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Theorem prdsval 13633
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 13626 . . . 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 11 . . 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 2919 . . . . . . . . . . . 12  |-  r  e. 
_V
54rnex 5092 . . . . . . . . . . 11  |-  ran  r  e.  _V
65uniex 4664 . . . . . . . . . 10  |-  U. ran  r  e.  _V
76rnex 5092 . . . . . . . . 9  |-  ran  U. ran  r  e.  _V
87uniex 4664 . . . . . . . 8  |-  U. ran  U.
ran  r  e.  _V
9 baseid 13466 . . . . . . . . . . . 12  |-  Base  = Slot  ( Base `  ndx )
109strfvss 13442 . . . . . . . . . . 11  |-  ( Base `  ( r `  x
) )  C_  U. ran  ( r `  x
)
11 fvssunirn 5713 . . . . . . . . . . . 12  |-  ( r `
 x )  C_  U.
ran  r
12 rnss 5057 . . . . . . . . . . . 12  |-  ( ( r `  x ) 
C_  U. ran  r  ->  ran  ( r `  x
)  C_  ran  U. ran  r )
13 uniss 3996 . . . . . . . . . . . 12  |-  ( ran  ( r `  x
)  C_  ran  U. ran  r  ->  U. ran  ( r `
 x )  C_  U.
ran  U. ran  r )
1411, 12, 13mp2b 10 . . . . . . . . . . 11  |-  U. ran  ( r `  x
)  C_  U. ran  U. ran  r
1510, 14sstri 3317 . . . . . . . . . 10  |-  ( Base `  ( r `  x
) )  C_  U. ran  U.
ran  r
1615rgenw 2733 . . . . . . . . 9  |-  A. x  e.  dom  r ( Base `  ( r `  x
) )  C_  U. ran  U.
ran  r
17 iunss 4092 . . . . . . . . 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 201 . . . . . . . 8  |-  U_ x  e.  dom  r ( Base `  ( r `  x
) )  C_  U. ran  U.
ran  r
198, 18ssexi 4308 . . . . . . 7  |-  U_ x  e.  dom  r ( Base `  ( r `  x
) )  e.  _V
20 ixpssmap2g 7050 . . . . . . 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 6065 . . . . . . 7  |-  ( U_ x  e.  dom  r (
Base `  ( r `  x ) )  ^m  dom  r )  e.  _V
2322ssex 4307 . . . . . 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 12 . . . . 5  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  dom  r ( Base `  ( r `  x
) )  e.  _V )
25 simpr 448 . . . . . . . . 9  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  r  =  R )
2625fveq1d 5689 . . . . . . . 8  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
r `  x )  =  ( R `  x ) )
2726fveq2d 5691 . . . . . . 7  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( Base `  ( r `  x ) )  =  ( Base `  ( R `  x )
) )
2827ixpeq2dv 7037 . . . . . 6  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  I  ( Base `  ( r `  x
) )  =  X_ x  e.  I  ( Base `  ( R `  x ) ) )
2925dmeqd 5031 . . . . . . . 8  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  dom  r  =  dom  R )
30 prdsval.i . . . . . . . . 9  |-  ( ph  ->  dom  R  =  I )
3130ad2antrr 707 . . . . . . . 8  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  dom  R  =  I )
3229, 31eqtrd 2436 . . . . . . 7  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  dom  r  =  I )
3332ixpeq1d 7033 . . . . . 6  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  dom  r ( Base `  ( r `  x
) )  =  X_ x  e.  I  ( Base `  ( r `  x ) ) )
34 prdsval.b . . . . . . 7  |-  ( ph  ->  B  =  X_ x  e.  I  ( Base `  ( R `  x
) ) )
3534ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  B  =  X_ x  e.  I 
( Base `  ( R `  x ) ) )
3628, 33, 353eqtr4d 2446 . . . . 5  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  dom  r ( Base `  ( r `  x
) )  =  B )
37 ovssunirn 6066 . . . . . . . . . . . . . . 15  |-  ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  U.
