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Theorem imasval 15125
Description: Value of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
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.i  |-  .,  =  ( .i `  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.w  |-  ( ph  ->  I  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  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)  >. ,  <. ( .i `  ndx ) ,  I >. } )  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)    ., ( x, y, g, h, i, n, q, p)    I( 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 15122 . . . 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 ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( p
( .i `  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 11 . . 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 ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( p
( .i `  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 5859 . . . . 5  |-  ( Base `  r )  e.  _V
54a1i 11 . . . 4  |-  ( (
ph  /\  ( f  =  F  /\  r  =  R ) )  -> 
( Base `  r )  e.  _V )
6 simplrl 762 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  f  =  F )
76rneqd 5051 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ran  f  =  ran  F )
8 imasval.f . . . . . . . . . . 11  |-  ( ph  ->  F : V -onto-> B
)
9 forn 5781 . . . . . . . . . . 11  |-  ( F : V -onto-> B  ->  ran  F  =  B )
108, 9syl 17 . . . . . . . . . 10  |-  ( ph  ->  ran  F  =  B )
1110ad2antrr 724 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ran  F  =  B )
127, 11eqtrd 2443 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ran  f  =  B )
1312opeq2d 4166 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( Base `  ndx ) ,  ran  f >.  =  <. ( Base `  ndx ) ,  B >. )
14 simplrr 763 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  r  =  R )
1514fveq2d 5853 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( Base `  r )  =  (
Base `  R )
)
16 simpr 459 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  v  =  ( Base `  r )
)
17 imasval.v . . . . . . . . . . . 12  |-  ( ph  ->  V  =  ( Base `  R ) )
1817ad2antrr 724 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  V  =  ( Base `  R )
)
1915, 16, 183eqtr4d 2453 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  v  =  V )
206fveq1d 5851 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  p )  =  ( F `  p ) )
216fveq1d 5851 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  q )  =  ( F `  q ) )
2220, 21opeq12d 4167 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( f `
 p ) ,  ( f `  q
) >.  =  <. ( F `  p ) ,  ( F `  q ) >. )
2314fveq2d 5853 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( +g  `  r )  =  ( +g  `  R ) )
24 imasval.p . . . . . . . . . . . . . . . 16  |-  .+  =  ( +g  `  R )
2523, 24syl6eqr 2461 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( +g  `  r )  =  .+  )
2625oveqd 6295 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( p
( +g  `  r ) q )  =  ( p  .+  q ) )
276, 26fveq12d 5855 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( p ( +g  `  r ) q ) )  =  ( F `
 ( p  .+  q ) ) )
2822, 27opeq12d 4167 . . . . . . . . . . . 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 3984 . . . . . . . . . . 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 4297 . . . . . . . . . 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 4297 . . . . . . . . 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 724 . . . . . . . . 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 2446 . . . . . . . 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 4166 . . . . . . 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 5853 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( .r `  r )  =  ( .r `  R ) )
37 imasval.m . . . . . . . . . . . . . . . 16  |-  .X.  =  ( .r `  R )
3836, 37syl6eqr 2461 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( .r `  r )  =  .X.  )
3938oveqd 6295 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( p
( .r `  r
) q )  =  ( p  .X.  q
) )
406, 39fveq12d 5855 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( p ( .r
`  r ) q ) )  =  ( F `  ( p 
.X.  q ) ) )
4122, 40opeq12d 4167 . . . . . . . . . . . 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 3984 . . . . . . . . . . 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 4297 . . . . . . . . . 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 4297 . . . . . . . . 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 724 . . . . . . . . 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 2446 . . . . . . . 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 4166 . . . . . . 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 4066 . . . . . 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 5853 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  (Scalar `  r
)  =  (Scalar `  R ) )
51 imasval.g . . . . . . . . 9  |-  G  =  (Scalar `  R )
5250, 51syl6eqr 2461 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  (Scalar `  r
)  =  G )
5352opeq2d 4166 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. (Scalar `  ndx ) ,  (Scalar `  r ) >.  =  <. (Scalar `  ndx ) ,  G >. )
5452fveq2d 5853 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( Base `  (Scalar `  r )
)  =  ( Base `  G ) )
55 imasval.k . . . . . . . . . . . 12  |-  K  =  ( Base `  G
)
