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Theorem prdsval 13565
Description: Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
prdsval.p  |-  P  =  ( S X_s R )
prdsval.k  |-  K  =  ( Base `  S
)
prdsval.i  |-  ( ph  ->  dom  R  =  I )
prdsval.b  |-  ( ph  ->  B  =  X_ x  e.  I  ( Base `  ( R `  x
) ) )
prdsval.a  |-  ( ph  ->  .+  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x )
) ( g `  x ) ) ) ) )
prdsval.t  |-  ( ph  ->  .X.  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( .r `  ( R `  x )
) ( g `  x ) ) ) ) )
prdsval.m  |-  ( ph  ->  .x.  =  ( f  e.  K ,  g  e.  B  |->  ( x  e.  I  |->  ( f ( .s `  ( R `  x )
) ( g `  x ) ) ) ) )
prdsval.o  |-  ( ph  ->  O  =  ( Xt_ `  ( TopOpen  o.  R )
) )
prdsval.l  |-  ( ph  -> 
.<_  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
prdsval.d  |-  ( ph  ->  D  =  ( f  e.  B ,  g  e.  B  |->  sup (
( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) )
prdsval.h  |-  ( ph  ->  H  =  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
(  Hom  `  ( R `
 x ) ) ( g `  x
) ) ) )
prdsval.x  |-  ( ph  -> 
.xb  =  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a
) ) ,  e  e.  ( H `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) )
prdsval.s  |-  ( ph  ->  S  e.  W )
prdsval.r  |-  ( ph  ->  R  e.  Z )
Assertion
Ref Expression
prdsval  |-  ( ph  ->  P  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
Distinct variable groups:    a, c,
d, e, f, g, B    x, I    H, a, c, d, e    x, a, ph, c, d, e, f, g    R, a, c, d, e, f, g, x    S, a, c, d, e, f, g, x
Allowed substitution hints:    B( x)    D( x, e, f, g, a, c, d)    P( x, e, f, g, a, c, d)    .+ ( x, e, f, g, a, c, d)    .xb ( x, e, f, g, a, c, d)    .x. ( x, e, f, g, a, c, d)    .X. ( x, e, f, g, a, c, d)    H( x, f, g)    I( e, f, g, a, c, d)    K( x, e, f, g, a, c, d)    .<_ ( x, e, f, g, a, c, d)    O( x, e, f, g, a, c, d)    W( x, e, f, g, a, c, d)    Z( x, e, f, g, a, c, d)

Proof of Theorem prdsval
Dummy variables  h  r  s  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsval.p . 2  |-  P  =  ( S X_s R )
2 df-prds 13558 . . . 4  |-  X_s  =  (
s  e.  _V , 
r  e.  _V  |->  [_ X_ x  e.  dom  r
( Base `  ( r `  x ) )  / 
v ]_ [_ ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) ) )  /  h ]_ (
( { <. ( Base `  ndx ) ,  v >. ,  <. ( +g  `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( x  e.  dom  r  |->  ( ( f `
 x ) ( +g  `  ( r `
 x ) ) ( g `  x
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  s
>. ,  <. ( .s
`  ndx ) ,  ( f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) ) >. } )  u.  ( { <. (TopSet `  ndx ) ,  (
Xt_ `  ( TopOpen  o.  r
) ) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  (
r `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. (  Hom  `  ndx ) ,  h >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } ) ) )
32a1i 10 . . 3  |-  ( ph  -> 
X_s 
=  ( s  e. 
_V ,  r  e. 
_V  |->  [_ X_ x  e.  dom  r ( Base `  (
r `  x )
)  /  v ]_ [_ ( f  e.  v ,  g  e.  v 
|->  X_ x  e.  dom  r ( ( f `
 x ) (  Hom  `  ( r `  x ) ) ( g `  x ) ) )  /  h ]_ ( ( { <. (
Base `  ndx ) ,  v >. ,  <. ( +g  `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( x  e.  dom  r  |->  ( ( f `
 x ) ( +g  `  ( r `
 x ) ) ( g `  x
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  s
>. ,  <. ( .s
`  ndx ) ,  ( f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) ) >. } )  u.  ( { <. (TopSet `  ndx ) ,  (
Xt_ `  ( TopOpen  o.  r
) ) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  (
r `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. (  Hom  `  ndx ) ,  h >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } ) ) ) )
4 vex 2876 . . . . . . . . . . . 12  |-  r  e. 