ran  (  Hom  `  (
r `  x )
)
38 df-hom 13508 . . . . . . . . . . . . . . . . . 18  |-  Hom  = Slot ; 1 4
3938strfvss 13442 . . . . . . . . . . . . . . . . 17  |-  (  Hom  `  ( r `  x
) )  C_  U. ran  ( r `  x
)
4039, 14sstri 3317 . . . . . . . . . . . . . . . 16  |-  (  Hom  `  ( r `  x
) )  C_  U. ran  U.
ran  r
41 rnss 5057 . . . . . . . . . . . . . . . 16  |-  ( (  Hom  `  ( r `  x ) )  C_  U.
ran  U. ran  r  ->  ran  (  Hom  `  (
r `  x )
)  C_  ran  U. ran  U.
ran  r )
42 uniss 3996 . . . . . . . . . . . . . . . 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 )
4340, 41, 42mp2b 10 . . . . . . . . . . . . . . 15  |-  U. ran  (  Hom  `  ( r `  x ) )  C_  U.
ran  U. ran  U. ran  r
4437, 43sstri 3317 . . . . . . . . . . . . . 14  |-  ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  U.
ran  U. ran  U. ran  r
4544rgenw 2733 . . . . . . . . . . . . 13  |-  A. x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  U.
ran  U. ran  U. ran  r
46 ss2ixp 7034 . . . . . . . . . . . . 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 )
4745, 46ax-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
484dmex 5091 . . . . . . . . . . . . 13  |-  dom  r  e.  _V
498rnex 5092 . . . . . . . . . . . . . 14  |-  ran  U. ran  U. ran  r  e. 
_V
5049uniex 4664 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  U. ran  r  e. 
_V
5148, 50ixpconst 7031 . . . . . . . . . . . 12  |-  X_ x  e.  dom  r U. ran  U.
ran  U. ran  r  =  ( U. ran  U. ran  U. ran  r  ^m  dom  r )
5247, 51sseqtri 3340 . . . . . . . . . . 11  |-  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) )  C_  ( U. ran  U. ran  U.
ran  r  ^m  dom  r )
53 ovex 6065 . . . . . . . . . . . 12  |-  ( U. ran  U. ran  U. ran  r  ^m  dom  r )  e.  _V
5453elpw2 4324 . . . . . . . . . . 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 ) )
5552, 54mpbir 201 . . . . . . . . . 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 )
5655rgen2w 2734 . . . . . . . . 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 )
57 eqid 2404 . . . . . . . . . 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 ) ) )
5857fmpt2 6377 . . . . . . . . 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 ) )
5956, 58mpbi 200 . . . . . . . 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
)
60 vex 2919 . . . . . . . . 9  |-  v  e. 
_V
6160, 60xpex 4949 . . . . . . . 8  |-  ( v  X.  v )  e. 
_V
6253pwex 4342 . . . . . . . 8  |-  ~P ( U. ran  U. ran  U. ran  r  ^m  dom  r
)  e.  _V
63 fex2 5562 . . . . . . . 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 )
6459, 61, 62, 63mp3an 1279 . . . . . . 7  |-  ( f  e.  v ,  g  e.  v  |->  X_ x  e.  dom  r ( ( f `  x ) (  Hom  `  (
r `  x )
) ( g `  x ) ) )  e.  _V
6564a1i 11 . . . . . 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 )
66 simpr 448 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  v  =  B )
6732adantr 452 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  dom  r  =  I )
6867ixpeq1d 7033 . . . . . . . . 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 ) ) )
6926fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (  Hom  `  ( r `  x ) )  =  (  Hom  `  ( R `  x )
) )
7069oveqd 6057 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) (  Hom  `  (
r `  x )
) ( g `  x ) )  =  ( ( f `  x ) (  Hom  `  ( R `  x
) ) ( g `
 x ) ) )
7170ixpeq2dv 7037 . . . . . . . . . 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 ) ) )
7271adantr 452 . . . . . . . . 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 ) ) )
7368, 72eqtrd 2436 . . . . . . . 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 ) ) )
7466, 66, 73mpt2eq123dv 6095 . . . . . . 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 ) ) ) )
75 prdsval.h . . . . . . . 8  |-  ( ph  ->  H  =  ( f  e.  B ,  g  e.  B  |->  X_ x  e.  I  ( (
f `  x )
(  Hom  `  ( R `
 x ) ) ( g `  x
) ) ) )
7675ad3antrrr 711 . . . . . . 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 ) ) ) )
7774, 76eqtr4d 2439 . . . . . 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 )
78 simplr 732 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  v  =  B )
7978opeq2d 3951 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. ( Base `  ndx ) ,  v >.  =  <. (
