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Theorem prdsval 14385
Description: Value of the structure product. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 7-Jan-2017.) (Revised by Thierry Arnoux, 16-Jun-2019.)
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.j  |-  ( ph  ->  .,  =  ( f  e.  B ,  g  e.  B  |->  ( S 
gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( 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.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  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    H, a, c, d, e    x, a, ph, c, d, e, f, g   
x, I    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)    ., ( x, e, f, g, a, c, d)    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 14378 . . . 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 ) ) ) ) >. ,  <. ( .i `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `  x
) ( .i `  ( 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 ) ) ) ) >. ,  <. ( .i `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `  x
) ( .i `  ( 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 2970 . . . . . . . . . . . 12  |-  r  e. 
_V
54rnex 6507 . . . . . . . . . . 11  |-  ran  r  e.  _V
65uniex 6371 . . . . . . . . . 10  |-  U. ran  r  e.  _V
76rnex 6507 . . . . . . . . 9  |-  ran  U. ran  r  e.  _V
87uniex 6371 . . . . . . . 8  |-  U. ran  U.
ran  r  e.  _V
9 baseid 14212 . . . . . . . . . . . 12  |-  Base  = Slot  ( Base `  ndx )
109strfvss 14184 . . . . . . . . . . 11  |-  ( Base `  ( r `  x
) )  C_  U. ran  ( r `  x
)
11 fvssunirn 5708 . . . . . . . . . . . 12  |-  ( r `
 x )  C_  U.
ran  r
12 rnss 5063 . . . . . . . . . . . 12  |-  ( ( r `  x ) 
C_  U. ran  r  ->  ran  ( r `  x
)  C_  ran  U. ran  r )
13 uniss 4107 . . . . . . . . . . . 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 3360 . . . . . . . . . 10  |-  ( Base `  ( r `  x
) )  C_  U. ran  U.
ran  r
1615rgenw 2778 . . . . . . . . 9  |-  A. x  e.  dom  r ( Base `  ( r `  x
) )  C_  U. ran  U.
ran  r
17 iunss 4206 . . . . . . . . 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 209 . . . . . . . 8  |-  U_ x  e.  dom  r ( Base `  ( r `  x
) )  C_  U. ran  U.
ran  r
198, 18ssexi 4432 . . . . . . 7  |-  U_ x  e.  dom  r ( Base `  ( r `  x
) )  e.  _V
20 ixpssmap2g 7284 . . . . . . 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 5 . . . . . 6  |-  X_ x  e.  dom  r ( Base `  ( r `  x
) )  C_  ( U_ x  e.  dom  r ( Base `  (
r `  x )
)  ^m  dom  r )
22 ovex 6111 . . . . . . 7  |-  ( U_ x  e.  dom  r (
Base `  ( r `  x ) )  ^m  dom  r )  e.  _V
2322ssex 4431 . . . . . 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 461 . . . . . . . . 9  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  r  =  R )
2625fveq1d 5688 . . . . . . . 8  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
r `  x )  =  ( R `  x ) )
2726fveq2d 5690 . . . . . . 7  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( Base `  ( r `  x ) )  =  ( Base `  ( R `  x )
) )
2827ixpeq2dv 7271 . . . . . 6  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  I  ( Base `  ( r `  x
) )  =  X_ x  e.  I  ( Base `  ( R `  x ) ) )
2925dmeqd 5037 . . . . . . . 8  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  dom  r  =  dom  R )
30 prdsval.i . . . . . . . . 9  |-  ( ph  ->  dom  R  =  I )
3130ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  dom  R  =  I )
3229, 31eqtrd 2470 . . . . . . 7  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  dom  r  =  I )
3332ixpeq1d 7267 . . . . . 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 725 . . . . . 6  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  B  =  X_ x  e.  I 
( Base `  ( R `  x ) ) )
3628, 33, 353eqtr4d 2480 . . . . 5  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  X_ x  e.  dom  r ( Base `  ( r `  x
) )  =  B )
37 ovssunirn 6112 . . . . . . . . . . . . . . 15  |-  ( ( f `  x ) ( Hom  `  (
r `  x )
) ( g `  x ) )  C_  U.
ran  ( Hom  `  (
r `  x )
)
38 df-hom 14254 . . . . . . . . . . . . . . . . . 18  |-  Hom  = Slot ; 1 4
3938strfvss 14184 . . . . . . . . . . . . . . . . 17  |-  ( Hom  `  ( r `  x
) )  C_  U. ran  ( r `  x
)
4039, 14sstri 3360 . . . . . . . . . . . . . . . 16  |-  ( Hom  `  ( r `  x
) )  C_  U. ran  U.