5654, 55syl6eqr 2461 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( Base `  (Scalar `  r )
)  =  K )
5721sneqd 3984 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  { (
f `  q ) }  =  { ( F `  q ) } )
5814fveq2d 5853 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( .s `  r )  =  ( .s `  R ) )
59 imasval.q . . . . . . . . . . . . . 14  |-  .x.  =  ( .s `  R )
6058, 59syl6eqr 2461 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( .s `  r )  =  .x.  )
6160oveqd 6295 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( p
( .s `  r
) q )  =  ( p  .x.  q
) )
626, 61fveq12d 5855 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( p ( .s
`  r ) q ) )  =  ( F `  ( p 
.x.  q ) ) )
6356, 57, 62mpt2eq123dv 6340 . . . . . . . . . 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 4298 . . . . . . . . 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 4296 . . . . . . . . 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 724 . . . . . . . . 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 2453 . . . . . . . 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 4166 . . . . . . 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)  >. )
7014fveq2d 5853 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( .i `  r )  =  ( .i `  R ) )
71 imasval.i . . . . . . . . . . . . . . 15  |-  .,  =  ( .i `  R )
7270, 71syl6eqr 2461 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( .i `  r )  =  .,  )
7372oveqd 6295 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( p
( .i `  r
) q )  =  ( p  .,  q
) )
7422, 73opeq12d 4167 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( p ( .i `  r ) q )
>.  =  <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. )
7574sneqd 3984 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( p ( .i `  r ) q )
>. }  =  { <. <.
( F `  p
) ,  ( F `
 q ) >. ,  ( p  .,  q ) >. } )
7619, 75iuneq12d 4297 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( p ( .i `  r ) q )
>. }  =  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } )
7719, 76iuneq12d 4297 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ p  e.  v  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( p ( .i `  r ) q )
>. }  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } )
78 imasval.w . . . . . . . . . 10  |-  ( ph  ->  I  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( p  .,  q )
>. } )
7978ad2antrr 724 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  I  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( p  .,  q ) >. } )
8077, 79eqtr4d 2446 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  U_ p  e.  v  U_ q  e.  v  { <. <. (
f `  p ) ,  ( f `  q ) >. ,  ( p ( .i `  r ) q )
>. }  =  I )
8180opeq2d 4166 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( .i
`  ndx ) ,  U_ p  e.  v  U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( p
( .i `  r
) q ) >. } >.  =  <. ( .i `  ndx ) ,  I >. )
8253, 69, 81tpeq123d 4066 . . . . . 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 ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( p
( .i `  r
) q ) >. } >. }  =  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. ,  <. ( .i `  ndx ) ,  I >. } )
8349, 82uneq12d 3598 . . . . 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 ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( p
( .i `  r
) q ) >. } >. } )  =  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. ,  <. ( .i `  ndx ) ,  I >. } ) )
8414fveq2d 5853 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( TopOpen `  r )  =  (
TopOpen `  R ) )
85 imasval.j . . . . . . . . . 10  |-  J  =  ( TopOpen `  R )
8684, 85syl6eqr 2461 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( TopOpen `  r )  =  J )
8786, 6oveq12d 6296 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( ( TopOpen
`  r ) qTop  f
)  =  ( J qTop 
F ) )
88 imasval.o . . . . . . . . 9  |-  ( ph  ->  O  =  ( J qTop 
F ) )
8988ad2antrr 724 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  O  =  ( J qTop  F )
)
9087, 89eqtr4d 2446 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( ( TopOpen
`  r ) qTop  f
)  =  O )
9190opeq2d 4166 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. (TopSet `  ndx ) ,  ( (
TopOpen `  r ) qTop  f
) >.  =  <. (TopSet ` 
ndx ) ,  O >. )
9214fveq2d 5853 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( le `  r )  =  ( le `  R ) )
93 imasval.n . . . . . . . . . . 11  |-  N  =  ( le `  R
)
9492, 93syl6eqr 2461 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( le `  r )  =  N )
956, 94coeq12d 4988 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f  o.  ( le `  r
) )  =  ( F  o.  N ) )
966cnveqd 4999 . . . . . . . . 9  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  `' f  =  `' F )
9795, 96coeq12d 4988 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
f  o.  ( le
`  r ) )  o.  `' f )  =  ( ( F  o.  N )  o.  `' F ) )
98 imasval.l . . . . . . . . 9  |-  ( ph  -> 
.<_  =  ( ( F  o.  N )  o.  `' F ) )
9998ad2antrr 724 . . . . . . . 8  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  .<_  =  ( ( F  o.  N
)  o.  `' F
) )
10097, 99eqtr4d 2446 . . . . . . 7  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
f  o.  ( le
`  r ) )  o.  `' f )  =  .<_  )
101100opeq2d 4166 . . . . . 6  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  <. ( le
`  ndx ) ,  ( ( f  o.  ( le `  r ) )  o.  `' f )
>.  =  <. ( le
`  ndx ) ,  .<_  >.