Base `  ndx ) ,  B >. )
8026fveq2d 5691 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( +g  `  ( r `  x ) )  =  ( +g  `  ( R `  x )
) )
8180oveqd 6057 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( +g  `  (
r `  x )
) ( g `  x ) )  =  ( ( f `  x ) ( +g  `  ( R `  x
) ) ( g `
 x ) ) )
8232, 81mpteq12dv 4247 . . . . . . . . . . . . . 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 ) ) ) )
8382adantr 452 . . . . . . . . . . . . 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 ) ) ) )
8466, 66, 83mpt2eq123dv 6095 . . . . . . . . . . . 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 ) ) ) ) )
8584adantr 452 . . . . . . . . . . 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 ) ) ) ) )
86 prdsval.a . . . . . . . . . . . 12  |-  ( ph  ->  .+  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( +g  `  ( R `  x )
) ( g `  x ) ) ) ) )
8786ad4antr 713 . . . . . . . . . . 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 ) ) ) ) )
8885, 87eqtr4d 2439 . . . . . . . . . 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 ) ) ) )  =  .+  )
8988opeq2d 3951 . . . . . . . . 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 ) ,  .+  >.
)
9026fveq2d 5691 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( .r `  ( r `  x ) )  =  ( .r `  ( R `  x )
) )
9190oveqd 6057 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( .r `  ( r `  x
) ) ( g `
 x ) )  =  ( ( f `
 x ) ( .r `  ( R `
 x ) ) ( g `  x
) ) )
9232, 91mpteq12dv 4247 . . . . . . . . . . . . . 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 ) ) ) )
9392adantr 452 . . . . . . . . . . . . 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 ) ) ) )
9466, 66, 93mpt2eq123dv 6095 . . . . . . . . . . . 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 ) ) ) ) )
9594adantr 452 . . . . . . . . . . 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 ) ) ) ) )
96 prdsval.t . . . . . . . . . . . 12  |-  ( ph  ->  .X.  =  ( f  e.  B ,  g  e.  B  |->  ( x  e.  I  |->  ( ( f `  x ) ( .r `  ( R `  x )
) ( g `  x ) ) ) ) )
9796ad4antr 713 . . . . . . . . . . 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 ) ) ) ) )
9895, 97eqtr4d 2439 . . . . . . . . . 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.  )
9998opeq2d 3951 . . . . . . . . 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.  >.
)
10079, 89, 99tpeq123d 3858 . . . . . . . 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.  >. } )
101 simp-4r 744 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  s  =  S )
102101opeq2d 3951 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. (Scalar ` 
ndx ) ,  s
>.  =  <. (Scalar `  ndx ) ,  S >. )
103 simpllr 736 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  s  =  S )
104103fveq2d 5691 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( Base `  s )  =  ( Base `  S
) )
105 prdsval.k . . . . . . . . . . . . . 14  |-  K  =  ( Base `  S
)
106104, 105syl6eqr 2454 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( Base `  s )  =  K )
10726fveq2d 5691 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( .s `  ( r `  x ) )  =  ( .s `  ( R `  x )
) )
108107oveqd 6057 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
f ( .s `  ( r `  x
) ) ( g `
 x ) )  =  ( f ( .s `  ( R `
 x ) ) ( g `  x
) ) )
10932, 108mpteq12dv 4247 . . . . . . . . . . . . . 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 ) ) ) )
110109adantr 452 . . . . . . . . . . . . 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 ) ) ) )
111106, 66, 110mpt2eq123dv 6095 . . . . . . . . . . . 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 ) ) ) ) )
112111adantr 452 . . . . . . . . . . 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 ) ) ) ) )
113 prdsval.m . . . . . . . . . . . 12  |-  ( ph  ->  .x.  =  ( f  e.  K ,  g  e.  B  |->  ( x  e.  I  |->  ( f ( .s `  ( R `  x )
) ( g `  x ) ) ) ) )
114113ad4antr 713 . . . . . . . . . . 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 ) ) ) ) )
115112, 114eqtr4d 2439 . . . . . . . . . 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.  )
116115opeq2d 3951 . . . . . . . . 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.  >.
