Step |
Hyp |
Ref |
Expression |
1 |
|
prdsval.p |
|- P = ( S Xs_ R ) |
2 |
|
prdsval.k |
|- K = ( Base ` S ) |
3 |
|
prdsval.i |
|- ( ph -> dom R = I ) |
4 |
|
prdsval.b |
|- ( ph -> B = X_ x e. I ( Base ` ( R ` x ) ) ) |
5 |
|
prdsval.a |
|- ( ph -> .+ = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( +g ` ( R ` x ) ) ( g ` x ) ) ) ) ) |
6 |
|
prdsval.t |
|- ( ph -> .X. = ( f e. B , g e. B |-> ( x e. I |-> ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) ) ) ) |
7 |
|
prdsval.m |
|- ( ph -> .x. = ( f e. K , g e. B |-> ( x e. I |-> ( f ( .s ` ( R ` x ) ) ( g ` x ) ) ) ) ) |
8 |
|
prdsval.j |
|- ( ph -> ., = ( f e. B , g e. B |-> ( S gsum ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) ) |
9 |
|
prdsval.o |
|- ( ph -> O = ( Xt_ ` ( TopOpen o. R ) ) ) |
10 |
|
prdsval.l |
|- ( ph -> .<_ = { <. f , g >. | ( { f , g } C_ B /\ A. x e. I ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) } ) |
11 |
|
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* , < ) ) ) |
12 |
|
prdsval.h |
|- ( ph -> H = ( f e. B , g e. B |-> X_ x e. I ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) ) |
13 |
|
prdsval.x |
|- ( ph -> .xb = ( a e. ( B X. B ) , c e. B |-> ( d e. ( ( 2nd ` a ) H c ) , e e. ( H ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. ( comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) ) |
14 |
|
prdsval.s |
|- ( ph -> S e. W ) |
15 |
|
prdsval.r |
|- ( ph -> R e. Z ) |
16 |
|
df-prds |
|- Xs_ = ( 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 gsum ( 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. ( ( 2nd ` a ) h c ) , e e. ( h ` a ) |-> ( x e. dom r |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. ( comp ` ( r ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } ) ) ) |
17 |
16
|
a1i |
|- ( ph -> Xs_ = ( 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 gsum ( 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. ( ( 2nd ` a ) h c ) , e e. ( h ` a ) |-> ( x e. dom r |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. ( comp ` ( r ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) >. } ) ) ) ) |
18 |
|
vex |
|- r e. _V |
19 |
18
|
rnex |
|- ran r e. _V |
20 |
19
|
uniex |
|- U. ran r e. _V |
21 |
20
|
rnex |
|- ran U. ran r e. _V |
22 |
21
|
uniex |
|- U. ran U. ran r e. _V |
23 |
|
baseid |
|- Base = Slot ( Base ` ndx ) |
24 |
23
|
strfvss |
|- ( Base ` ( r ` x ) ) C_ U. ran ( r ` x ) |
25 |
|
fvssunirn |
|- ( r ` x ) C_ U. ran r |
26 |
|
rnss |
|- ( ( r ` x ) C_ U. ran r -> ran ( r ` x ) C_ ran U. ran r ) |
27 |
|
uniss |
|- ( ran ( r ` x ) C_ ran U. ran r -> U. ran ( r ` x ) C_ U. ran U. ran r ) |
28 |
25 26 27
|
mp2b |
|- U. ran ( r ` x ) C_ U. ran U. ran r |
29 |
24 28
|
sstri |
|- ( Base ` ( r ` x ) ) C_ U. ran U. ran r |
30 |
29
|
rgenw |
|- A. x e. dom r ( Base ` ( r ` x ) ) C_ U. ran U. ran r |
31 |
|
iunss |
|- ( 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 ) |
32 |
30 31
|
mpbir |
|- U_ x e. dom r ( Base ` ( r ` x ) ) C_ U. ran U. ran r |
33 |
22 32
|
ssexi |
|- U_ x e. dom r ( Base ` ( r ` x ) ) e. _V |
34 |
|
ixpssmap2g |
|- ( 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 ) ) |
35 |
33 34
|
ax-mp |
|- X_ x e. dom r ( Base ` ( r ` x ) ) C_ ( U_ x e. dom r ( Base ` ( r ` x ) ) ^m dom r ) |
36 |
|
ovex |
|- ( U_ x e. dom r ( Base ` ( r ` x ) ) ^m dom r ) e. _V |
37 |
36
|
ssex |
|- ( 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 ) |
38 |
35 37
|
mp1i |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> X_ x e. dom r ( Base ` ( r ` x ) ) e. _V ) |
39 |
|
simpr |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> r = R ) |
40 |
39
|
fveq1d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( r ` x ) = ( R ` x ) ) |
41 |
40
|
fveq2d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( Base ` ( r ` x ) ) = ( Base ` ( R ` x ) ) ) |
42 |
41
|
ixpeq2dv |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> X_ x e. I ( Base ` ( r ` x ) ) = X_ x e. I ( Base ` ( R ` x ) ) ) |
43 |
39
