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Theorem psrval 16110
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 16098 . . . 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 10 . . 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 732 . . . . . . . 8  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  -> 
i  =  I )
54oveq2d 5874 . . . . . . 7  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  -> 
( NN0  ^m  i
)  =  ( NN0 
^m  I ) )
6 rabeq 2782 . . . . . . 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 15 . . . . . 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 2333 . . . . 5  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  ->  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  =  D )
109csbeq1d 3087 . . . 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 5883 . . . . . . 7  |-  ( NN0 
^m  i )  e. 
_V
1211rabex 4165 . . . . . 6  |-  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  e.  _V
139, 12syl6eqelr 2372 . . . . 5  |-  ( (
ph  /\  ( i  =  I  /\  r  =  R ) )  ->  D  e.  _V )
14 simplrr 737 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  r  =  R )
1514fveq2d 5529 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  ( Base `  r )  =  ( Base `  R
) )
16 psrval.k . . . . . . . . . 10  |-  K  =  ( Base `  R
)
1715, 16syl6eqr 2333 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  ( Base `  r )  =  K )
18 simpr 447 . . . . . . . . 9  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  d  =  D )
1917, 18oveq12d 5876 . . . . . . . 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 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  B  =  ( K  ^m  D ) )
2219, 21eqtr4d 2318 . . . . . . 7  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  (
( Base `  r )  ^m  d )  =  B )
2322csbeq1d 3087 . . . . . 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 5883 . . . . . . . 8  |-  ( (
Base `  r )  ^m  d )  e.  _V
2522, 24syl6eqelr 2372 . . . . . . 7  |-  ( ( ( ph  /\  (
i  =  I  /\  r  =  R )
)  /\  d  =  D )  ->  B  e.  _V )
26 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  b  =  B )
2726opeq2d 3803 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  <. ( Base `  ndx ) ,  b >.  =  <. (
Base `  ndx ) ,  B >. )
2814adantr 451 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  r  =  R )
2928fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( +g  `  r )  =  ( +g  `  R
) )
30 psrval.a . . . . . . . . . . . . . 14  |-  .+  =  ( +g  `  R )
3129, 30syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( +g  `  r )  = 
.+  )
32 ofeq 6080 . . . . . . . . . . . . 13  |-  ( ( +g  `  r )  =  .+  ->  o F ( +g  `  r
)  =  o F 
.+  )
3331, 32syl 15 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  o F ( +g  `  r
)  =  o F 
.+  )
3426, 26xpeq12d 4714 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
b  X.  b )  =  ( B  X.  B ) )
3533, 34reseq12d 4956 . . . . . . . . . . 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 2333 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (  o F ( +g  `  r
)  |`  ( b  X.  b ) )  = 
.+b  )
3837opeq2d 3803 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  <. ( +g  `  ndx ) ,  (  o F ( +g  `  r )  |`  ( b  X.  b
) ) >.  =  <. ( +g  `  ndx ) ,  .+b  >. )
3918adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  d  =  D )
40 rabeq 2782 . . . . . . . . . . . . . . . 16  |-  ( d  =  D  ->  { y  e.  d  |  y  o R  <_  k }  =  { y  e.  D  |  y  o R  <_  k } )
4139, 40syl 15 . . . . . . . . . . . . . . 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 5529 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( .r `  r )  =  ( .r `  R
) )
43 psrval.m . . . . . . . . . . . . . . . . 17  |-  .x.  =  ( .r `  R )
4442, 43syl6eqr 2333 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( .r `  r )  = 
.x.  )
4544oveqd 5875 . . . . . . . . . . . . . . 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 4098 . . . . . . . . . . . . . 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 5876 . . . . . . . . . . . . 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 4098 . . . . . . . . . . . 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 5910 . . . . . . . . . . 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 2333 . . . . . . . . . 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 3803 . . . . . . . . 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 3721 . . . . . . . 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 3803 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  <. (Scalar ` 
ndx ) ,  r
>.  =  <. (Scalar `  ndx ) ,  R >. )
5517adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( Base `  r )  =  K )
56 ofeq 6080 . . . . . . . . . . . . . 14  |-  ( ( .r `  r )  =  .x.  ->  o F ( .r `  r )  =  o F  .x.  )
5744, 56syl 15 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  o F ( .r `  r )  =  o F  .x.  )
5839xpeq1d 4712 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
d  X.  { x } )  =  ( D  X.  { x } ) )
59 eqidd 2284 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  f  =  f )
6057, 58, 59oveq123d 5879 . . . . . . . . . . . 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 5910 . . . . . . . . . . 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 2333 . . . . . . . . . 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 3803 . . . . . . . . 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 5529 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( TopOpen
`  r )  =  ( TopOpen `  R )
)
66 psrval.o . . . . . . . . . . . . . . 15  |-  O  =  ( TopOpen `  R )
6765, 66syl6eqr 2333 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( TopOpen
`  r )  =  O )
6867sneqd 3653 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  { (
TopOpen `  r ) }  =  { O }
)
6939, 68xpeq12d 4714 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  (
d  X.  { (
TopOpen `  r ) } )  =  ( D  X.  { O }
) )
7069fveq2d 5529 . . . . . . . . . . 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 710 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  J  =  ( Xt_ `  ( D  X.  { O }
) ) )
7370, 72eqtr4d 2318 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  ( Xt_ `  ( d  X. 
{ ( TopOpen `  r
) } ) )  =  J )
7473opeq2d 3803 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( i  =  I  /\  r  =  R ) )  /\  d  =  D )  /\  b  =  B )  ->  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( d  X.  { ( TopOpen `  r
) } ) )
>.  =  <. (TopSet `  ndx ) ,  J >. )
7554, 64, 74tpeq123d 3721 . . . . . . . 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 3330 . . . . . . 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 3123 . . . . . 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 2315 . . . . 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 3123 . . . 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 2315 . . 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 2796 . . . 4  |-  ( I  e.  W  ->  I  e.  _V )
8381, 82syl 15 . . 3  |-  ( ph  ->  I  e.  _V )
84 psrval.r . . . 4  |-  ( ph  ->  R  e.  X )
85 elex 2796 . . . 4  |-  ( R  e.  X  ->  R  e.  _V )
8684, 85syl 15 . . 3  |-  ( ph  ->  R  e.  _V )
87 tpex 4519 . . . . 5  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  e.  _V
88 tpex 4519 . . . . 5  |-  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  .xb  >. ,  <. (TopSet `  ndx ) ,  J >. }  e.  _V
8987, 88unex 4518 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) , 
.xb  >. ,  <. (TopSet ` 
ndx ) ,  J >. } )  e.  _V
9089a1i 10 . . 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 5975 . 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 2327 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 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788   [_csb 3081    u. cun 3150   {csn 3640   {ctp 3642   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    o Fcof 6076    o Rcofr 6077    ^m cmap 6772   Fincfn 6863    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212  TopSetcts 13214   TopOpenctopn 13326   Xt_cpt 13343    gsumg cgsu 13401   mPwSer cmps 16087
This theorem is referenced by:  psrbas  16124  psrplusg  16126  psrmulr  16129  psrsca  16134  psrvscafval  16135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-psr 16098
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