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Theorem imasval 13414
Description: Value of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.)
Hypotheses
Ref Expression
imasval.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasval.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasval.p  |-  .+  =  ( +g  `  R )
imasval.m  |-  .X.  =  ( .r `  R )
imasval.g  |-  G  =  (Scalar `  R )
imasval.k  |-  K  =  ( Base `  G
)
imasval.q  |-  .x.  =  ( .s `  R )
imasval.j  |-  J  =  ( TopOpen `  R )
imasval.e  |-  E  =  ( dist `  R
)
imasval.n  |-  N  =  ( le `  R
)
imasval.a  |-  ( ph  -> 
.+b  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.+  q ) )
>. } )
imasval.t  |-  ( ph  -> 
.xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.X.  q ) )
>. } )
imasval.s  |-  ( ph  -> 
.(x)  =  U_ q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) ) )
imasval.o  |-  ( ph  ->  O  =  ( J qTop 
F ) )
imasval.d  |-  ( ph  ->  D  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  ( ( F `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( h `  n
) ) )  =  y  /\  A. i  e.  ( 1 ... (
n  -  1 ) ) ( F `  ( 2nd `  ( h `
 i ) ) )  =  ( F `
 ( 1st `  (
h `  ( i  +  1 ) ) ) ) ) } 
|->  ( RR* s  gsumg  ( E  o.  g ) ) ) ,  RR* ,  `'  <  ) ) )
imasval.l  |-  ( ph  -> 
.<_  =  ( ( F  o.  N )  o.  `' F ) )
imasval.f  |-  ( ph  ->  F : V -onto-> B
)
imasval.r  |-  ( ph  ->  R  e.  Z )
Assertion
Ref Expression
imasval  |-  ( ph  ->  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } ) )
Distinct variable groups:    g, h, i, n, p, q, x, y, F    R, g, h, i, n, p, q, x, y    h, V, p, q    ph, g, h, i, n, p, q, x, y
Allowed substitution hints:    B( x, y, g, h, i, n, q, p)    D( x, y, g, h, i, n, q, p)    .+ ( x, y, g, h, i, n, q, p)    .+b ( x, y, g, h, i, n, q, p)    .xb ( x, y, g, h, i, n, q, p)    .x. ( x, y, g, h, i, n, q, p)    .X. ( x, y, g, h, i, n, q, p)    .(x) ( x, y, g, h, i, n, q, p)    U( x, y, g, h, i, n, q, p)    E( x, y, g, h, i, n, q, p)    G( x, y, g, h, i, n, q, p)    J( x, y, g, h, i, n, q, p)    K( x, y, g, h, i, n, q, p)    .<_ ( x, y, g, h, i, n, q, p)    N( x, y, g, h, i, n, q, p)    O( x, y, g, h, i, n, q, p)    V( x, y, g, i, n)    Z( x, y, g, h, i, n, q, p)

Proof of Theorem imasval
Dummy variables  f 
r  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasval.u . 2  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 df-imas 13411 . . . 4  |-  "s  =  (
f  e.  _V , 
r  e.  _V  |->  [_ ( Base `  r )  /  v ]_ (
( { <. ( Base `  ndx ) ,  ran  f >. ,  <. ( +g  `  ndx ) ,  U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( +g  `  r ) q ) ) >. } >. ,  <. ( .r `  ndx ) ,  U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( .r
`  r ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  r ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  v  ( p  e.  (
Base `  (Scalar `  r
) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s `  r
) q ) ) ) >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  r ) qTop  f
) >. ,  <. ( le `  ndx ) ,  ( ( f  o.  ( le `  r
) )  o.  `' f ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ran  f ,  y  e.  ran  f  |->  sup ( U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( v  X.  v )  ^m  ( 1 ... n ) )  |  ( ( f `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( f `  ( 2nd `  ( h `  n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } ) )
32a1i 10 . . 3  |-  ( ph  ->  "s  =  ( f  e. 
_V ,  r  e. 