)
117102, 116preq12d 3851 . . . . . . . 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.  >. } )
118100, 117uneq12d 3462 . . . . . . 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.  >. } ) )
119 simpllr 736 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  r  =  R )
120119coeq2d 4994 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( TopOpen  o.  r )  =  ( TopOpen  o.  R )
)
121120fveq2d 5691 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( Xt_ `  ( TopOpen  o.  r
) )  =  (
Xt_ `  ( TopOpen  o.  R
) ) )
122 prdsval.o . . . . . . . . . . . 12  |-  ( ph  ->  O  =  ( Xt_ `  ( TopOpen  o.  R )
) )
123122ad4antr 713 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  O  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
124121, 123eqtr4d 2439 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( Xt_ `  ( TopOpen  o.  r
) )  =  O )
125124opeq2d 3951 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( TopOpen  o.  r
) ) >.  =  <. (TopSet `  ndx ) ,  O >. )
12666sseq2d 3336 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( { f ,  g }  C_  v  <->  { f ,  g }  C_  B ) )
12726fveq2d 5691 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( le `  ( r `  x ) )  =  ( le `  ( R `  x )
) )
128127breqd 4183 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( le `  ( r `  x
) ) ( g `
 x )  <->  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) ) )
12932, 128raleqbidv 2876 . . . . . . . . . . . . . . 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 ) ) )
130129adantr 452 . . . . . . . . . . . . . 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 ) ) )
131126, 130anbi12d 692 . . . . . . . . . . . . 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 ) ) ) )
132131opabbidv 4231 . . . . . . . . . . . 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 ) ) } )
133132adantr 452 . . . . . . . . . . 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 ) ) } )
134 prdsval.l . . . . . . . . . . . 12  |-  ( ph  -> 
.<_  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
135134ad4antr 713 . . . . . . . . . . 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 ) ) } )
136133, 135eqtr4d 2439 . . . . . . . . . 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 ) ) }  =  .<_  )
137136opeq2d 3951 . . . . . . . . 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 ) ,  .<_  >.
)
13826fveq2d 5691 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( dist `  ( r `  x ) )  =  ( dist `  ( R `  x )
) )
139138oveqd 6057 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( dist `  (
r `  x )
) ( g `  x ) )  =  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )
14032, 139mpteq12dv 4247 . . . . . . . . . . . . . . . . 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 ) ) ) )
141140adantr 452 . . . . . . . . . . . . . . . 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 ) ) ) )
142141rneqd 5056 . . . . . . . . . . . . . . 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 ) ) ) )
143142uneq1d 3460 . . . . . . . . . . . . . 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 } ) )
144143supeq1d 7409 . . . . . . . . . . . . 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* ,  <  ) )
14566, 66, 144mpt2eq123dv 6095 . . . . . . . . . . . 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* ,  <  ) ) )
146145adantr 452 . . . . . . . . . . 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* ,  <  ) ) )
147 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* ,  <  ) ) )
148147ad4antr 713 . . . . . . . . . . 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* ,  <  ) ) )
149146, 148eqtr4d 2439 . . . . . . . . . 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 )
150149opeq2d 3951 . . . . . . . . 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 >. )
151125, 137, 150tpeq123d 3858 . . . . . . . 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 >. } )
152 simpr 448 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  h  =  H )
153152opeq2d 3951 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. (  Hom  `  ndx ) ,  h >.  =  <. (  Hom  `  ndx ) ,  H >. )
15478, 78xpeq12d 4862 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
v  X.  v )  =  ( B  X.  B ) )
155152oveqd 6057 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
c h ( 2nd `  a ) )  =  ( c H ( 2nd `  a ) ) )
156152fveq1d 5689 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
