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Theorem prdsval 16115
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 16108 . . . 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 3203 . . . . . . . . . . . 12 |- r e. _V
54rnex 7100 . . . . . . . . . . 11 |- ran r e. _V
65uniex 6953 . . . . . . . . . 10 |- U. ran r e. _V
76rnex 7100 . . . . . . . . 9 |- ran U. ran r e. _V
87uniex 6953 . . . . . . . 8 |- U. ran U. ran r e. _V
9 baseid 15919 . . . . . . . . . . . 12 |- Base = Slot ( Base ` ndx )
109strfvss 15880 . . . . . . . . . . 11 |- ( Base ` ( r ` x ) ) C_ U. ran ( r ` x )
11 fvssunirn 6217 . . . . . . . . . . . 12 |- ( r ` x ) C_ U. ran r
12 rnss 5354 . . . . . . . . . . . 12 |- ( ( r ` x ) C_ U. ran r -> ran ( r ` x ) C_ ran U. ran r )
13 uniss 4458 . . . . . . . . . . . 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 3612 . . . . . . . . . 10 |- ( Base ` ( r ` x ) ) C_ U. ran U. ran r
1615rgenw 2924 . . . . . . . . 9 |- A. x e. dom r ( Base ` ( r ` x ) ) C_ U. ran U. ran r
17 iunss 4561 . . . . . . . . 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 221 . . . . . . . 8 |- U_ x e. dom r ( Base ` ( r ` x ) ) C_ U. ran U. ran r
198, 18ssexi 4803 . . . . . . 7 |- U_ x e. dom r ( Base ` ( r ` x ) ) e. _V
20 ixpssmap2g 7937 . . . . . . 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 6678 . . . . . . 7 |- ( U_ x e. dom r ( Base ` ( r ` x ) ) ^m dom r ) e. _V
2322ssex 4802 . . . . . 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 13 . . . . 5 |- ( ( ( ph /\ s = S ) /\ r = R ) -> X_ x e. dom r ( Base ` ( r ` x ) ) e. _V )
25 simpr 477 . . . . . . . . 9 |- ( ( ( ph /\ s = S ) /\ r = R ) -> r = R )
2625fveq1d 6193 . . . . . . . 8 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( r ` x ) = ( R ` x ) )
2726fveq2d 6195 . . . . . . 7 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( Base ` ( r ` x ) ) = ( Base ` ( R ` x ) ) )
2827ixpeq2dv 7924 . . . . . 6 |- ( ( ( ph /\ s = S ) /\ r = R ) -> X_ x e. I ( Base ` ( r ` x ) ) = X_ x e. I ( Base ` ( R ` x ) ) )
2925dmeqd 5326 . . . . . . . 8 |- ( ( ( ph /\ s = S ) /\ r = R ) -> dom r = dom R )
30 prdsval.i . . . . . . . . 9 |- ( ph -> dom R = I )
3130ad2antrr 762 . . . . . . . 8 |- ( ( ( ph /\ s = S ) /\ r = R ) -> dom R = I )
3229, 31eqtrd 2656 . . . . . . 7 |- ( ( ( ph /\ s = S ) /\ r = R ) -> dom r = I )
3332ixpeq1d 7920 . . . . . 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 762 . . . . . 6 |- ( ( ( ph /\ s = S ) /\ r = R ) -> B = X_ x e. I ( Base ` ( R ` x ) ) )
3628, 33, 353eqtr4d 2666 . . . . 5 |- ( ( ( ph /\ s = S ) /\ r = R ) -> X_ x e. dom r ( Base ` ( r ` x ) ) = B )
37 ovssunirn 6681 . . . . . . . . . . . . . . 15 |- ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) C_ U. ran ( Hom ` ( r ` x ) )
38 df-hom 15966 . . . . . . . . . . . . . . . . . 18 |- Hom = Slot ; 1 4
3938strfvss 15880 . . . . . . . . . . . . . . . . 17 |- ( Hom ` ( r ` x ) ) C_ U. ran ( r ` x )
4039, 14sstri 3612 . . . . . . . . . . . . . . . 16 |- ( Hom ` ( r ` x ) ) C_ U. ran U. ran r
41 rnss 5354 . . . . . . . . . . . . . . . 16 |- ( ( Hom ` ( r ` x ) ) C_ U. ran U. ran r -> ran ( Hom ` ( r ` x ) ) C_ ran U. ran U. ran r )
42 uniss 4458 . . . . . . . . . . . . . . . 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 3612 . . . . . . . . . . . . . 14 |- ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) C_ U. ran U. ran U. ran r
4544rgenw 2924 . . . . . . . . . . . . 13 |- A. x e. dom r ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) C_ U. ran U. ran U. ran r
46 ss2ixp 7921 . . . . . . . . . . . . 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 7099 . . . . . . . . . . . . 13 |- dom r e. _V
498rnex 7100 . . . . . . . . . . . . . 14 |- ran U. ran U. ran r e. _V
5049uniex 6953 . . . . . . . . . . . . 13 |- U. ran U. ran U. ran r e. _V
5148, 50ixpconst 7918 . . . . . . . . . . . 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 3637 . . . . . . . . . . 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 6678 . . . . . . . . . . . 12 |- ( U. ran U. ran U. ran r ^m dom r ) e. _V
5453elpw2 4828 . . . . . . . . . . 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 221 . . . . . . . . . 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 2925 . . . . . . . . 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 2622 . . . . . . . . . 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 7237 . . . . . . . . 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 220 . . . . . . . 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 3203 . . . . . . . . 9 |- v e. _V
6160, 60xpex 6962 . . . . . . . 8 |- ( v X. v ) e. _V
6253pwex 4848 . . . . . . . 8 |- ~P ( U. ran U. ran U. ran r ^m dom r ) e. _V
63 fex2 7121 . . . . . . . 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 1424 . . . . . . 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 477 . . . . . . . 8 |- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> v = B )
6732adantr 481 . . . . . . . . . 10 |- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> dom r = I )
6867ixpeq1d 7920 . . . . . . . . 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 6195 . . . . . . . . . . . 12 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( Hom ` ( r ` x ) ) = ( Hom ` ( R ` x ) ) )
7069oveqd 6667 . . . . . . . . . . 11 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) = ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) )
