopOpen
`  r )  =  O )
6867sneqd 3829 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  { (
TopOpen `  r ) }  =  { O }
)
6939, 68xpeq12d 4906 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
d  X.  { (
TopOpen `  r ) } )  =  ( D  X.  { O }
) )
7069fveq2d 5735 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( Xt_ `  ( d  X. 
{ ( TopOpen `  r
) } ) )  =  ( Xt_ `  ( D  X.  { O }
) ) )
71 psrval.j . . . . . . . . . . . 12  |-  ( ph  ->  J  =  ( Xt_ `  ( D  X.  { O } ) ) )
7271ad3antrrr 712 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  J  =  ( Xt_ `  ( D  X.  { O }
) ) )
7370, 72eqtr4d 2473 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( Xt_ `  ( d  X. 
{ ( TopOpen `  r
) } ) )  =  J )
7473opeq2d 3993 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( d  X.  { ( TopOpen `  r
) } ) )
>.  =  <. (TopSet `  ndx ) ,  J >. )
7554, 64, 74tpeq123d 3900 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  { <. (Scalar `  ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. }  =  { <. (Scalar ` 
ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  .xb  >. ,  <. (TopSet `  ndx ) ,  J >. } )
7653, 75uneq12d 3504 . . . . . . 7  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  (  o F ( +g  `  r
)  |`  ( b  X.  b ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r 
gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `
 x ) ( .r `  r ) ( g `  (
k  o F  -  x ) ) ) ) ) ) )
>. }  u.  { <. (Scalar `  ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } ) )
7725, 76csbied 3295 . . . . . 6  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  [_ B  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  (  o F ( +g  `  r
)  |`  ( b  X.  b ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r 
gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `
 x ) ( .r `  r ) ( g `  (
k  o F  -  x ) ) ) ) ) ) )
>. }  u.  { <. (Scalar `  ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } ) )
7823, 77eqtrd 2470 . . . . 5  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  [_ (
( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  o F  -  x
) ) ) ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } ) )
7913, 78csbied 3295 . . . 4  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  ->  [_ D  /  d ]_ [_ ( ( Base `  r )  ^m  d
)  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  (  o F ( +g  `  r
)  |`  ( b  X.  b ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r 
gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `
 x ) ( .r `  r ) ( g `  (
k  o F  -  x ) ) ) ) ) ) )
>. }  u.  { <. (Scalar `  ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } ) )
8010, 79eqtrd 2470 . . 3  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  ->  [_ { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  /  d ]_ [_ (
( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  o R  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  o F  -  x
) ) ) ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  o F ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } ) )
81 psrval.i . . . 4  |-  ( ph  ->  I  e.  W )
82 elex 2966 . . . 4  |-  ( I  e.  W  ->  I  e.  _V )
8381, 82syl 16 . . 3  |-  ( ph  ->  I  e.  _V )
84 psrval.r . . . 4  |-  ( ph  ->  R  e.  X )
85 elex 2966 . . . 4  |-  ( R  e.  X  ->  R  e.  _V )
8684, 85syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
87 tpex 4711 . . . . 5  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  e.  _V
88 tpex 4711 . . . . 5  |-  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  .xb  >. ,  <. (TopSet `  ndx ) ,  J >. }  e.  _V
8987, 88unex 4710 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } )  e.  _V
9089a1i 11 . . 3  |-  ( ph  ->  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } )  e.  _V )
913, 80, 83, 86, 90ovmpt2d 6204 . 2  |-  ( ph  ->  ( I mPwSer  R )  =  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } ) )
921, 91syl5eq 2482 1  |-  ( ph  ->  S  =  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958   [_csb 3253    u. cun 3320   {csn 3816   {ctp 3818   <.cop 3819   class class class wbr 4215    e. cmpt 4269    X. cxp 4879   `'ccnv 4880    |` cres 4883   "cima 4884   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086    o Fcof 6306    o Rcofr 6307    ^m cmap 7021   Fincfn 7112    <_ cle 9126    - cmin 9296   NNcn 10005   NN0cn0 10226   ndxcnx 13471   Basecbs 13474   +g cplusg 13534   .rcmulr 13535  Scalarcsca 13537   .scvsca 13538  TopSetcts 13540   TopOpenctopn 13654   Xt_cpt 13671    gsumg cgsu 13729   mPwSer cmps 16411
This theorem is referenced by:  psrbas  16448  psrplusg  16450  psrmulr  16453  psrsca  16458  psrvscafval  16459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-psr 16422
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