_V  |->  [_ ( Base `  r
)  /  v ]_ ( ( { <. (
Base `  ndx ) ,  ran  f >. ,  <. ( +g  `  ndx ) ,  U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( +g  `  r ) q ) ) >. } >. ,  <. ( .r `  ndx ) ,  U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( .r
`  r ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  r ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  v  ( p  e.  (
Base `  (Scalar `  r
) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s `  r
) q ) ) ) >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  r ) qTop  f
) >. ,  <. ( le `  ndx ) ,  ( ( f  o.  ( le `  r
) )  o.  `' f ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ran  f ,  y  e.  ran  f  |->  sup ( U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( v  X.  v )  ^m  ( 1 ... n ) )  |  ( ( f `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( f `  ( 2nd `  ( h `  n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } ) ) )
4 fvex 5539 . . . . 5  |-  ( Base `  r )  e.  _V
54a1i 10 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  r  =  R ) )  -> 
( Base `  r )  e.  _V )
6 simplrl 736 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  f  =  F )
76rneqd 4906 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ran  f  =  ran  F )
8 imasval.f . . . . . . . . . . 11  |-  ( ph  ->  F : V -onto-> B
)
9 forn 5454 . . . . . . . . . . 11  |-  ( F : V -onto-> B  ->  ran  F  =  B )
108, 9syl 15 . . . . . . . . . 10  |-  ( ph  ->  ran  F  =  B )
1110ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ran  F  =  B )
127, 11eqtrd 2315 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ran  f  =  B )
1312opeq2d 3803 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( Base `  ndx ) ,  ran  f >.  =  <. ( Base `  ndx ) ,  B >. )
14 simplrr 737 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  r  =  R )
1514fveq2d 5529 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( Base `  r )  =  (
Base `  R )
)
16 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  v  =  ( Base `  r )
)
17 imasval.v . . . . . . . . . . . 12  |-  ( ph  ->  V  =  ( Base `  R ) )
1817ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  V  =  ( Base `  R )
)
1915, 16, 183eqtr4d 2325 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  v  =  V )
206fveq1d 5527 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  p )  =  ( F `  p ) )
216fveq1d 5527 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  q )  =  ( F `  q ) )
2220, 21opeq12d 3804 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( f `
 p ) ,  ( f `  q
) >.  =  <. ( F `  p ) ,  ( F `  q ) >. )
2314fveq2d 5529 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( +g  `  r )  =  ( +g  `  R ) )
24 imasval.p . . . . . . . . . . . . . . . 16  |-  .+  =  ( +g  `  R )
2523, 24syl6eqr 2333 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( +g  `  r )  =  .+  )
2625oveqd 5875 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( p
( +g  `  r ) q )  =  ( p  .+  q ) )
276, 26fveq12d 5531 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( p ( +g  `  r ) q ) )  =  ( F `
 ( p  .+  q ) ) )
2822, 27opeq12d 3804 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( +g  `  r
) q ) )
>.  =  <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.+  q ) )
>. )
2928sneqd 3653 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( +g  `  r
) q ) )
>. }  =  { <. <.
( F `  p
) ,  ( F `
 q ) >. ,  ( F `  ( p  .+  q ) ) >. } )
3019, 29iuneq12d 3929 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( +g  `  r
) q ) )
>. }  =  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.+  q ) )
>. } )
3119, 30iuneq12d 3929 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ p  e.  v  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( +g  `  r
) q ) )
>. }  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.+  q ) )
>. } )
32 imasval.a . . . . . . . . . 10  |-  ( ph  -> 
.+b  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.+  q ) )
>. } )
3332ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  .+b  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.+  q ) )
>. } )
3431, 33eqtr4d 2318 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ p  e.  v  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( +g  `  r
) q ) )
>. }  =  .+b  )
3534opeq2d 3803 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( +g  ` 
ndx ) ,  U_ p  e.  v  U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( +g  `  r ) q ) ) >. } >.  =  <. ( +g  `  ndx ) ,  .+b  >. )
3614fveq2d 5529 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( .r `  r )  =  ( .r `  R ) )
37 imasval.m . . . . . . . . . . . . . . . 16  |-  .X.  =  ( .r `  R )
3836, 37syl6eqr 2333 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( .r `  r )  =  .X.  )
3938oveqd 5875 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( p
( .r `  r
) q )  =  ( p  .X.  q
) )
406, 39fveq12d 5531 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( p ( .r
`  r ) q ) )  =  ( F `  ( p 
.X.  q ) ) )
4122, 40opeq12d 3804 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( .r `  r
) q ) )
>.  =  <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.X.  q ) )
>. )
4241sneqd 3653 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( .r `  r
) q ) )
>. }  =  { <. <.