ran  r
41 rnss 5063 . . . . . . . . . . . . . . . 16  |-  ( ( Hom  `  ( r `  x ) )  C_  U.
ran  U. ran  r  ->  ran  ( Hom  `  (
r `  x )
)  C_  ran  U. ran  U.
ran  r )
42 uniss 4107 . . . . . . . . . . . . . . . 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 3360 . . . . . . . . . . . . . 14  |-  ( ( f `  x ) ( Hom  `  (
r `  x )
) ( g `  x ) )  C_  U.
ran  U. ran  U. ran  r
4544rgenw 2778 . . . . . . . . . . . . 13  |-  A. x  e.  dom  r ( ( f `  x ) ( Hom  `  (
r `  x )
) ( g `  x ) )  C_  U.
ran  U. ran  U. ran  r
46 ss2ixp 7268 . . . . . . . . . . . . 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 5 . . . . . . . . . . . 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 6506 . . . . . . . . . . . . 13  |-  dom  r  e.  _V
498rnex 6507 . . . . . . . . . . . . . 14  |-  ran  U. ran  U. ran  r  e. 
_V
5049uniex 6371 . . . . . . . . . . . . 13  |-  U. ran  U.
ran  U. ran  r  e. 
_V
5148, 50ixpconst 7265 . . . . . . . . . . . 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 3383 . . . . . . . . . . 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 6111 . . . . . . . . . . . 12  |-  ( U. ran  U. ran  U. ran  r  ^m  dom  r )  e.  _V
5453elpw2 4451 . . . . . . . . . . 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 209 . . . . . . . . . 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 2779 . . . . . . . . 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 2438 . . . . . . . . . 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 6636 . . . . . . . . 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 208 . . . . . . . 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 2970 . . . . . . . . 9  |-  v  e. 
_V
6160, 60xpex 6503 . . . . . . . 8  |-  ( v  X.  v )  e. 
_V
6253pwex 4470 . . . . . . . 8  |-  ~P ( U. ran  U. ran  U. ran  r  ^m  dom  r
)  e.  _V
63 fex2 6527 . . . . . . . 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 1314 . . . . . . 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 461 . . . . . . . 8  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  v  =  B )
6732adantr 465 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  dom  r  =  I )
6867ixpeq1d 7267 . . . . . . . . 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 5690 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( Hom  `  ( r `  x ) )  =  ( Hom  `  ( R `  x )
) )
7069oveqd 6103 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( Hom  `  (
r `  x )
) ( g `  x ) )  =  ( ( f `  x ) ( Hom  `  ( R `  x
) ) ( g `
 x ) ) )
7170ixpeq2dv 7271 . . . . . . . . . 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 465 . . . . . . . . 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 2470 . . . . . . . 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 6143 . . . . . . 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 729 . . . . . . 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 2473 . . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  v  =  B )
7978opeq2d 4061 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. ( Base `  ndx ) ,  v >.  =  <. (
Base `  ndx ) ,  B >. )
8026fveq2d 5690 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( +g  `  ( r `  x ) )  =  ( +g  `  ( R `  x )
) )
8180oveqd 6103 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( +g  `  (
r `  x )
) ( g `  x ) )  =  ( ( f `  x ) ( +g  `  ( R `  x
) ) ( g `
 x ) ) )
8232, 81mpteq12dv 4365 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 6143 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 731 . . . . . . . . . . 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 2473 . . . . . . . . . 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 4061 . . . . . . . . 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 5690 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( .r `  ( r `  x ) )  =  ( .r `  ( R `  x )
) )
9190oveqd 6103 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( .r `  ( r `  x
) ) ( g `
 x ) )  =  ( ( f `
 x ) ( .r `  ( R `
 x ) ) ( g `  x
) ) )
9232, 91mpteq12dv 4365 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 6143 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 731 . . . . . . . . . . 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 2473 . . . . . . . . . 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 4061 . . . . . . . . 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 3964 . . . . . . . 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 766 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  s  =  S )
102101opeq2d 4061 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. (Scalar ` 
ndx ) ,  s
>.  =  <. (Scalar `  ndx ) ,  S >. )
103 simpllr 758 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  s  =  S )
104103fveq2d 5690 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( Base `  s )  =  ( Base `  S
) )
105 prdsval.k . . . . . . . . . . . . . 14  |-  K  =  ( Base `  S
)
106104, 105syl6eqr 2488 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( Base `  s )  =  K )
10726fveq2d 5690 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( .s `  ( r `  x ) )  =  ( .s `  ( R `  x )
) )
108107oveqd 6103 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
f ( .s `  ( r `  x
) ) ( g `
 x ) )  =  ( f ( .s `  ( R `
 x ) ) ( g `  x
) ) )
10932, 108mpteq12dv 4365 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . 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 6143 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 731 . . . . . . . . . . 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 2473 . . . . . . . . . 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 4061 . . . . . . . . 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.  >.