|
dmeqd |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> dom r = dom R ) |
44 |
3
|
ad2antrr |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> dom R = I ) |
45 |
43 44
|
eqtrd |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> dom r = I ) |
46 |
45
|
ixpeq1d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> X_ x e. dom r ( Base ` ( r ` x ) ) = X_ x e. I ( Base ` ( r ` x ) ) ) |
47 |
4
|
ad2antrr |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> B = X_ x e. I ( Base ` ( R ` x ) ) ) |
48 |
42 46 47
|
3eqtr4d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> X_ x e. dom r ( Base ` ( r ` x ) ) = B ) |
49 |
|
prdsvallem |
|- ( f e. v , g e. v |-> X_ x e. dom r ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) ) e. _V |
50 |
49
|
a1i |
|- ( ( ( ( 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 ) |
51 |
|
simpr |
|- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> v = B ) |
52 |
45
|
adantr |
|- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> dom r = I ) |
53 |
52
|
ixpeq1d |
|- ( ( ( ( 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 ) ) ) |
54 |
40
|
fveq2d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( Hom ` ( r ` x ) ) = ( Hom ` ( R ` x ) ) ) |
55 |
54
|
oveqd |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( Hom ` ( r ` x ) ) ( g ` x ) ) = ( ( f ` x ) ( Hom ` ( R ` x ) ) ( g ` x ) ) ) |
56 |
55
|
ixpeq2dv |
|- ( ( ( 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 ) ) ) |
57 |
56
|
adantr |
|- ( ( ( ( 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 ) ) ) |
58 |
53 57
|
eqtrd |
|- ( ( ( ( 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 ) ) ) |
59 |
51 51 58
|
mpoeq123dv |
|- ( ( ( ( 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 ) ) ) ) |
60 |
12
|
ad3antrrr |
|- ( ( ( ( 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 ) ) ) ) |
61 |
59 60
|
eqtr4d |
|- ( ( ( ( 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 ) |
62 |
|
simplr |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> v = B ) |
63 |
62
|
opeq2d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> <. ( Base ` ndx ) , v >. = <. ( Base ` ndx ) , B >. ) |
64 |
40
|
fveq2d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( +g ` ( r ` x ) ) = ( +g ` ( R ` x ) ) ) |
65 |
64
|
oveqd |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( +g ` ( r ` x ) ) ( g ` x ) ) = ( ( f ` x ) ( +g ` ( R ` x ) ) ( g ` x ) ) ) |
66 |
45 65
|
mpteq12dv |
|- ( ( ( 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 ) ) ) ) |
67 |
66
|
adantr |
|- ( ( ( ( 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 ) ) ) ) |
68 |
51 51 67
|
mpoeq123dv |
|- ( ( ( ( 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 ) ) ) ) ) |
69 |
68
|
adantr |
|- ( ( ( ( ( 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 ) ) ) ) ) |
70 |
5
|
ad4antr |
|- ( ( ( ( ( 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 ) ) ) ) ) |
71 |
69 70
|
eqtr4d |
|- ( ( ( ( ( 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 ) ) ) ) = .+ ) |
72 |
71
|
opeq2d |
|- ( ( ( ( ( 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 ) , .+ >. ) |
73 |
40
|
fveq2d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( .r ` ( r ` x ) ) = ( .r ` ( R ` x ) ) ) |
74 |
73
|
oveqd |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( .r ` ( r ` x ) ) ( g ` x ) ) = ( ( f ` x ) ( .r ` ( R ` x ) ) ( g ` x ) ) ) |
75 |
45 74
|
mpteq12dv |
|- ( ( ( 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 ) ) ) ) |
76 |
75
|
adantr |
|- ( ( ( ( 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 ) ) ) ) |
77 |
51 51 76
|
mpoeq123dv |
|- ( ( ( ( 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 ) ) ) ) ) |
78 |
77
|
adantr |
|- ( ( ( ( ( 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 ) ) ) ) ) |
79 |
6
|
ad4antr |
|- ( ( ( ( ( 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 ) ) ) ) ) |
80 |
78 79
|
eqtr4d |
|- ( ( ( ( ( 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. ) |