_V
54rnex 5045 . . . . . . . . . . 11  |-  ran  r  e.  _V
65uniex 4619 . . . . . . . . . 10  |-  U. ran  r  e.  _V
76rnex 5045 . . . . . . . . 9  |-  ran  U. ran  r  e.  _V
87uniex 4619 . . . . . . . 8  |-  U. ran  U.
ran  r  e.  _V
9 baseid 13398 . . . . . . . . . . . 12  |-  Base  = Slot  ( Base `  ndx )
109strfvss 13374 . . . . . . . . . . 11  |-  ( Base `  ( r `  x
) )  C_  U. ran  ( r `  x
)
11 fvssunirn 5658 . . . . . . . . . . . 12  |-  ( r `
 x )  C_  U.
ran  r
12 rnss 5010 . . . . . . . . . . . 12  |-  ( ( r `  x ) 
C_  U. ran  r  ->  ran  ( r `  x
)  C_  ran  U. ran  r )
13 uniss 3950 . . . . . . . . . . . 12  |-  ( ran  ( r `  x
)  C_  ran  U. ran  r  ->  U. ran  ( r `
 x )  C_  U.
ran  U. ran  r )
1411, 12, 13mp2b 9 . . . . . . . . . . 11  |-  U. ran  ( r `  x
)  C_  U. ran  U. ran  r
1510, 14sstri 3274 . . . . . . . . . 10  |-  ( Base `  ( r `  x
) )  C_  U. ran  U.
ran  r
1615rgenw 2695 . . . . . . . . 9  |-  A. x  e.  dom  r ( Base `  ( r `  x
) )  C_  U. ran  U.
ran  r
17 iunss 4045 . . . . . . . . 9  |-  ( U_ x  e.  dom  r (
Base `  ( r `  x ) )  C_  U.
ran  U. ran  r  <->  A. x  e.  dom  r ( Base `  ( r `  x
) )  C_  U. ran  U.
ran  r )
1816, 17mpbir 200 . . . . . . . 8  |-  U_ x  e.  dom  r ( Base `  ( r `  x
) )  C_  U. ran  U.
ran  r
198, 18ssexi 4261 . . . . . . 7  |-  U_ x  e.  dom  r ( Base `  ( r `  x
) )  e.  _V
20 ixpssmap2g 6988 . . . . . . 7  |-  ( U_ x  e.  dom  r (
Base `  ( r `  x ) )  e. 
_V  ->  X_ x  e.  dom  r ( Base `  (
r `  x )
)  C_  ( U_ x  e.  dom  r (
Base `  ( r `  x ) )  ^m  dom  r ) )
2119, 20ax-mp 8 . . . . . 6  |-  X_ x  e.  dom  r ( Base `  ( r `  x
) )  C_  ( U_ x  e.  dom  r ( Base `  (
r `  x )
)  ^m  dom  r )
22 ovex 6006 . . . . . . 7  |-  ( U_ x  e.  dom  r (
Base `  ( r `  x ) )  ^m  dom  r )  e.  _V
2322ssex 4260 . . . . . 6  |-  ( X_ x  e.  dom  r (
Base `  ( r `  x ) )  C_  ( U_ x  e.  dom  r ( Base `  (
r `  x )
)  ^m  dom  r )  ->  X_ x  e.  dom  r ( Base `  (
r `  x )
)  e.  _V )
2421, 23mp1i 11 . . . . 5  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  dom  r ( Base `  ( r `  x
) )  e.  _V )
25 simpr 447 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  r  =  R )
2625dmeqd 4984 . . . . . . . . 9  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  dom  r  =  dom  R )
27 prdsval.i . . . . . . . . . 10  |-  ( ph  ->  dom  R  =  I )
2827ad2antrr 706 . . . . . . . . 9  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  dom  R  =  I )
2926, 28eqtrd 2398 . . . . . . . 8  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  dom  r  =  I )
3029ixpeq1d 6971 . . . . . . 7  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  dom  r ( Base `  ( r `  x
) )  =  X_ x  e.  I  ( Base `  ( r `  x ) ) )
3125fveq1d 5634 . . . . . . . . 9  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
r `  x )  =  ( R `  x ) )
3231fveq2d 5636 . . . . . . . 8  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( Base `  ( r `  x ) )  =  ( Base `  ( R `  x )
) )
3332ixpeq2dv 6975 . . . . . . 7  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  I  ( Base `  ( r `  x
) )  =  X_ x  e.  I  ( Base `  ( R `  x ) ) )
3430, 33eqtrd 2398 . . . . . 6  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  dom  r ( Base `  ( r `  x
) )  =  X_ x  e.  I  ( Base `  ( R `  x ) ) )
35 prdsval.b . . . . . . 7  |-  ( ph  ->  B  =  X_ x  e.  I  ( Base `  ( R `  x
) ) )
3635ad2antrr 706 . . . . . 6  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  B  =  X_ x  e.  I 
( Base `  ( R `  x ) ) )
3734, 36eqtr4d 2401 . . . . 5  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  dom  r ( Base `  ( r `  x
) )  =  B )
38 df-ov 5984 . . . . . . . . . . . . . . . 16  |-  ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  =  ( (  Hom  `  (
r `  x )
) `  <. ( f `
 x ) ,  ( g `  x
) >. )
39 fvssunirn 5658 . . . . . . . . . . . . . . . 16  |-  ( (  Hom  `  ( r `  x ) ) `  <. ( f `  x
) ,  ( g `
 x ) >.