( F `  p
) ,  ( F `
 q ) >. ,  ( F `  ( p  .X.  q ) ) >. } )
4319, 42iuneq12d 3929 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( .r `  r
) q ) )
>. }  =  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.X.  q ) )
>. } )
4419, 43iuneq12d 3929 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ p  e.  v  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( .r `  r
) q ) )
>. }  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.X.  q ) )
>. } )
45 imasval.t . . . . . . . . . 10  |-  ( ph  -> 
.xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.X.  q ) )
>. } )
4645ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  .xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.X.  q ) )
>. } )
4744, 46eqtr4d 2318 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ p  e.  v  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( .r `  r
) q ) )
>. }  =  .xb  )
4847opeq2d 3803 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( .r
`  ndx ) ,  U_ p  e.  v  U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( .r
`  r ) q ) ) >. } >.  = 
<. ( .r `  ndx ) ,  .xb  >. )
4913, 35, 48tpeq123d 3721 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  { <. ( Base `  ndx ) ,  ran  f >. ,  <. ( +g  `  ndx ) ,  U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( +g  `  r ) q ) ) >. } >. ,  <. ( .r `  ndx ) ,  U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( .r
`  r ) q ) ) >. } >. }  =  { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.xb  >. } )
5014fveq2d 5529 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  (Scalar `  r
)  =  (Scalar `  R ) )
51 imasval.g . . . . . . . . 9  |-  G  =  (Scalar `  R )
5250, 51syl6eqr 2333 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  (Scalar `  r
)  =  G )
5352opeq2d 3803 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. (Scalar `  ndx ) ,  (Scalar `  r ) >.  =  <. (Scalar `  ndx ) ,  G >. )
5452fveq2d 5529 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( Base `  (Scalar `  r )
)  =  ( Base `  G ) )
55 imasval.k . . . . . . . . . . . 12  |-  K  =  ( Base `  G
)
5654, 55syl6eqr 2333 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( Base `  (Scalar `  r )
)  =  K )
5721sneqd 3653 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  { (
f `  q ) }  =  { ( F `  q ) } )
5814fveq2d 5529 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( .s `  r )  =  ( .s `  R ) )
59 imasval.q . . . . . . . . . . . . . 14  |-  .x.  =  ( .s `  R )
6058, 59syl6eqr 2333 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( .s `  r )  =  .x.  )
6160oveqd 5875 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( p
( .s `  r
) q )  =  ( p  .x.  q
) )
626, 61fveq12d 5531 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( p ( .s
`  r ) q ) )  =  ( F `  ( p 
.x.  q ) ) )
6356, 57, 62mpt2eq123dv 5910 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( p  e.  ( Base `  (Scalar `  r ) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s
`  r ) q ) ) )  =  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) )
6463iuneq2d 3930 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  r ) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s
`  r ) q ) ) )  = 
U_ q  e.  V  ( p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) )
6519iuneq1d 3928 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ q  e.  v  ( p  e.  ( Base `  (Scalar `  r ) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s
`  r ) q ) ) )  = 
U_ q  e.  V  ( p  e.  ( Base `  (Scalar `  r
) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s `  r
) q ) ) ) )
66 imasval.s . . . . . . . . . 10  |-  ( ph  -> 
.(x)  =  U_ q  e.  V  ( p  e.  K ,  x  e. 