81 |
80
|
opeq2d |
|- ( ( ( ( ( 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. >. ) |
82 |
63 72 81
|
tpeq123d |
|- ( ( ( ( ( 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. >. } ) |
83 |
|
simp-4r |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> s = S ) |
84 |
83
|
opeq2d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> <. ( Scalar ` ndx ) , s >. = <. ( Scalar ` ndx ) , S >. ) |
85 |
|
simpllr |
|- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> s = S ) |
86 |
85
|
fveq2d |
|- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> ( Base ` s ) = ( Base ` S ) ) |
87 |
86 2
|
eqtr4di |
|- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> ( Base ` s ) = K ) |
88 |
40
|
fveq2d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( .s ` ( r ` x ) ) = ( .s ` ( R ` x ) ) ) |
89 |
88
|
oveqd |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( f ( .s ` ( r ` x ) ) ( g ` x ) ) = ( f ( .s ` ( R ` x ) ) ( g ` x ) ) ) |
90 |
45 89
|
mpteq12dv |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( x e. dom r |-> ( f ( .s ` ( r ` x ) ) ( g ` x ) ) ) = ( x e. I |-> ( f ( .s ` ( R ` x ) ) ( g ` x ) ) ) ) |
91 |
90
|
adantr |
|- ( ( ( ( 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 ) ) ) ) |
92 |
87 51 91
|
mpoeq123dv |
|- ( ( ( ( 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 ) ) ) ) ) |
93 |
92
|
adantr |
|- ( ( ( ( ( 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 ) ) ) ) ) |
94 |
7
|
ad4antr |
|- ( ( ( ( ( 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 ) ) ) ) ) |
95 |
93 94
|
eqtr4d |
|- ( ( ( ( ( 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. ) |
96 |
95
|
opeq2d |
|- ( ( ( ( ( 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. >. ) |
97 |
40
|
fveq2d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( .i ` ( r ` x ) ) = ( .i ` ( R ` x ) ) ) |
98 |
97
|
oveqd |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) = ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) |
99 |
45 98
|
mpteq12dv |
|- ( ( ( 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 ) ) ) ) |
100 |
99
|
adantr |
|- ( ( ( ( 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 ) ) ) ) |
101 |
85 100
|
oveq12d |
|- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> ( s gsum ( x e. dom r |-> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) ) ) = ( S gsum ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) |
102 |
51 51 101
|
mpoeq123dv |
|- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> ( f e. v , g e. v |-> ( s gsum ( x e. dom r |-> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) ) ) ) = ( f e. B , g e. B |-> ( S gsum ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) ) |
103 |
102
|
adantr |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( f e. v , g e. v |-> ( s gsum ( x e. dom r |-> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) ) ) ) = ( f e. B , g e. B |-> ( S gsum ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) ) |
104 |
8
|
ad4antr |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ., = ( f e. B , g e. B |-> ( S gsum ( x e. I |-> ( ( f ` x ) ( .i ` ( R ` x ) ) ( g ` x ) ) ) ) ) ) |
105 |
103 104
|
eqtr4d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( f e. v , g e. v |-> ( s gsum ( x e. dom r |-> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) ) ) ) = ., ) |
106 |
105
|
opeq2d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> <. ( .i ` ndx ) , ( f e. v , g e. v |-> ( s gsum ( x e. dom r |-> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) ) ) ) >. = <. ( .i ` ndx ) , ., >. ) |
107 |
84 96 106
|
tpeq123d |
|- ( ( ( ( ( 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 gsum ( x e. dom r |-> ( ( f ` x ) ( .i ` ( r ` x ) ) ( g ` x ) ) ) ) ) >. } = { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) |
108 |
82 107
|
uneq12d |
|- ( ( ( ( ( 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 gsum ( 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 ) , ., >. } ) ) |