7170ixpeq2dv 7924 . . . . . . . . . 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 481 . . . . . . . . 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 2656 . . . . . . . 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 6717 . . . . . . 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 766 . . . . . . 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 2659 . . . . . 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 792 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> v = B )
7978opeq2d 4409 . . . . . . . . 9 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> <. ( Base ` ndx ) , v >. = <. ( Base ` ndx ) , B >. )
8026fveq2d 6195 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( +g ` ( r ` x ) ) = ( +g ` ( R ` x ) ) )
8180oveqd 6667 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( +g ` ( r ` x ) ) ( g ` x ) ) = ( ( f ` x ) ( +g ` ( R ` x ) ) ( g ` x ) ) )
8232, 81mpteq12dv 4733 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 6717 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 768 . . . . . . . . . . 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 2659 . . . . . . . . . 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 4409 . . . . . . . . 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 6195 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( .r ` ( r ` x ) ) = ( .r ` ( R ` x ) ) )
9190oveqd 6667 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( .r ` ( r ` x ) ) ( g ` x ) ) = ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) )
9232, 91mpteq12dv 4733 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 6717 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 768 . . . . . . . . . . 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 2659 . . . . . . . . . 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 4409 . . . . . . . . 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 4283 . . . . . . . 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 807 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> s = S )
102101opeq2d 4409 . . . . . . . . 9 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> <. (Scalar ` ndx ) , s >. = <. (Scalar ` ndx ) , S >. )
103 simpllr 799 . . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> s = S )
104103fveq2d 6195 . . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> ( Base ` s ) = ( Base ` S ) )
105 prdsval.k . . . . . . . . . . . . . 14 |- K = ( Base ` S )
106104, 105syl6eqr 2674 . . . . . . . . . . . . 13 |- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> ( Base ` s ) = K )
10726fveq2d 6195 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( .s ` ( r ` x ) ) = ( .s ` ( R ` x ) ) )
108107oveqd 6667 . . . . . . . . . . . . . . 15 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( f ( .s ` ( r ` x ) ) ( g ` x ) ) = ( f ( .s ` ( R ` x ) ) ( g ` x ) ) )
10932, 108mpteq12dv 4733 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . 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 6717 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 768 . . . . . . . . . . 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 2659 . . . . . . . . . 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 4409 . . . . . . . . 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 6195 . . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( .i ` ( r ` x ) ) = ( .i ` ( R ` x ) ) )
118117oveqd 6667 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) = ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) )
11932, 118mpteq12dv 4733 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 6668 . . . . . . . . . . . . 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 6717 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 768 . . . . . . . . . . 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 2659 . . . . . . . . . 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 4409 . . . . . . . . 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 4283 . . . . . . . 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 3768 . . . . . . 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 799 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> r = R )
131130coeq2d 5284 . . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( TopOpen o. r ) = ( TopOpen o. R ) )
132131fveq2d 6195 . . . . . . . . . . 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 768 . . . . . . . . . . 11 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> O = ( Xt_ ` ( TopOpen o. R ) ) )
135132, 134eqtr4d 2659 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( Xt_ ` ( TopOpen o. r ) ) = O )
136135opeq2d 4409 . . . . . . . . 9 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> <. (TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. r ) ) >. = <. (TopSet ` ndx ) , O >. )
13766sseq2d 3633 . . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> ( { f , g } C_ v <-> { f , g } C_ B ) )
13826fveq2d 6195 . . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( le ` ( r ` x ) ) = ( le ` ( R ` x ) ) )
139138breqd 4664 . . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( le ` ( r ` x ) ) ( g ` x ) <-> ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) )