)
11726fveq2d 5690 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( .i `  ( r `  x ) )  =  ( .i `  ( R `  x )
) )
118117oveqd 6103 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( .i `  ( r `  x
) ) ( g `
 x ) )  =  ( ( f `
 x ) ( .i `  ( R `
 x ) ) ( g `  x
) ) )
11932, 118mpteq12dv 4365 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
x  e.  dom  r  |->  ( ( f `  x ) ( .i
`  ( r `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `  x
) ( .i `  ( R `  x ) ) ( g `  x ) ) ) )
120119adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
x  e.  dom  r  |->  ( ( f `  x ) ( .i
`  ( r `  x ) ) ( g `  x ) ) )  =  ( x  e.  I  |->  ( ( f `  x
) ( .i `  ( R `  x ) ) ( g `  x ) ) ) )
121103, 120oveq12d 6104 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
s  gsumg  ( x  e.  dom  r  |->  ( ( f `
 x ) ( .i `  ( r `
 x ) ) ( g `  x
) ) ) )  =  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i `  ( R `  x )
) ( g `  x ) ) ) ) )
12266, 66, 121mpt2eq123dv 6143 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  (
f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `
 x ) ( .i `  ( r `
 x ) ) ( g `  x
) ) ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( S 
gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) ) )
123122adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `
 x ) ( .i `  ( r `
 x ) ) ( g `  x
) ) ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( S 
gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) ) )
124 prdsval.j . . . . . . . . . . . 12  |-  ( ph  ->  .,  =  ( f  e.  B ,  g  e.  B  |->  ( S 
gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) ) )
125124ad4antr 731 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  .,  =  ( f  e.  B ,  g  e.  B  |->  ( S  gsumg  ( x  e.  I  |->  ( ( f `  x ) ( .i
`  ( R `  x ) ) ( g `  x ) ) ) ) ) )
126123, 125eqtr4d 2473 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `
 x ) ( .i `  ( r `
 x ) ) ( g `  x
) ) ) ) )  =  .,  )
127126opeq2d 4061 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. ( .i `  ndx ) ,  ( f  e.  v ,  g  e.  v 
|->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `
 x ) ( .i `  ( r `
 x ) ) ( g `  x
) ) ) ) ) >.  =  <. ( .i `  ndx ) ,  .,  >. )
128102, 116, 127tpeq123d 3964 . . . . . . . 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 ) ) ) ) >. ,  <. ( .i `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `  x
) ( .i `  ( r `  x
) ) ( g `
 x ) ) ) ) ) >. }  =  { <. (Scalar ` 
ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. } )
129100, 128uneq12d 3506 . . . . . . 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 ) ) ) ) >. ,  <. ( .i `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `  x
) ( .i `  ( r `  x
) ) ( g `
 x ) ) ) ) ) >. } )  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } ) )
130 simpllr 758 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  r  =  R )
131130coeq2d 4997 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( TopOpen  o.  r )  =  ( TopOpen  o.  R )
)
132131fveq2d 5690 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( Xt_ `  ( TopOpen  o.  r
) )  =  (
Xt_ `  ( TopOpen  o.  R
) ) )
133 prdsval.o . . . . . . . . . . . 12  |-  ( ph  ->  O  =  ( Xt_ `  ( TopOpen  o.  R )
) )
134133ad4antr 731 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  O  =  ( Xt_ `  ( TopOpen  o.  R ) ) )
135132, 134eqtr4d 2473 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  ( Xt_ `  ( TopOpen  o.  r
) )  =  O )
136135opeq2d 4061 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( TopOpen  o.  r
) ) >.  =  <. (TopSet `  ndx ) ,  O >. )
13766sseq2d 3379 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  s  =  S )  /\  r  =  R
)  /\  v  =  B )  ->  ( { f ,  g }  C_  v  <->  { f ,  g }  C_  B ) )
13826fveq2d 5690 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( le `  ( r `  x ) )  =  ( le `  ( R `  x )
) )
139138breqd 4298 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( le `  ( r `  x
) ) ( g `
 x )  <->  ( f `  x ) ( le
`  ( R `  x ) ) ( g `  x ) ) )
14032, 139raleqbidv 2926 . . . . . . . . . . . . . . 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 ) ) )
141140adantr 465 . . . . . . . . . . . . . 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 ) ) )