)  C_  U. ran  (  Hom  `  ( r `  x ) )
4038, 39eqsstri 3294 . . . . . . . . . . . . . . 15  |-  ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  U.
ran  (  Hom  `  (
r `  x )
)
41 df-hom 13440 . . . . . . . . . . . . . . . . . 18  |-  Hom  = Slot ; 1 4
4241strfvss 13374 . . . . . . . . . . . . . . . . 17  |-  (  Hom  `  ( r `  x
) )  C_  U. ran  ( r `  x
)
4342, 14sstri 3274 . . . . . . . . . . . . . . . 16  |-  (  Hom  `  ( r `  x
) )  C_  U. ran  U.
ran  r
44 rnss 5010 . . . . . . . . . . . . . . . 16  |-  ( (  Hom  `  ( r `  x ) )  C_  U.
ran  U. ran  r  ->  ran  (  Hom  `  (
r `  x )
)  C_  ran  U. ran  U.
ran  r )
45 uniss 3950 . . . . . . . . . . . . . . . 16  |-  ( ran  (  Hom  `  (
r `  x )
)  C_  ran  U. ran  U.
ran  r  ->  U. ran  (  Hom  `  ( r `  x ) )  C_  U.
ran  U. ran  U. ran  r )
4643, 44, 45mp2b 9 . . . . . . . . . . . . . . 15  |-  U. ran  (  Hom  `  ( r `  x ) )  C_  U.
ran  U. ran  U. ran  r
4740, 46sstri 3274 . . . . . . . . . . . . . 14  |-  ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  U.
ran  U. ran  U. ran  r
4847rgenw 2695 . . . . . . . . . . . . 13  |-  A. x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  U.
ran  U. ran  U. ran  r
49 ss2ixp 6972 . . . . . . . . . . . . 13  |-  ( A. x  e.  dom  r ( ( f `  x
) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  U.
ran  U. ran  U. ran  r  ->  X_ x  e.  dom  r ( ( f `
 x ) (  Hom  `  ( r `  x ) ) ( g `  x ) )  C_  X_ x  e. 
dom  r U. ran  U.
ran  U. ran  r )
5048, 49ax-mp 8 . . . . . . . . . . . 12  |-  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  X_ x  e.  dom  r U. ran  U. ran  U. ran  r
514dmex 5044 . . . . . . . . . . . . 13  |-  dom  r  e.  _V
528rnex 5045 . . . . . . . . . . . . . 14  |-  ran  U. ran  U. ran  r  e. 
_V
5352uniex 4619 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  U. ran  r  e. 
_V
5451, 53ixpconst 6969 . . . . . . . . . . . 12  |-  X_ x  e.  dom  r U. ran  U.
ran  U. ran  r  =  ( U. ran  U. ran  U. ran  r  ^m  dom  r )
5550, 54sseqtri 3296 . . . . . . . . . . 11  |-  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  ( U. ran  U. ran  U.
ran  r  ^m  dom  r )
56 ovex 6006 . . . . . . . . . . . 12  |-  ( U. ran  U. ran  U. ran  r  ^m  dom  r )  e.  _V
5756elpw2 4277 . . . . . . . . . . 11  |-  ( X_ x  e.  dom  r ( ( f `  x
) (  Hom  `  (
r `  x )
) ( g `  x ) )  e. 
~P ( U. ran  U.
ran  U. ran  r  ^m  dom  r )  <->  X_ x  e. 
dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  ( U. ran  U. ran  U.
ran  r  ^m  dom  r ) )
5855, 57mpbir 200 . . . . . . . . . 10  |-  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  e. 
~P ( U. ran  U.
ran  U. ran  r  ^m  dom  r )
5958rgen2w 2696 . . . . . . . . 9  |-  A. f  e.  v  A. g  e.  v  X_ x  e. 
dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  e. 
~P ( U. ran  U.
ran  U. ran  r  ^m  dom  r )
60 eqid 2366 . . . . . . . . . 10  |-  ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) ) )  =  ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `
 x ) (  Hom  `  ( r `  x ) ) ( g `  x ) ) )
6160fmpt2 6318 . . . . . . . . 9  |-  ( A. f  e.  v  A. g  e.  v  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  e. 
~P ( U. ran  U.
ran  U. ran  r  ^m  dom  r )  <->  ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `
 x ) (  Hom  `  ( r `  x ) ) ( g `  x ) ) ) : ( v  X.  v ) --> ~P ( U. ran  U.
ran  U. ran  r  ^m  dom  r ) )
6259, 61mpbi 199 . . . . . . . 8  |-  ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) ) ) : ( v  X.  v ) --> ~P ( U. ran  U. ran  U. ran  r  ^m  dom  r
)
63 vex 2876 . . . . . . . . 9  |-  v  e. 
_V
6463, 63xpex 4904 . . . . . . . 8  |-  ( v  X.  v )  e. 