{ ( F `  q ) }  |->  ( F `  ( p 
.x.  q ) ) ) )
6766ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  .(x)  =  U_ q  e.  V  (
p  e.  K ,  x  e.  { ( F `  q ) }  |->  ( F `  ( p  .x.  q ) ) ) )
6864, 65, 673eqtr4d 2325 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ q  e.  v  ( p  e.  ( Base `  (Scalar `  r ) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s
`  r ) q ) ) )  = 
.(x)  )
6968opeq2d 3803 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( .s
`  ndx ) ,  U_ q  e.  v  (
p  e.  ( Base `  (Scalar `  r )
) ,  x  e. 
{ ( f `  q ) }  |->  ( f `  ( p ( .s `  r
) q ) ) ) >.  =  <. ( .s `  ndx ) ,  .(x)  >. )
7053, 69preq12d 3714 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  { <. (Scalar ` 
ndx ) ,  (Scalar `  r ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  v  ( p  e.  (
Base `  (Scalar `  r
) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s `  r
) q ) ) ) >. }  =  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. } )
7149, 70uneq12d 3330 . . . . 5  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( { <. ( Base `  ndx ) ,  ran  f >. ,  <. ( +g  `  ndx ) ,  U_ p  e.  v  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( +g  `  r
) q ) )
>. } >. ,  <. ( .r `  ndx ) , 
U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( .r
`  r ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  r ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  v  ( p  e.  (
Base `  (Scalar `  r
) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s `  r
) q ) ) ) >. } )  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. } ) )
7214fveq2d 5529 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( TopOpen `  r )  =  (
TopOpen `  R ) )
73 imasval.j . . . . . . . . . 10  |-  J  =  ( TopOpen `  R )
7472, 73syl6eqr 2333 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( TopOpen `  r )  =  J )
7574, 6oveq12d 5876 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( ( TopOpen
`  r ) qTop  f
)  =  ( J qTop 
F ) )
76 imasval.o . . . . . . . . 9  |-  ( ph  ->  O  =  ( J qTop 
F ) )
7776ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  O  =  ( J qTop  F )
)
7875, 77eqtr4d 2318 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( ( TopOpen
`  r ) qTop  f
)  =  O )
7978opeq2d 3803 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. (TopSet `  ndx ) ,  ( (
TopOpen `  r ) qTop  f
) >.  =  <. (TopSet ` 
ndx ) ,  O >. )
8014fveq2d 5529 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( le `  r )  =  ( le `  R ) )
81 imasval.n . . . . . . . . . . 11  |-  N  =  ( le `  R
)
8280, 81syl6eqr 2333 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( le `  r )  =  N )
836, 82coeq12d 4848 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f  o.  ( le `  r
) )  =  ( F  o.  N ) )
846cnveqd 4857 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  `' f  =  `' F )
8583, 84coeq12d 4848 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
f  o.  ( le
`  r ) )  o.  `' f )  =  ( ( F  o.  N )  o.  `' F ) )
86 imasval.l . . . . . . . . 9  |-  ( ph  -> 
.<_  =  ( ( F  o.  N )  o.  `' F ) )
8786ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  .<_  =  ( ( F  o.  N
)  o.  `' F
) )
8885, 87eqtr4d 2318 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
f  o.  ( le
`  r ) )  o.  `' f )  =  .<_  )
8988opeq2d 3803 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( le
`  ndx ) ,  ( ( f  o.  ( le `  r ) )  o.  `' f )
>.  =  <. ( le
`  ndx ) ,  .<_  >.