142137, 141anbi12d 710 . . . . . . . . . . . . 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 ) ) ) )
143142opabbidv 4350 . . . . . . . . . . . 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 ) ) } )
144143adantr 465 . . . . . . . . . . 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 ) ) } )
145 prdsval.l . . . . . . . . . . . 12  |-  ( ph  -> 
.<_  =  { <. f ,  g >.  |  ( { f ,  g }  C_  B  /\  A. x  e.  I  ( f `  x ) ( le `  ( R `  x )
) ( g `  x ) ) } )
146145ad4antr 731 . . . . . . . . . . 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 ) ) } )
147144, 146eqtr4d 2473 . . . . . . . . . 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 ) ) }  =  .<_  )
148147opeq2d 4061 . . . . . . . . 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 ) ,  .<_  >.
)
14926fveq2d 5690 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  ( dist `  ( r `  x ) )  =  ( dist `  ( R `  x )
) )
150149oveqd 6103 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (
( f `  x
) ( dist `  (
r `  x )
) ( g `  x ) )  =  ( ( f `  x ) ( dist `  ( R `  x
) ) ( g `
 x ) ) )
15132, 150mpteq12dv 4365 . . . . . . . . . . . . . . . . 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 ) ) ) )
152151adantr 465 . . . . . . . . . . . . . . . 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 ) ) ) )
153152rneqd 5062 . . . . . . . . . . . . . . 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 ) ) ) )
154153uneq1d 3504 . . . . . . . . . . . . . 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 } ) )
155154supeq1d 7688 . . . . . . . . . . . . 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* ,  <  ) )
15666, 66, 155mpt2eq123dv 6143 . . . . . . . . . . . 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* ,  <  ) ) )
157156adantr 465 . . . . . . . . . . 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* ,  <  ) ) )
158 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* ,  <  ) ) )
159158ad4antr 731 . . . . . . . . . . 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* ,  <  ) ) )
160157, 159eqtr4d 2473 . . . . . . . . . 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 )
161160opeq2d 4061 . . . . . . . . 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 >. )
162136, 148, 161tpeq123d 3964 . . . . . . . 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 >. } )
163 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  h  =  H )
164163opeq2d 4061 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  <. ( Hom  `  ndx ) ,  h >.  =  <. ( Hom  `  ndx ) ,  H >. )
16578, 78xpeq12d 4860 . . . . . . . . . . . 12  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
v  X.  v )  =  ( B  X.  B ) )
166163oveqd 6103 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
c h ( 2nd `  a ) )  =  ( c H ( 2nd `  a ) ) )
167163fveq1d 5688 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ph  /\  s  =  S )  /\  r  =  R )  /\  v  =  B )  /\  h  =  H )  ->  (
h `  a )  =  ( H `  a ) )
16826fveq2d 5690 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  s  =  S )  /\  r  =  R )  ->  (comp `  ( r `  x
) )  =  (comp `  ( R `  x
) ) )
169168oveqd 6103 . . . . . . . . . . . . . . . 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 ) ) )
170169oveqd 6103 . . . . . . . . . . . . . . 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 ) ) )
17132, 170mpteq12dv 4365 . . . . . . . . . . . . . 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 ) ) ) )
172171ad2antrr 725 . . . . . . . . . . . . 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 ) ) ) )
173166, 167, 172mpt2eq123dv 6143 . . . . . . . . . . . 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 ) ) ) ) )
174165, 78, 173mpt2eq123dv 6143 . . . . . . . . . . 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 ) ) ) ) ) )
175 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 ) ) ) ) ) )
176175ad4antr 731 . . . . . . . . . . 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 ) ) ) ) ) )
177174, 176eqtr4d 2473 . . . . . . . . . 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  )
178177opeq2d 4061 . . . . . . . . 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  >. )
179164, 178preq12d 3957 . . . . . . . 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  >. } )
180162, 179uneq12d 3506 . . . . . . 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  >. } ) )