_V
6556pwex 4295 . . . . . . . 8  |-  ~P ( U. ran  U. ran  U. ran  r  ^m  dom  r
)  e.  _V
66 fex2 5507 . . . . . . . 8  |-  ( ( ( f  e.  v ,  g  e.  v 
|->  X_ x  e.  dom  r ( ( f `
 x ) (  Hom  `  ( r `  x ) ) ( g `  x ) ) ) : ( v  X.  v ) --> ~P ( U. ran  U.
ran  U. ran  r  ^m  dom  r )  /\  (
v  X.  v )  e.  _V  /\  ~P ( U. ran  U. ran  U.
ran  r  ^m  dom  r )  e.  _V )  ->  ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `
 x ) (  Hom  `  ( r `  x ) ) ( g `  x ) ) )  e.  _V )
6762, 64, 65, 66mp3an 1278 . . . . . . 7  |-  ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) ) )  e.  _V
6867a1i 10 . . . . . 6  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `  x
) (  Hom  `  (
r `  x )
) ( g `  x ) ) )  e.  _V )
69 simpr 447 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  v  =  B )
7029adantr 451 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  dom  r  =  I )
7170ixpeq1d 6971 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  = 
X_ x  e.  I 
( ( f `  x ) (  Hom  `  ( r `  x
) ) ( g `
 x ) ) )
7231fveq2d 5636 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (  Hom  `  ( r `  x ) )  =  (  Hom  `  ( R `  x )
) )
7372oveqd 5998 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) (  Hom  `  (
r `  x )
) ( g `  x ) )  =  ( ( f `  x ) (  Hom  `  ( R `  x
) ) ( g `
 x ) ) )
7473ixpeq2dv 6975 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  I  ( (
f `  x )
(  Hom  `  ( r `
 x ) ) ( g `  x
) )  =  X_ x  e.  I  (
( f `  x
) (  Hom  `  ( R `  x )
) ( g `  x ) ) )
7574adantr 451 . . . . . . . . 9  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  X_ x  e.  I  ( (
f `  x )
(  Hom  `  ( r `
 x ) ) ( g `  x
) )  =  X_ x  e.  I  (
( f `  x
) (  Hom  `  ( R `  x )
) ( g `  x ) ) )
7671, 75eqtrd 2398 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  = 
X_ x  e.  I 
( ( f `  x ) (  Hom  `  ( R `  x
) ) ( g `
 x ) ) )
7769, 69, 76mpt2eq123dv 6036 . . . . . . 7  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `  x
) (  Hom  `  (
r `  x )
) ( g `  x ) ) )  =  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I 
( ( f `  x ) (  Hom  `  ( R `  x
) ) ( g `
 x ) ) ) )
78 prdsval.h . . . . . . . 8  |-  ( ph  ->  H  =  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
(  Hom  `  ( R `
 x ) ) ( g `  x
) ) ) )
7978ad3antrrr 710 . . . . . . 7  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  H  =  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I 
( ( f `  x ) (  Hom  `  ( R `  x
) ) ( g `
 x ) ) ) )
8077, 79eqtr4d 2401 . . . . . 6  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `  x
) (  Hom  `  (
r `  x )
) ( g `  x ) ) )  =  H )
81 simplr 731 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  v  =  B )
8281opeq2d 3905 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. ( Base `  ndx ) ,  v >.  =  <. (
Base `  ndx ) ,  B >. )
8331fveq2d 5636 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( +g  `  ( r `  x ) )  =  ( +g  `  ( R `  x )
) )
8483oveqd 5998 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( +g  `  (
r `  x )
) ( g `  x ) )  =  ( ( f `  x ) ( +g  `  ( R `  x
) ) ( g `
 x ) ) )
8529, 84mpteq12dv 4200 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
x  e.  dom  r  |->  ( ( f `  x ) ( +g  `  ( r `  x
) ) ( g `
 x ) ) )  =  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x )
) ( g `  x ) ) ) )
8685adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
x  e.  dom  r  |->  ( ( f `  x ) ( +g  `  ( r `  x
) ) ( g `
 x ) ) )  =  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x )
) ( g `  x ) ) ) )
8769, 69, 86mpt2eq123dv 6036 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( +g  `  ( r `  x
) ) ( g `
 x ) ) ) )  =  ( f  e.  B , 
g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x
) ( +g  `  ( R `  x )
) ( g `  x ) ) ) ) )
8887adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( +g  `  ( r `  x
) ) ( g `
 x ) ) ) )  =  ( f  e.  B , 
g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x
) ( +g  `  ( R `  x )
) ( g `  x ) ) ) ) )
89 prdsval.a . . . . . . . . . . . 12  |-  ( ph  ->  .+  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x )
) ( g `  x ) ) ) ) )
9089ad4antr 712 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  .+  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x
) ) ( g `
 x ) ) ) ) )
9188, 90eqtr4d 2401 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( +g  `  ( r `  x
) ) ( g `
 x ) ) ) )  =  .+  )
9291opeq2d 3905 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. ( +g  `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( x  e.  dom  r  |->  ( ( f `
 x ) ( +g  `  ( r `
 x ) ) ( g `  x
) ) ) )
>.  =  <. ( +g  ` 
ndx ) ,  .+  >.