)
9019, 19xpeq12d 4714 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( v  X.  v )  =  ( V  X.  V ) )
9190oveq1d 5873 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
v  X.  v )  ^m  ( 1 ... n ) )  =  ( ( V  X.  V )  ^m  (
1 ... n ) ) )
926fveq1d 5527 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( 1st `  (
h `  1 )
) )  =  ( F `  ( 1st `  ( h `  1
) ) ) )
9392eqeq1d 2291 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
f `  ( 1st `  ( h `  1
) ) )  =  x  <->  ( F `  ( 1st `  ( h `
 1 ) ) )  =  x ) )
946fveq1d 5527 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( 2nd `  (
h `  n )
) )  =  ( F `  ( 2nd `  ( h `  n
) ) ) )
9594eqeq1d 2291 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
f `  ( 2nd `  ( h `  n
) ) )  =  y  <->  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y ) )
966fveq1d 5527 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( 2nd `  (
h `  i )
) )  =  ( F `  ( 2nd `  ( h `  i
) ) ) )
976fveq1d 5527 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( 1st `  (
h `  ( i  +  1 ) ) ) )  =  ( F `  ( 1st `  ( h `  (
i  +  1 ) ) ) ) )
9896, 97eqeq12d 2297 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) )  <-> 
( F `  ( 2nd `  ( h `  i ) ) )  =  ( F `  ( 1st `  ( h `
 ( i  +  1 ) ) ) ) ) )
9998ralbidv 2563 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) )  <->  A. i  e.  (
1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) )
10093, 95, 993anbi123d 1252 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
( f `  ( 1st `  ( h ` 
1 ) ) )  =  x  /\  (
f `  ( 2nd `  ( h `  n
) ) )  =  y  /\  A. i  e.  ( 1 ... (
n  -  1 ) ) ( f `  ( 2nd `  ( h `
 i ) ) )  =  ( f `
 ( 1st `  (
h `  ( i  +  1 ) ) ) ) )  <->  ( ( F `  ( 1st `  ( h `  1
) ) )  =  x  /\  ( F `
 ( 2nd `  (
h `  n )
) )  =  y  /\  A. i  e.  ( 1 ... (
n  -  1 ) ) ( F `  ( 2nd `  ( h `
 i ) ) )  =  ( F `
 ( 1st `  (
h `  ( i  +  1 ) ) ) ) ) ) )
10191, 100rabeqbidv 2783 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  { h  e.  ( ( v  X.  v )  ^m  (
1 ... n ) )  |  ( ( f `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( f `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  =  {
h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  ( ( F `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( h `  n
) ) )  =  y  /\  A. i  e.  ( 1 ... (
n  -  1 ) ) ( F `  ( 2nd `  ( h `
 i ) ) )  =  ( F `
 ( 1st `  (
h `  ( i  +  1 ) ) ) ) ) } )
10214fveq2d 5529 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( dist `  r )  =  (
dist `  R )
)
103 imasval.e . . . . . . . . . . . . . . . 16  |-  E  =  ( dist `  R
)
104102, 103syl6eqr 2333 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( dist `  r )  =  E )
105104coeq1d 4845 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( ( dist `  r )  o.  g )  =  ( E  o.  g ) )
106105oveq2d 5874 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) )  =  ( RR* s  gsumg  ( E  o.  g
) ) )
107101, 106mpteq12dv 4098 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( g  e.  { h  e.  ( ( v  X.  v
)  ^m  ( 1 ... n ) )  |  ( ( f `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( f `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) )  =  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( E  o.  g
) ) ) )
108107rneqd 4906 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ran  ( g  e.  { h  e.  ( ( v  X.  v )  ^m  (
1 ... n ) )  |  ( ( f `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( f `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) )  =  ran  ( g  e.  {
h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  ( ( F `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( h `  n
) ) )  =  y  /\  A. i  e.  ( 1 ... (
n  -  1 ) ) ( F `  ( 2nd `  ( h `
 i ) ) )  =  ( F `
 ( 1st `  (
h `  ( i  +  1 ) ) ) ) ) } 
|->  ( RR* s  gsumg  ( E  o.  g ) ) ) )
109108iuneq2d 3930 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( v  X.  v )  ^m  (
1 ... n ) )  |  ( ( f `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( f `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) )  =  U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  ( ( F `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( h `  n
) ) )  =  y  /\  A. i  e.  ( 1 ... (
n  -  1 ) ) ( F `  ( 2nd `  ( h `
 i ) ) )  =  ( F `
 ( 1st `  (
h `  ( i  +  1 ) ) ) ) ) } 
|->  ( RR* s  gsumg  ( E  o.  g ) ) ) )
