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Theorem imasds 15019
Description: The distance function 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
imasbas.u  |-  ( ph  ->  U  =  ( F 
"s  R ) )
imasbas.v  |-  ( ph  ->  V  =  ( Base `  R ) )
imasbas.f  |-  ( ph  ->  F : V -onto-> B
)
imasbas.r  |-  ( ph  ->  R  e.  Z )
imasds.e  |-  E  =  ( dist `  R
)
imasds.d  |-  D  =  ( dist `  U
)
Assertion
Ref Expression
imasds  |-  ( 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* ,  `'  <  ) ) )
Distinct variable groups:    g, h, i, n, x, y, F    R, g, h, i, n, x, y    x, U   
x, B, y    x, E, y    ph, g, h, i, n, x, y   
g, V, h
Allowed substitution hints:    B( g, h, i, n)    D( x, y, g, h, i, n)    U( y, g, h, i, n)    E( g, h, i, n)    V( x, y, i, n)    Z( x, y, g, h, i, n)

Proof of Theorem imasds
Dummy variables  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . 3  |-  ( ph  ->  U  =  ( F 
"s  R ) )
2 imasbas.v . . 3  |-  ( ph  ->  V  =  ( Base `  R ) )
3 eqid 2402 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2402 . . 3  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2402 . . 3  |-  (Scalar `  R )  =  (Scalar `  R )
6 eqid 2402 . . 3  |-  ( Base `  (Scalar `  R )
)  =  ( Base `  (Scalar `  R )
)
7 eqid 2402 . . 3  |-  ( .s
`  R )  =  ( .s `  R
)
8 eqid 2402 . . 3  |-  ( .i
`  R )  =  ( .i `  R
)
9 eqid 2402 . . 3  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
10 imasds.e . . 3  |-  E  =  ( dist `  R
)
11 eqid 2402 . . 3  |-  ( le
`  R )  =  ( le `  R
)
12 eqidd 2403 . . 3  |-  ( ph  ->  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 ( +g  `  R
) q ) )
>. } )
13 eqidd 2403 . . 3  |-  ( ph  ->  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 ( .r
`  R ) q ) ) >. } )
14 eqidd 2403 . . 3  |-  ( ph  ->  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 ) ) ) )
15 eqidd 2403 . . 3  |-  ( ph  ->  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 ( .i `  R ) q )
>. } )
16 eqidd 2403 . . 3  |-  ( ph  ->  ( ( TopOpen `  R
) qTop  F )  =  ( ( TopOpen `  R ) qTop  F ) )
17 eqidd 2403 . . 3  |-  ( ph  ->  ( 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* ,  `'  <  ) )  =  ( 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* ,  `'  <  ) ) )
18 eqidd 2403 . . 3  |-  ( ph  ->  ( ( F  o.  ( le `  R ) )  o.  `' F
)  =  ( ( F  o.  ( le
`  R ) )  o.  `' F ) )
19 imasbas.f . . 3  |-  ( ph  ->  F : V -onto-> B
)
20 imasbas.r . . 3  |-  ( ph  ->  R  e.  Z )
211, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20imasval 15017 . 2  |-  ( ph  ->  U  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +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.  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* ,  `'  <  ) ) >. } ) )
22 eqid 2402 . . 3  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +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.  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* ,  `'  <  ) ) >. } )  =  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +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.  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* ,  `'  <  ) ) >. } )
2322imasvalstr 14958 . 2  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +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.  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* ,  `'  <  ) ) >. } ) Struct  <. 1 , ; 1 2 >.
24 dsid 14909 . 2  |-  dist  = Slot  ( dist `  ndx )
25 snsstp3 4124 . . 3  |-  { <. (
dist `  ndx ) ,  ( 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* ,  `'  <  ) ) >. }  C_  { <. (TopSet `  ndx ) ,  ( ( TopOpen `  R ) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( 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* ,  `'  <  ) ) >. }
26 ssun2 3606 . . 3  |-  { <. (TopSet `  ndx ) ,  ( ( TopOpen `  R ) qTop  F ) >. ,  <. ( le `  ndx ) ,  ( ( F  o.  ( le `  R ) )  o.  `' F
) >. ,  <. ( dist `  ndx ) ,  ( 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* ,  `'  <  ) ) >. }  C_  (
( { <. ( Base `  ndx ) ,  B >. ,  <. ( +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.  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* ,  `'  <  ) ) >. } )
2725, 26sstri 3450 . 2  |-  { <. (
dist `  ndx ) ,  ( 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* ,  `'  <  ) ) >. }  C_  (
( { <. ( Base `  ndx ) ,  B >. ,  <. ( +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.  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* ,  `'  <  ) ) >. } )
28 fvex 5815 . . . . 5  |-  ( Base `  R )  e.  _V
292, 28syl6eqel 2498 . . . 4  |-  ( ph  ->  V  e.  _V )
30 fornex 6707 . . . 4  |-  ( V  e.  _V  ->  ( F : V -onto-> B  ->  B  e.  _V )
)
3129, 19, 30sylc 59 . . 3  |-  ( ph  ->  B  e.  _V )
32 mpt2exga 6814 . . 3  |-  ( ( B  e.  _V  /\  B  e.  _V )  ->  ( 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* ,  `'  <  ) )  e.  _V )
3331, 31, 32syl2anc 659 . 2  |-  ( ph  ->  ( 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* ,  `'  <  ) )  e.  _V )
34 imasds.d . 2  |-  D  =  ( dist `  U
)
3521, 23, 24, 27, 33, 34strfv3 14770 1  |-  ( 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* ,  `'  <  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842   A.wral 2753   {crab 2757   _Vcvv 3058    u. cun 3411   {csn 3971   {ctp 3975   <.cop 3977   U_ciun 4270    |-> cmpt 4452    X. cxp 4940   `'ccnv 4941   ran crn 4943    o. ccom 4946   -onto->wfo 5523   ` cfv 5525  (class class class)co 6234    |-> cmpt2 6236   1stc1st 6736   2ndc2nd 6737    ^m cmap 7377   supcsup 7854   1c1 9443    + caddc 9445   RR*cxr 9577    < clt 9578    - cmin 9761   NNcn 10496   2c2 10546  ;cdc 10939   ...cfz 11643   ndxcnx 14730   Basecbs 14733   +g cplusg 14801   .rcmulr 14802  Scalarcsca 14804   .scvsca 14805   .icip 14806  TopSetcts 14807   lecple 14808   distcds 14810   TopOpenctopn 14928    gsumg cgsu 14947   RR*scxrs 15006   qTop cqtop 15009    "s cimas 15010
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 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  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-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-lim 4826  df-suc 4827  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-om 6639  df-1st 6738  df-2nd 6739  df-recs 6999  df-rdg 7033  df-1o 7087  df-oadd 7091  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-fin 7478  df-sup 7855  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-nn 10497  df-2 10555  df-3 10556  df-4 10557  df-5 10558  df-6 10559  df-7 10560  df-8 10561  df-9 10562  df-10 10563  df-n0 10757  df-z 10826  df-dec 10940  df-uz 11046  df-fz 11644  df-struct 14735  df-ndx 14736  df-slot 14737  df-base 14738  df-plusg 14814  df-mulr 14815  df-sca 14817  df-vsca 14818  df-ip 14819  df-tset 14820  df-ple 14821  df-ds 14823  df-imas 15014
This theorem is referenced by:  imasdsfn  15020  imasdsval  15021  imasplusg  15023  imasmulr  15024  imassca  15025  imasvsca  15026  imasip  15027  imastset  15028  imasle  15029
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