h `  a )  =  ( H `  a ) )
15726fveq2d 5691 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (comp `  ( r `  x
) )  =  (comp `  ( R `  x
) ) )
158157oveqd 6057 . . . . . . . . . . . . . . . 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 ) ) )
159158oveqd 6057 . . . . . . . . . . . . . . 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 ) ) )
16032, 159mpteq12dv 4247 . . . . . . . . . . . . . 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 ) ) ) )
161160ad2antrr 707 . . . . . . . . . . . . 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 ) ) ) )
162155, 156, 161mpt2eq123dv 6095 . . . . . . . . . . . 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 ) ) ) ) )
163154, 78, 162mpt2eq123dv 6095 . . . . . . . . . . 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 ) ) ) ) ) )
164 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 ) ) ) ) ) )
165164ad4antr 713 . . . . . . . . . . 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 ) ) ) ) ) )
166163, 165eqtr4d 2439 . . . . . . . . . 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  )
167166opeq2d 3951 . . . . . . . . 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  >. )
168153, 167preq12d 3851 . . . . . . . 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  >. } )
169151, 168uneq12d 3462 . . . . . . 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  >. } ) )
170118, 169uneq12d 3462 . . . . . 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  >. } ) ) )
17165, 77, 170csbied2 3254 . . . . 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  >. } ) ) )
17224, 36, 171csbied2 3254 . . . 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  >. } ) ) )
173172anasss 629 . . 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  >. } ) ) )
174 prdsval.s . . . 4  |-  ( ph  ->  S  e.  W )
175 elex 2924 . . . 4  |-  ( S  e.  W  ->  S  e.  _V )
176174, 175syl 16 . . 3  |-  ( ph  ->  S  e.  _V )
177 prdsval.r . . . 4  |-  ( ph  ->  R  e.  Z )
178 elex 2924 . . . 4  |-  ( R  e.  Z  ->  R  e.  _V )
179177, 178syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
180 tpex 4667 . . . . . 6  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  e.  _V
181 prex 4366 . . . . . 6  |-  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  .x.  >. }  e.  _V
182180, 181unex 4666 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. } )  e. 
_V
183 tpex 4667 . . . . . 6  |-  { <. (TopSet `  ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  e.  _V
184 prex 4366 . . . . . 6  |-  { <. (  Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .xb  >. }  e.  _V
185183, 184unex 4666 . . . . 5  |-  ( {
<. (TopSet `  ndx ) ,  O >. ,  <. ( le `  ndx ) , 
.<_  >. ,  <. ( dist `  ndx ) ,  D >. }  u.  { <. (  Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } )  e. 
_V
186182, 185unex 4666 . . . 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
187186a1i 11 . . 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 )
1883, 173, 176, 179, 187ovmpt2d 6160 . 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  >. } ) ) )
1891, 188syl5eq 2448 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916   [_csb 3211    u. cun 3278    C_ wss 3280   ~Pcpw 3759   {csn 3774   {cpr 3775   {ctp 3776   <.cop 3777   U.cuni 3975   U_ciun 4053   class class class wbr 4172   {copab 4225    e. cmpt 4226    X. cxp 4835   dom cdm 4837   ran crn 4838    o. ccom 4841   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307    ^m cmap 6977   X_cixp 7022   supcsup 7403   0cc0 8946   1c1 8947   RR*cxr 9075    < clt 9076   4c4 10007  ;cdc 10338   ndxcnx 13421   Basecbs 13424   +g cplusg 13484   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488  TopSetcts 13490   lecple 13491   distcds 13493    Hom chom 13495  compcco 13496   TopOpenctopn 13604   Xt_cpt 13621   X_scprds 13624
This theorem is referenced by:  prdssca  13634  prdsbas  13635  prdsplusg  13636  prdsmulr  13637  prdsvsca  13638  prdsle  13639  prdsds  13641  prdstset  13643  prdshom  13644  prdsco  13645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-i2m1 9014  ax-1ne0 9015  ax-rrecex 9018  ax-cnre 9019
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-map 6979  df-ixp 7023  df-sup 7404  df-nn 9957  df-ndx 13427  df-slot 13428  df-base 13429  df-hom 13508  df-prds 13626
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