110109supeq1d 7199 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  sup ( U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( v  X.  v )  ^m  ( 1 ... n ) )  |  ( ( f `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( f `  ( 2nd `  ( h `  n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) ) ,  RR* ,  `'  <  )  =  sup ( U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  ( ( F `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( h `  n
) ) )  =  y  /\  A. i  e.  ( 1 ... (
n  -  1 ) ) ( F `  ( 2nd `  ( h `
 i ) ) )  =  ( F `
 ( 1st `  (
h `  ( i  +  1 ) ) ) ) ) } 
|->  ( RR* s  gsumg  ( E  o.  g ) ) ) ,  RR* ,  `'  <  ) )
11112, 12, 110mpt2eq123dv 5910 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( x  e.  ran  f ,  y  e.  ran  f  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( v  X.  v )  ^m  (
1 ... n ) )  |  ( ( f `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( f `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) ) ,  RR* ,  `'  <  ) )  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( E  o.  g
) ) ) , 
RR* ,  `'  <  ) ) )
112 imasval.d . . . . . . . . 9  |-  ( ph  ->  D  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( V  X.  V )  ^m  ( 1 ... n ) )  |  ( ( F `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( F `  ( 2nd `  ( h `  n
) ) )  =  y  /\  A. i  e.  ( 1 ... (
n  -  1 ) ) ( F `  ( 2nd `  ( h `
 i ) ) )  =  ( F `
 ( 1st `  (
h `  ( i  +  1 ) ) ) ) ) } 
|->  ( RR* s  gsumg  ( E  o.  g ) ) ) ,  RR* ,  `'  <  ) ) )
113112ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  D  =  ( x  e.  B ,  y  e.  B  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( V  X.  V )  ^m  (
1 ... n ) )  |  ( ( F `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( F `  ( 2nd `  ( h `  i
) ) )  =  ( F `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( E  o.  g
) ) ) , 
RR* ,  `'  <  ) ) )
114111, 113eqtr4d 2318 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( x  e.  ran  f ,  y  e.  ran  f  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( v  X.  v )  ^m  (
1 ... n ) )  |  ( ( f `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( f `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) ) ,  RR* ,  `'  <  ) )  =  D )
115114opeq2d 3803 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( dist `  ndx ) ,  ( x  e.  ran  f ,  y  e.  ran  f  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( v  X.  v )  ^m  (
1 ... n ) )  |  ( ( f `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( f `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >.  =  <. ( dist `  ndx ) ,  D >. )
11679, 89, 115tpeq123d 3721 . . . . 5  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  { <. (TopSet ` 
ndx ) ,  ( ( TopOpen `  r ) qTop  f ) >. ,  <. ( le `  ndx ) ,  ( ( f  o.  ( le `  r ) )  o.  `' f ) >. ,  <. ( dist `  ndx ) ,  ( x  e.  ran  f ,  y  e.  ran  f  |->  sup ( U_ n  e.  NN  ran  ( g  e.  { h  e.  ( ( v  X.  v )  ^m  (
1 ... n ) )  |  ( ( f `
 ( 1st `  (
h `  1 )
) )  =  x  /\  ( f `  ( 2nd `  ( h `
 n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. }  =  { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } )
11771, 116uneq12d 3330 . . . 4  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( ( { <. ( Base `  ndx ) ,  ran  f >. ,  <. ( +g  `  ndx ) ,  U_ p  e.  v  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( f `  ( p ( +g  `  r
) q ) )
>. } >. ,  <. ( .r `  ndx ) , 
U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( .r
`  r ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  r ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  v  ( p  e.  (
Base `  (Scalar `  r
) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s `  r
) q ) ) ) >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  r ) qTop  f
) >. ,  <. ( le `  ndx ) ,  ( ( f  o.  ( le `  r
) )  o.  `' f ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ran  f ,  y  e.  ran  f  |->  sup ( U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( v  X.  v )  ^m  ( 1 ... n ) )  |  ( ( f `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( f `  ( 2nd `  ( h `  n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } )  =  ( ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } ) )
1185, 117csbied 3123 . . 3  |-  ( (
ph  /\  ( f  =  F  /\  r  =  R ) )  ->  [_ ( Base `  r
)  /  v ]_ ( ( { <. (
Base `  ndx ) ,  ran  f >. ,  <. ( +g  `  ndx ) ,  U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( +g  `  r ) q ) ) >. } >. ,  <. ( .r `  ndx ) ,  U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( f `  ( p ( .r
`  r ) q ) ) >. } >. }  u.  { <. (Scalar ` 
ndx ) ,  (Scalar `  r ) >. ,  <. ( .s `  ndx ) ,  U_ q  e.  v  ( p  e.  (
Base `  (Scalar `  r
) ) ,  x  e.  { ( f `  q ) }  |->  ( f `  ( p ( .s `  r
) q ) ) ) >. } )  u. 