)
9331fveq2d 5636 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( .r `  ( r `  x ) )  =  ( .r `  ( R `  x )
) )
9493oveqd 5998 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( .r `  ( r `  x
) ) ( g `
 x ) )  =  ( ( f `
 x ) ( .r `  ( R `
 x ) ) ( g `  x
) ) )
9529, 94mpteq12dv 4200 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `  x
) ( .r `  ( R `  x ) ) ( g `  x ) ) ) )
9695adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `  x
) ( .r `  ( R `  x ) ) ( g `  x ) ) ) )
9769, 69, 96mpt2eq123dv 6036 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( .r
`  ( R `  x ) ) ( g `  x ) ) ) ) )
9897adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( .r
`  ( R `  x ) ) ( g `  x ) ) ) ) )
99 prdsval.t . . . . . . . . . . . 12  |-  ( ph  ->  .X.  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( .r `  ( R `  x )
) ( g `  x ) ) ) ) )
10099ad4antr 712 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  .X.  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( .r
`  ( R `  x ) ) ( g `  x ) ) ) ) )
10198, 100eqtr4d 2401 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) ) )  = 
.X.  )
102101opeq2d 3905 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. ( .r `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( x  e.  dom  r  |->  ( ( f `
 x ) ( .r `  ( r `
 x ) ) ( g `  x
) ) ) )
>.  =  <. ( .r
`  ndx ) ,  .X.  >.
)
10382, 92, 102tpeq123d 3813 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  { <. (
Base `  ndx ) ,  v >. ,  <. ( +g  `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( x  e.  dom  r  |->  ( ( f `
 x ) ( +g  `  ( r `
 x ) ) ( g `  x
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) ) ) >. }  =  { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. } )
104 simp-4r 743 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  s  =  S )
105104opeq2d 3905 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. (Scalar ` 
ndx ) ,  s
>.  =  <. (Scalar `  ndx ) ,  S >. )
106 simpllr 735 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  s  =  S )
107106fveq2d 5636 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( Base `  s )  =  ( Base `  S
) )
108 prdsval.k . . . . . . . . . . . . . 14  |-  K  =  ( Base `  S
)
109107, 108syl6eqr 2416 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( Base `  s )  =  K )
11031fveq2d 5636 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( .s `  ( r `  x ) )  =  ( .s `  ( R `  x )
) )
111110oveqd 5998 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
f ( .s `  ( r `  x
) ) ( g `
 x ) )  =  ( f ( .s `  ( R `
 x ) ) ( g `  x
) ) )
11229, 111mpteq12dv 4200 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
x  e.  dom  r  |->  ( f ( .s
`  ( r `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( f ( .s `  ( R `  x ) ) ( g `  x ) ) ) )
113112adantr 451 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
x  e.  dom  r  |->  ( f ( .s
`  ( r `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( f ( .s `  ( R `  x ) ) ( g `  x ) ) ) )
114109, 69, 113mpt2eq123dv 6036 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) )  =  ( f  e.  K , 
g  e.  B  |->  ( x  e.  I  |->  ( f ( .s `  ( R `  x ) ) ( g `  x ) ) ) ) )
115114adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) )  =  ( f  e.  K , 
g  e.  B  |->  ( x  e.  I  |->  ( f ( .s `  ( R `  x ) ) ( g `  x ) ) ) ) )
116 prdsval.m . . . . . . . . . . . 12  |-  ( ph  ->  .x.  =  ( f  e.  K ,  g  e.  B  |->  ( x  e.  I  |->  ( f ( .s `  ( R `  x )
) ( g `  x ) ) ) ) )
117116ad4antr 712 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  .x.  =  ( f  e.  K ,  g  e.  B  |->  ( x  e.  I  |->  ( f ( .s
`  ( R `  x ) ) ( g `  x ) ) ) ) )
118115, 117eqtr4d 2401 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) )  =  .x.  )
119118opeq2d 3905 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. ( .s `  ndx ) ,  ( f  e.  (
Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `
 x ) ) ( g `  x
) ) ) )
>.  =  <. ( .s
`  ndx ) ,  .x.  >.
)
120105, 119preq12d 3806 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  { <. (Scalar `  ndx ) ,  s
>. ,  <. ( .s
`  ndx ) ,  ( f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) ) >. }  =  { <. (Scalar `  ndx ) ,  S >. , 
<. ( .s `  ndx ) ,  .x.  >. } )
121103, 120uneq12d 3418 . . . . . . 7  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( { <. ( Base `  ndx ) ,  v >. , 
<. ( +g  `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e. 