181129, 180uneq12d 3506 . . . . . 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 ) ) ) ) >. ,  <. ( .i `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `  x
) ( .i `  ( 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.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
18265, 77, 181csbied2 3310 . . . . 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 ) ) ) ) >. ,  <. ( .i `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `  x
) ( .i `  ( 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.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
18324, 36, 182csbied2 3310 . . . 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 ) ) ) ) >. ,  <. ( .i `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `  x
) ( .i `  ( 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.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
184183anasss 647 . . 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 ) ) ) ) >. ,  <. ( .i `  ndx ) ,  ( f  e.  v ,  g  e.  v  |->  ( s  gsumg  ( x  e.  dom  r  |->  ( ( f `  x
) ( .i `  ( 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.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
185 prdsval.s . . . 4  |-  ( ph  ->  S  e.  W )
186 elex 2976 . . . 4  |-  ( S  e.  W  ->  S  e.  _V )
187185, 186syl 16 . . 3  |-  ( ph  ->  S  e.  _V )
188 prdsval.r . . . 4  |-  ( ph  ->  R  e.  Z )
189 elex 2976 . . . 4  |-  ( R  e.  Z  ->  R  e.  _V )
190188, 189syl 16 . . 3  |-  ( ph  ->  R  e.  _V )
191 tpex 6374 . . . . . 6  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  e.  _V
192 tpex 6374 . . . . . 6  |-  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) ,  .x.  >. ,  <. ( .i `  ndx ) ,  .,  >. }  e.  _V
193191, 192unex 6373 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  e. 
_V
194 tpex 6374 . . . . . 6  |-  { <. (TopSet `  ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  e.  _V
195 prex 4529 . . . . . 6  |-  { <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .xb  >. }  e.  _V
196194, 195unex 6373 . . . . 5  |-  ( {
<. (TopSet `  ndx ) ,  O >. ,  <. ( le `  ndx ) , 
.<_  >. ,  <. ( dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } )  e. 
_V
197193, 196unex 6373 . . . 4  |-  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) )  e.  _V
198197a1i 11 . . 3  |-  ( ph  ->  ( ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) )  e.  _V )
1993, 184, 187, 190, 198ovmpt2d 6213 . 2  |-  ( ph  ->  ( S X_s R )  =  ( ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) , 
.+  >. ,  <. ( .r `  ndx ) , 
.X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
2001, 199syl5eq 2482 1  |-  ( ph  ->  P  =  ( ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  S >. ,  <. ( .s `  ndx ) , 
.x.  >. ,  <. ( .i `  ndx ) , 
.,  >. } )  u.  ( { <. (TopSet ` 
ndx ) ,  O >. ,  <. ( le `  ndx ) ,  .<_  >. ,  <. (
dist `  ndx ) ,  D >. }  u.  { <. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.xb  >. } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967   [_csb 3283    u. cun 3321    C_ wss 3323   ~Pcpw 3855   {csn 3872   {cpr 3874   {ctp 3876   <.cop 3878   U.cuni 4086   U_ciun 4166   class class class wbr 4287   {copab 4344    e. cmpt 4345    X. cxp 4833   dom cdm 4835   ran crn 4836    o. ccom 4839   -->wf 5409   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1stc1st 6570   2ndc2nd 6571    ^m cmap 7206   X_cixp 7255   supcsup 7682   0cc0 9274   1c1 9275   RR*cxr 9409    < clt 9410   4c4 10365  ;cdc 10747   ndxcnx 14163   Basecbs 14166   +g cplusg 14230   .rcmulr 14231  Scalarcsca 14233   .scvsca 14234   .icip 14235  TopSetcts 14236   lecple 14237   distcds 14239   Hom chom 14241  compcco 14242   TopOpenctopn 14352   Xt_cpt 14369    gsumg cgsu 14371   X_scprds 14376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-i2m1 9342  ax-1ne0 9343  ax-rrecex 9346  ax-cnre 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-map 7208  df-ixp 7256  df-sup 7683  df-nn 10315  df-ndx 14169  df-slot 14170  df-base 14171  df-hom 14254  df-prds 14378
This theorem is referenced by:  prdssca  14386  prdsbas  14387  prdsplusg  14388  prdsmulr  14389  prdsvsca  14390  prdsip  14391  prdsle  14392  prdsds  14394  prdstset  14396  prdshom  14397  prdsco  14398
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