)
10219sqxpeqd 4849 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( v  X.  v )  =  ( V  X.  V ) )
103102oveq1d 6293 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
v  X.  v )  ^m  ( 1 ... n ) )  =  ( ( V  X.  V )  ^m  (
1 ... n ) ) )
1046fveq1d 5851 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( 1st `  (
h `  1 )
) )  =  ( F `  ( 1st `  ( h `  1
) ) ) )
105104eqeq1d 2404 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
f `  ( 1st `  ( h `  1
) ) )  =  x  <->  ( F `  ( 1st `  ( h `
 1 ) ) )  =  x ) )
1066fveq1d 5851 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( 2nd `  (
h `  n )
) )  =  ( F `  ( 2nd `  ( h `  n
) ) ) )
107106eqeq1d 2404 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( (
f `  ( 2nd `  ( h `  n
) ) )  =  y  <->  ( F `  ( 2nd `  ( h `
 n ) ) )  =  y ) )
1086fveq1d 5851 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( 2nd `  (
h `  i )
) )  =  ( F `  ( 2nd `  ( h `  i
) ) ) )
1096fveq1d 5851 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( f `  ( 1st `  (
h `  ( i  +  1 ) ) ) )  =  ( F `  ( 1st `  ( h `  (
i  +  1 ) ) ) ) )
110108, 109eqeq12d 2424 . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
111110ralbidv 2843 . . . . . . . . . . . . . . 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 ) ) ) ) ) )
112105, 107, 1113anbi123d 1301 . . . . . . . . . . . . . 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 ) ) ) ) ) ) )
113103, 112rabeqbidv 3054 . . . . . . . . . . . . 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 ) ) ) ) ) } )
11414fveq2d 5853 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( dist `  r )  =  (
dist `  R )
)
115 imasval.e . . . . . . . . . . . . . . . 16  |-  E  =  ( dist `  R
)
116114, 115syl6eqr 2461 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( dist `  r )  =  E )
117116coeq1d 4985 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( ( dist `  r )  o.  g )  =  ( E  o.  g ) )
118117oveq2d 6294 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
f  =  F  /\  r  =  R )
)  /\  v  =  ( Base `  r )
)  ->  ( RR*s  gsumg  ( ( dist `  r
)  o.  g ) )  =  ( RR*s  gsumg  ( E  o.  g
) ) )
119113, 118mpteq12dv 4473 . . . . . . . . . . . 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
) ) ) )
120119rneqd 5051 . . . . . . . . . . 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 ) ) ) )
121120iuneq2d 4298 . . . . . . . . . 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 ) ) ) )
122121supeq1d 7939 . . . . . . . . 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* ,  `'  <  ) )
12312, 12, 122mpt2eq123dv 6340 . . . . . . . 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* ,  `'  <  ) ) )
124 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* ,  `'  <  ) ) )
125124ad2antrr 724 . . . . . . . 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* ,  `'  <  ) ) )
126123, 125eqtr4d 2446 . . . . . . 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 )
127126opeq2d 4166 . . . . . 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 >. )
12891, 101, 127tpeq123d 4066 . . . . 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 >. } )
12983, 128uneq12d 3598 . . . 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 ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( p