109 |
|
simpllr |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> r = R ) |
110 |
109
|
coeq2d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( TopOpen o. r ) = ( TopOpen o. R ) ) |
111 |
110
|
fveq2d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( Xt_ ` ( TopOpen o. r ) ) = ( Xt_ ` ( TopOpen o. R ) ) ) |
112 |
9
|
ad4antr |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> O = ( Xt_ ` ( TopOpen o. R ) ) ) |
113 |
111 112
|
eqtr4d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( Xt_ ` ( TopOpen o. r ) ) = O ) |
114 |
113
|
opeq2d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> <. ( TopSet ` ndx ) , ( Xt_ ` ( TopOpen o. r ) ) >. = <. ( TopSet ` ndx ) , O >. ) |
115 |
51
|
sseq2d |
|- ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) -> ( { f , g } C_ v <-> { f , g } C_ B ) ) |
116 |
40
|
fveq2d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( le ` ( r ` x ) ) = ( le ` ( R ` x ) ) ) |
117 |
116
|
breqd |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( le ` ( r ` x ) ) ( g ` x ) <-> ( f ` x ) ( le ` ( R ` x ) ) ( g ` x ) ) ) |
118 |
45 117
|
raleqbidv |
|- ( ( ( 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 ) ) ) |
119 |
118
|
adantr |
|- ( ( ( ( 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 ) ) ) |
120 |
115 119
|
anbi12d |
|- ( ( ( ( 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 ) ) ) ) |
121 |
120
|
opabbidv |
|- ( ( ( ( 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 ) ) } ) |
122 |
121
|
adantr |
|- ( ( ( ( ( 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 ) ) } ) |
123 |
10
|
ad4antr |
|- ( ( ( ( ( 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 ) ) } ) |
124 |
122 123
|
eqtr4d |
|- ( ( ( ( ( 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 ) ) } = .<_ ) |
125 |
124
|
opeq2d |
|- ( ( ( ( ( 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 ) , .<_ >. ) |
126 |
40
|
fveq2d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( dist ` ( r ` x ) ) = ( dist ` ( R ` x ) ) ) |
127 |
126
|
oveqd |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( ( f ` x ) ( dist ` ( r ` x ) ) ( g ` x ) ) = ( ( f ` x ) ( dist ` ( R ` x ) ) ( g ` x ) ) ) |
128 |
45 127
|
mpteq12dv |
|- ( ( ( 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 ) ) ) ) |
129 |
128
|
adantr |
|- ( ( ( ( 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 ) ) ) ) |
130 |
129
|
rneqd |
|- ( ( ( ( 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 ) ) ) ) |
131 |
130
|
uneq1d |
|- ( ( ( ( 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 } ) ) |
132 |
131
|
supeq1d |
|- ( ( ( ( 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* , < ) ) |
133 |
51 51 132
|
mpoeq123dv |
|- ( ( ( ( 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* , < ) ) ) |
134 |
133
|
adantr |
|- ( ( ( ( ( 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* , < ) ) ) |
135 |
11
|
ad4antr |
|- ( ( ( ( ( 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* , < ) ) ) |
136 |
134 135
|
eqtr4d |
|- ( ( ( ( ( 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 ) |
137 |
136
|
opeq2d |
|- ( ( ( ( ( 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 >. ) |
138 |
114 125 137
|
tpeq123d |
|- ( ( ( ( ( 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 >. } ) |
139 |
|
simpr |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> h = H ) |
140 |
139
|
opeq2d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> <. ( Hom ` ndx ) , h >. = <. ( Hom ` ndx ) , H >. ) |
141 |
62
|
sqxpeqd |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( v X. v ) = ( B X. B ) ) |
142 |
139
|
oveqd |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( ( 2nd ` a ) h c ) = ( ( 2nd ` a ) H c ) ) |
143 |
139
|
fveq1d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( h ` a ) = ( H ` a ) ) |
144 |
40
|
fveq2d |
|- ( ( ( ph /\ s = S ) /\ r = R ) -> ( comp ` ( r ` x ) ) = ( comp ` ( R ` x ) ) ) |
145 |
144
|
oveqd |