dom  r  |->  ( ( f `  x ) ( +g  `  (
r `  x )
) ( g `  x ) ) ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( x  e.  dom  r  |->  ( ( f `
 x ) ( .r `  ( r `
 x ) ) ( g `  x
) ) ) )
>. }  u.  { <. (Scalar `  ndx ) ,  s
>. ,  <. ( .s
`  ndx ) ,  ( f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) ) >. } )  =  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } ) )
122 simpllr 735 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  r  =  R )
123122coeq2d 4949 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( TopOpen  o.  r )  =  ( TopOpen  o.  R )
)
124123fveq2d 5636 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( Xt_ `  ( TopOpen  o.  r
) )  =  (
Xt_ `  ( TopOpen  o.  R
) ) )
125 prdsval.o . . . . . . . . . . . 12  |-  ( ph  ->  O  =  ( Xt_ `  ( TopOpen  o.  R )
) )
126125ad4antr 712 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  O  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
127124, 126eqtr4d 2401 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( Xt_ `  ( TopOpen  o.  r
) )  =  O )
128127opeq2d 3905 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( TopOpen  o.  r
) ) >.  =  <. (TopSet `  ndx ) ,  O >. )
12969sseq2d 3292 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( { f ,  g }  C_  v  <->  { f ,  g }  C_  B ) )
13031fveq2d 5636 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( le `  ( r `  x ) )  =  ( le `  ( R `  x )
) )
131130breqd 4136 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( le `  ( r `  x
) ) ( g `
 x )  <->  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) ) )
13229, 131raleqbidv 2833 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( A. x  e.  dom  r ( f `  x ) ( le
`  ( r `  x ) ) ( g `  x )  <->  A. x  e.  I 
( f `  x
) ( le `  ( R `  x ) ) ( g `  x ) ) )
133132adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( A. x  e.  dom  r ( f `  x ) ( le
`  ( r `  x ) ) ( g `  x )  <->  A. x  e.  I 
( f `  x
) ( le `  ( R `  x ) ) ( g `  x ) ) )
134129, 133anbi12d 691 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
( { f ,  g }  C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le
`  ( r `  x ) ) ( g `  x ) )  <->  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) ) ) )
135134opabbidv 4184 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  { <. f ,  g >.  |  ( { f ,  g }  C_  v  /\  A. x  e.  dom  r
( f `  x
) ( le `  ( r `  x
) ) ( g `
 x ) ) }  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
136135adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  { <. f ,  g >.  |  ( { f ,  g }  C_  v  /\  A. x  e.  dom  r
( f `  x
) ( le `  ( r `  x
) ) ( g `
 x ) ) }  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
137 prdsval.l . . . . . . . . . . . 12  |-  ( ph  -> 
.<_  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
138137ad4antr 712 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  .<_  =  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  B  /\  A. x  e.  I  (
f `  x )
( le `  ( R `  x )
) ( g `  x ) ) } )
139136, 138eqtr4d 2401 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  { <. f ,  g >.  |  ( { f ,  g }  C_  v  /\  A. x  e.  dom  r
( f `  x
) ( le `  ( r `  x
) ) ( g `
 x ) ) }  =  .<_  )
140139opeq2d 3905 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  (
r `  x )
) ( g `  x ) ) }
>.  =  <. ( le
`  ndx ) ,  .<_  >.
)
14131fveq2d 5636 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( dist `  ( r `  x ) )  =  ( dist `  ( R `  x )
) )
142141oveqd 5998 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( dist `  (
r `  x )
) ( g `  x ) )  =  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )
14329, 142mpteq12dv 4200 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  =  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x )
) ( g `  x ) ) ) )
144143adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  =  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x )
) ( g `  x ) ) ) )
145144rneqd 5009 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ran  ( x  e.  dom  r  |->  ( ( f `
 x ) (
dist `  ( r `  x ) ) ( g `  x ) ) )  =  ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) ) )
146145uneq1d 3416 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( ran  ( x  e.  dom  r  |->  ( ( f `
 x ) (
dist `  ( r `  x ) ) ( g `  x ) ) )  u.  {
0 } )  =  ( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) )
147146supeq1d 7346 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  )  =  sup ( ( ran  (
x  e.  I  |->  ( ( f `  x
) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )
14869, 69, 147mpt2eq123dv 6036 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )  =  ( f  e.  B ,  g  e.  B  |->  sup ( ( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) )
149148adantr 451 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )  =  ( f  e.  B ,  g  e.  B  |->  sup ( ( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) )
150 prdsval.d . . . . . . . . . . . 12  |-  ( ph  ->  D  =  ( f  e.  B ,  g  e.  B  |->  sup (
( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x )
) ( g `  x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) )
151150ad4antr 712 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  D  =  ( f  e.  B ,  g  e.  B  |->  sup ( ( ran  ( x  e.  I  |->  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) )
152149, 151eqtr4d 2401 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) )  =  D )
153152opeq2d 3905 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. ( dist `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  sup ( ( ran  ( x  e.  dom  r  |->  ( ( f `
 x ) (
dist `  ( r `  x ) ) ( g `  x ) ) )  u.  {
0 } ) , 
RR* ,  <  ) )
>.  =  <. ( dist `  ndx ) ,  D >. )
154128, 140, 153tpeq123d 3813 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  { <. (TopSet `  ndx ) ,  (
Xt_ `  ( TopOpen  o.  r
) ) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  (
r `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  =  { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. } )
155 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  h  =  H )
156155opeq2d 3905 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. (  Hom  `  ndx ) ,  h >.  =  <. (  Hom  `  ndx ) ,  H >. )
15781, 81xpeq12d 4817 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
v  X.  v )  =  ( B  X.  B ) )
158155oveqd 5998 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
c h ( 2nd `  a ) )  =  ( c H ( 2nd `  a ) ) )
159155fveq1d 5634 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
h `  a )  =  ( H `  a ) )
16031fveq2d 5636 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (comp `  ( r `  x
) )  =  (comp `  ( R `  x
) ) )
161160oveqd 5998 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( <. ( ( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) )  =  ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) )
162161oveqd 5998 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) )  =  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) )
16329, 162mpteq12dv 4200 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
x  e.  dom  r  |->  ( ( d `  x ) ( <.
( ( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) )  =  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) )
164163ad2antrr 706 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
x  e.  dom  r  |->  ( ( d `  x ) ( <.
( ( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) )  =  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) )
165158, 159, 164mpt2eq123dv 6036 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) )  =  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
166157, 81, 165mpt2eq123dv 6036 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )  =  ( a  e.  ( B  X.  B
) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `  a
)  |->  ( x  e.  I  |->  ( ( d `
 x ) (
<. ( ( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) ) )
167 prdsval.x . . . . . . . . . . . 12  |-  ( ph  -> 
.xb  =  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a
) ) ,  e  e.  ( H `  a )  |->  ( x  e.  I  |->  ( ( d `  x ) ( <. ( ( 1st `  a ) `  x
) ,  ( ( 2nd `  a ) `
 x ) >.
(comp `  ( R `  x ) ) ( c `  x ) ) ( e `  x ) ) ) ) ) )
168167ad4antr 712 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  .xb  =  ( a  e.  ( B  X.  B ) ,  c  e.  B  |->  ( d  e.  ( c H ( 2nd `  a ) ) ,  e  e.  ( H `
 a )  |->  ( x  e.  I  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  ( R `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) ) )
169166, 168eqtr4d 2401 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )  =  .xb  )
170169opeq2d 3905 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. (comp ` 
ndx ) ,  ( a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>.  =  <. (comp `  ndx ) ,  .xb  >. )
171156, 170preq12d 3806 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  { <. (  Hom  `  ndx ) ,  h >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. }  =  { <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .xb  >. } )
172154, 171uneq12d 3418 . . . . . . 7  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( { <. (TopSet `  ndx ) ,  ( Xt_ `  ( TopOpen  o.  r )
) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  (
r `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. (  Hom  `  ndx ) ,  h >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } )  =  ( { <. (TopSet `  ndx ) ,  O >. , 
<. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) )
173121, 172uneq12d 3418 . . . . . 6  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
( { <. ( Base `  ndx ) ,  v >. ,  <. ( +g  `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( x  e.  dom  r  |->  ( ( f `
 x ) ( +g  `  ( r `
 x ) ) ( g `  x
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  s
>. ,  <. ( .s
`  ndx ) ,  ( f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) ) >. } )  u.  ( { <. (TopSet `  ndx ) ,  (
Xt_ `  ( TopOpen  o.  r
) ) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  (
r `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. (  Hom  `  ndx ) ,  h >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } ) )  =  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
17468, 80, 173csbied2 3210 . . . . 5  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  [_ (
f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `  x
) (  Hom  `  (
r `  x )
) ( g `  x ) ) )  /  h ]_ (
( { <. ( Base `  ndx ) ,  v >. ,  <. ( +g  `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( x  e.  dom  r  |->  ( ( f `
 x ) ( +g  `  ( r `
 x ) ) ( g `  x
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  s
>. ,  <. ( .s
`  ndx ) ,  ( f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) ) >. } )  u.  ( { <. (TopSet `  ndx ) ,  (
Xt_ `  ( TopOpen  o.  r
) ) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  (
r `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. (  Hom  `  ndx ) ,  h >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } ) )  =  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
17524, 37, 174csbied2 3210 . . . 4  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  [_ X_ x  e.  dom  r ( Base `  ( r `  x
) )  /  v ]_ [_ ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `
 x ) (  Hom  `  ( r `  x ) ) ( g `  x ) ) )  /  h ]_ ( ( { <. (
Base `  ndx ) ,  v >. ,  <. ( +g  `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( x  e.  dom  r  |->  ( ( f `
 x ) ( +g  `  ( r `
 x ) ) ( g `  x
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  s
>. ,  <. ( .s
`  ndx ) ,  ( f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) ) >. } )  u.  ( { <. (TopSet `  ndx ) ,  (