{ <. (TopSet `  ndx ) ,  ( ( TopOpen
`  r ) qTop  f
) >. ,  <. ( le `  ndx ) ,  ( ( f  o.  ( le `  r
) )  o.  `' f ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ran  f ,  y  e.  ran  f  |->  sup ( U_ n  e.  NN  ran  ( g  e.  {
h  e.  ( ( v  X.  v )  ^m  ( 1 ... n ) )  |  ( ( f `  ( 1st `  ( h `
 1 ) ) )  =  x  /\  ( f `  ( 2nd `  ( h `  n ) ) )  =  y  /\  A. i  e.  ( 1 ... ( n  - 
1 ) ) ( f `  ( 2nd `  ( h `  i
) ) )  =  ( f `  ( 1st `  ( h `  ( i  +  1 ) ) ) ) ) }  |->  ( RR* s  gsumg  ( ( dist `  r
)  o.  g ) ) ) ,  RR* ,  `'  <  ) ) >. } )  =  ( ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } ) )
119 fof 5451 . . . . 5  |-  ( F : V -onto-> B  ->  F : V --> B )
1208, 119syl 15 . . . 4  |-  ( ph  ->  F : V --> B )
121 fvex 5539 . . . . 5  |-  ( Base `  R )  e.  _V
12217, 121syl6eqel 2371 . . . 4  |-  ( ph  ->  V  e.  _V )
123 fex 5749 . . . 4  |-  ( ( F : V --> B  /\  V  e.  _V )  ->  F  e.  _V )
124120, 122, 123syl2anc 642 . . 3  |-  ( ph  ->  F  e.  _V )
125 imasval.r . . . 4  |-  ( ph  ->  R  e.  Z )
126 elex 2796 . . . 4  |-  ( R  e.  Z  ->  R  e.  _V )
127125, 126syl 15 . . 3  |-  ( ph  ->  R  e.  _V )
128 tpex 4519 . . . . . 6  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.xb  >. }  e.  _V
129 prex 4217 . . . . . 6  |-  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) ,  .(x)  >. }  e.  _V
130128, 129unex 4518 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. } )  e. 
_V
131 tpex 4519 . . . . 5  |-  { <. (TopSet `  ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  e.  _V
132130, 131unex 4518 . . . 4  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } )  e. 
_V
133132a1i 10 . . 3  |-  ( ph  ->  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } )  e. 
_V )
1343, 118, 124, 127, 133ovmpt2d 5975 . 2  |-  ( ph  ->  ( F  "s  R )  =  ( ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } ) )
1351, 134eqtrd 2315 1  |-  ( ph  ->  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   [_csb 3081    u. cun 3150   {csn 3640   {cpr 3641   {ctp 3642   <.cop 3643   U_ciun 3905    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   ran crn 4690    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   supcsup 7193   1c1 8738    + caddc 8740   RR*cxr 8866    < clt 8867    - cmin 9037   NNcn 9746   ...cfz 10782   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212  TopSetcts 13214   lecple 13215   distcds 13217   TopOpenctopn 13326   RR* scxrs 13399    gsumg cgsu 13401   qTop cqtop 13406    "s cimas 13407
This theorem is referenced by:  imasbas  13415  imasds  13416  imasplusg  13420  imasmulr  13421  imassca  13422  imasvsca  13423  imastset  13424  imasle  13425
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-rep 4131  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-reu 2550  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-iun 3907  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-sup 7194  df-imas 13411
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