( .i `  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)  >. ,  <. ( .i `  ndx ) ,  I >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } ) )
1305, 129csbied 3400 . . 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 ) ) ) >. ,  <. ( .i `  ndx ) , 
U_ p  e.  v 
U_ q  e.  v  { <. <. ( f `  p ) ,  ( f `  q )
>. ,  ( p
( .i `  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)  >. ,  <. ( .i `  ndx ) ,  I >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } ) )
131 fof 5778 . . . . 5  |-  ( F : V -onto-> B  ->  F : V --> B )
1328, 131syl 17 . . . 4  |-  ( ph  ->  F : V --> B )
133 fvex 5859 . . . . 5  |-  ( Base `  R )  e.  _V
13417, 133syl6eqel 2498 . . . 4  |-  ( ph  ->  V  e.  _V )
135 fex 6126 . . . 4  |-  ( ( F : V --> B  /\  V  e.  _V )  ->  F  e.  _V )
136132, 134, 135syl2anc 659 . . 3  |-  ( ph  ->  F  e.  _V )
137 imasval.r . . . 4  |-  ( ph  ->  R  e.  Z )
138 elex 3068 . . . 4  |-  ( R  e.  Z  ->  R  e.  _V )
139137, 138syl 17 . . 3  |-  ( ph  ->  R  e.  _V )
140 tpex 6581 . . . . . 6  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.xb  >. }  e.  _V
141 tpex 6581 . . . . . 6  |-  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) ,  .(x)  >. ,  <. ( .i `  ndx ) ,  I >. }  e.  _V
142140, 141unex 6580 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. ,  <. ( .i `  ndx ) ,  I >. } )  e. 
_V
143 tpex 6581 . . . . 5  |-  { <. (TopSet `  ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  e.  _V
144142, 143unex 6580 . . . 4  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. ,  <. ( .i `  ndx ) ,  I >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } )  e. 
_V
145144a1i 11 . . 3  |-  ( ph  ->  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+b  >. ,  <. ( .r `  ndx ) , 
.xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. ,  <. ( .i `  ndx ) ,  I >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } )  e. 
_V )
1463, 130, 136, 139, 145ovmpt2d 6411 . 2  |-  ( ph  ->  ( F  "s  R )  =  ( ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. ,  <. ( .i `  ndx ) ,  I >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } ) )
1471, 146eqtrd 2443 1  |-  ( ph  ->  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .xb  >. }  u.  { <. (Scalar `  ndx ) ,  G >. ,  <. ( .s `  ndx ) , 
.(x)  >. ,  <. ( .i `  ndx ) ,  I >. } )  u. 
{ <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2754   {crab 2758   _Vcvv 3059   [_csb 3373    u. cun 3412   {csn 3972   {ctp 3976   <.cop 3978   U_ciun 4271    |-> cmpt 4453    X. cxp 4821   `'ccnv 4822   ran crn 4824    o. ccom 4827   -->wf 5565   -onto->wfo 5567   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280   1stc1st 6782   2ndc2nd 6783    ^m cmap 7457   supcsup 7934   1c1 9523    + caddc 9525   RR*cxr 9657    < clt 9658    - cmin 9841   NNcn 10576   ...cfz 11726   ndxcnx 14838   Basecbs 14841   +g cplusg 14909   .rcmulr 14910  Scalarcsca 14912   .scvsca 14913   .icip 14914  TopSetcts 14915   lecple 14916   distcds 14918   TopOpenctopn 15036    gsumg cgsu 15055   RR*scxrs 15114   qTop cqtop 15117    "s cimas 15118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-sup 7935  df-imas 15122
This theorem is referenced by:  imasbas  15126  imasds  15127  imasplusg  15131  imasmulr  15132  imassca  15133  imasvsca  15134  imasip  15135  imastset  15136  imasle  15137
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