Xt_ `  ( TopOpen  o.  r
) ) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  (
r `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. (  Hom  `  ndx ) ,  h >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } ) )  =  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
176175anasss 628 . . 3  |-  ( (
ph  /\  ( s  =  S  /\  r  =  R ) )  ->  [_ X_ x  e.  dom  r ( Base `  (
r `  x )
)  /  v ]_ [_ ( f  e.  v ,  g  e.  v 
|->  X_ x  e.  dom  r ( ( f `
 x ) (  Hom  `  ( r `  x ) ) ( g `  x ) ) )  /  h ]_ ( ( { <. (
Base `  ndx ) ,  v >. ,  <. ( +g  `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( x  e.  dom  r  |->  ( ( f `
 x ) ( +g  `  ( r `
 x ) ) ( g `  x
) ) ) )
>. ,  <. ( .r
`  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( ( f `  x ) ( .r
`  ( r `  x ) ) ( g `  x ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  s
>. ,  <. ( .s
`  ndx ) ,  ( f  e.  ( Base `  s ) ,  g  e.  v  |->  ( x  e.  dom  r  |->  ( f ( .s `  ( r `  x
) ) ( g `
 x ) ) ) ) >. } )  u.  ( { <. (TopSet `  ndx ) ,  (
Xt_ `  ( TopOpen  o.  r
) ) >. ,  <. ( le `  ndx ) ,  { <. f ,  g
>.  |  ( {
f ,  g } 
C_  v  /\  A. x  e.  dom  r ( f `  x ) ( le `  (
r `  x )
) ( g `  x ) ) }
>. ,  <. ( dist `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  sup ( ( ran  (
x  e.  dom  r  |->  ( ( f `  x ) ( dist `  ( r `  x
) ) ( g `
 x ) ) )  u.  { 0 } ) ,  RR* ,  <  ) ) >. }  u.  { <. (  Hom  `  ndx ) ,  h >. ,  <. (comp ` 
ndx ) ,  ( a  e.  ( v  X.  v ) ,  c  e.  v  |->  ( d  e.  ( c h ( 2nd `  a
) ) ,  e  e.  ( h `  a )  |->  ( x  e.  dom  r  |->  ( ( d `  x
) ( <. (
( 1st `  a
) `  x ) ,  ( ( 2nd `  a ) `  x
) >. (comp `  (
r `  x )
) ( c `  x ) ) ( e `  x ) ) ) ) )
>. } ) )  =  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
177 prdsval.s . . . 4  |-  ( ph  ->  S  e.  W )
178 elex 2881 . . . 4  |-  ( S  e.  W  ->  S  e.  _V )
179177, 178syl 15 . . 3  |-  ( ph  ->  S  e.  _V )
180 prdsval.r . . . 4  |-  ( ph  ->  R  e.  Z )
181 elex 2881 . . . 4  |-  ( R  e.  Z  ->  R  e.  _V )
182180, 181syl 15 . . 3  |-  ( ph  ->  R  e.  _V )
183 tpex 4622 . . . . . 6  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  e.  _V
184 prex 4319 . . . . . 6  |-  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  .x.  >. }  e.  _V
185183, 184unex 4621 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  e. 
_V
186 tpex 4622 . . . . . 6  |-  { <. (TopSet `  ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  e.  _V
187 prex 4319 . . . . . 6  |-  { <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .xb  >. }  e.  _V
188186, 187unex 4621 . . . . 5  |-  ( {
<. (TopSet `  ndx ) ,  O >. ,  <. ( le `  ndx ) , 
.<_  >. ,  <. ( dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } )  e. 
_V
189185, 188unex 4621 . . . 4  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) )  e.  _V
190189a1i 10 . . 3  |-  ( ph  ->  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) )  e.  _V )
1913, 176, 179, 182, 190ovmpt2d 6101 . 2  |-  ( ph  ->  ( S X_s R )  =  ( ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
1921, 191syl5eq 2410 1  |-  ( ph  ->  P  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   _Vcvv 2873   [_csb 3167    u. cun 3236    C_ wss 3238   ~Pcpw 3714   {csn 3729   {cpr 3730   {ctp 3731   <.cop 3732   U.cuni 3929   U_ciun 4007   class class class wbr 4125   {copab 4178    e. cmpt 4179    X. cxp 4790   dom cdm 4792   ran crn 4793    o. ccom 4796   -->wf 5354   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983   1stc1st 6247   2ndc2nd 6248    ^m cmap 6915   X_cixp 6960   supcsup 7340   0cc0 8884   1c1 8885   RR*cxr 9013    < clt 9014   4c4 9944  ;cdc 10275   ndxcnx 13353   Basecbs 13356   +g cplusg 13416   .rcmulr 13417  Scalarcsca 13419   .scvsca 13420  TopSetcts 13422   lecple 13423   distcds 13425    Hom chom 13427  compcco 13428   TopOpenctopn 13536   Xt_cpt 13553   X_scprds 13556
This theorem is referenced by:  prdssca  13566  prdsbas  13567  prdsplusg  13568  prdsmulr  13569  prdsvsca  13570  prdsle  13571  prdsds  13573  prdstset  13575  prdshom  13576  prdsco  13577
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-i2m1 8952  ax-1ne0 8953  ax-rrecex 8956  ax-cnre 8957
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-1st 6249  df-2nd 6250  df-recs 6530  df-rdg 6565  df-map 6917  df-ixp 6961  df-sup 7341  df-nn 9894  df-ndx 13359  df-slot 13360  df-base 13361  df-hom 13440  df-prds 13558
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