14032, 139raleqbidv 3152 . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . 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 4716 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 768 . . . . . . . . . . 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 2659 . . . . . . . . . 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 4409 . . . . . . . . 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 6195 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( dist ` ( r ` x ) ) = ( dist ` ( R ` x ) ) )
150149oveqd 6667 . . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( dist ` ( r ` x ) ) ( g ` x ) ) = ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) )
15132, 150mpteq12dv 4733 . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . 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 5353 . . . . . . . . . . . . . . 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 3766 . . . . . . . . . . . . . 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 8352 . . . . . . . . . . . . 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 6717 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 768 . . . . . . . . . . 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 2659 . . . . . . . . . 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 4409 . . . . . . . . 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 4283 . . . . . . . 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 477 . . . . . . . . . 10 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> h = H )
164163opeq2d 4409 . . . . . . . . 9 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> <. ( Hom ` ndx ) , h >. = <. ( Hom ` ndx ) , H >. )
16578sqxpeqd 5141 . . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( v X. v ) = ( B X. B ) )
166163oveqd 6667 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( c h ( 2nd ` a ) ) = ( c H ( 2nd ` a ) ) )
167163fveq1d 6193 . . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( h ` a ) = ( H ` a ) )
16826fveq2d 6195 . . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ s = S ) /\ r = R ) -> (comp ` ( r ` x ) ) = (comp ` ( R ` x ) ) )
169168oveqd 6667 . . . . . . . . . . . . . . . 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 6667 . . . . . . . . . . . . . . 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 4733 . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . 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 6717 . . . . . . . . . . . 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 6717 . . . . . . . . . . 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 768 . . . . . . . . . . 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 2659 . . . . . . . . . 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 4409 . . . . . . . . 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 4276 . . . . . . . 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 3768 . . . . . . 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 3768 . . . . . 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 3561 . . . . 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 3561 . . . 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 679 . . 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 3212 . . . 4 |- ( S e. W -> S e. _V )
187185, 186syl 17 . . 3 |- ( ph -> S e. _V )
188 prdsval.r . . . 4 |- ( ph -> R e. Z )
189 elex 3212 . . . 4 |- ( R e. Z -> R e. _V )
190188, 189syl 17 . . 3 |- ( ph -> R e. _V )
191 tpex 6957 . . . . . 6 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } e. _V
192 tpex 6957 . . . . . 6 |- { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } e. _V
193191, 192unex 6956 . . . . 5 |- ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) e. _V
194 tpex 6957 . . . . . 6 |- { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } e. _V
195 prex 4909 . . . . . 6 |- { <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .xb >. } e. _V
196194, 195unex 6956 . . . . 5 |- ( { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } u. { <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .xb >. } ) e. _V
197193, 196unex 6956 . . . 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 6788 . 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 2668 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 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   [_csb 3533   u. cun 3572   C_ wss 3574   ~Pcpw 4158   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   U.cuni 4436   U_ciun 4520   class class class wbr 4653   {copab 4712   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   X_cixp 7908   supcsup 8346   0cc0 9936   1c1 9937   RR*cxr 10073   < clt 10074   4c4 11072  ;cdc 11493   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   .icip 15946  TopSetcts 15947   lecple 15948   distcds 15950   Hom chom 15952  compcco 15953   TopOpenctopn 16082   Xt_cpt 16099   gsumg cgsu 16101   X_scprds 16106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rrecex 10008  ax-cnre 10009
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-map 7859  df-ixp 7909  df-sup 8348  df-nn 11021  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-prds 16108
This theorem is referenced by:  prdssca  16116  prdsbas  16117  prdsplusg  16118  prdsmulr  16119  prdsvsca  16120  prdsip  16121  prdsle  16122  prdsds  16124  prdstset  16126  prdshom  16127  prdsco  16128
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