|- ( ( ( 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 ) ) ) |
146 |
145
|
oveqd |
|- ( ( ( 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 ) ) ) |
147 |
45 146
|
mpteq12dv |
|- ( ( ( 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 ) ) ) ) |
148 |
147
|
ad2antrr |
|- ( ( ( ( ( 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 ) ) ) ) |
149 |
142 143 148
|
mpoeq123dv |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( d e. ( ( 2nd ` a ) h c ) , 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. ( ( 2nd ` a ) H c ) , e e. ( H ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. ( comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) |
150 |
141 62 149
|
mpoeq123dv |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( a e. ( v X. v ) , c e. v |-> ( d e. ( ( 2nd ` a ) h c ) , 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. ( ( 2nd ` a ) H c ) , e e. ( H ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. ( comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) ) |
151 |
13
|
ad4antr |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> .xb = ( a e. ( B X. B ) , c e. B |-> ( d e. ( ( 2nd ` a ) H c ) , e e. ( H ` a ) |-> ( x e. I |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. ( comp ` ( R ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) ) |
152 |
150 151
|
eqtr4d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> ( a e. ( v X. v ) , c e. v |-> ( d e. ( ( 2nd ` a ) h c ) , e e. ( h ` a ) |-> ( x e. dom r |-> ( ( d ` x ) ( <. ( ( 1st ` a ) ` x ) , ( ( 2nd ` a ) ` x ) >. ( comp ` ( r ` x ) ) ( c ` x ) ) ( e ` x ) ) ) ) ) = .xb ) |
153 |
152
|
opeq2d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> <. ( comp ` ndx ) , ( a e. ( v X. v ) , c e. v |-> ( d e. ( ( 2nd ` a ) h c ) , 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 >. ) |
154 |
140 153
|
preq12d |
|- ( ( ( ( ( ph /\ s = S ) /\ r = R ) /\ v = B ) /\ h = H ) -> { <. ( Hom ` ndx ) , h >. , <. ( comp ` ndx ) , ( a e. ( v X. v ) , c e. v |-> ( d e. ( ( 2nd ` a ) h c ) , 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 >. } ) |
155 |
138 154
|
uneq12d |
|- ( ( ( ( ( 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. ( ( 2nd ` a ) h c ) , 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 >. } ) ) |
156 |
108 155
|
uneq12d |
|- ( ( ( ( ( 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 gsum ( 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. ( ( 2nd ` a ) h c ) , 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 >. } ) ) ) |
157 |
50 61 156
|
csbied2 |
|- ( ( ( ( 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 gsum ( 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. ( ( 2nd ` a ) h c ) , 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 >. } ) ) ) |
158 |
38 48 157
|
csbied2 |
|- ( ( ( 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 gsum ( 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. ( ( 2nd ` a ) h c ) , 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 >. } ) ) ) |
159 |
158
|
anasss |
|- ( ( 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 gsum ( 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. ( ( 2nd ` a ) h c ) , 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 >. } ) ) ) |
160 |
14
|
elexd |
|- ( ph -> S e. _V ) |
161 |
15
|
elexd |
|- ( ph -> R e. _V ) |
162 |
|
tpex |
|- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } e. _V |
163 |
|
tpex |
|- { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } e. _V |
164 |
162 163
|
unex |
|- ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. ( Scalar ` ndx ) , S >. , <. ( .s ` ndx ) , .x. >. , <. ( .i ` ndx ) , ., >. } ) e. _V |
165 |
|
tpex |
|- { <. ( TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } e. _V |
166 |
|
prex |
|- { <. ( Hom ` ndx ) , H >. , <. ( comp ` ndx ) , .xb >. } e. _V |
167 |
165 166
|
unex |
|- ( { <. ( TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } u. { <. ( Hom ` ndx ) , H >. , <. ( comp ` ndx ) , .xb >. } ) e. _V |
168 |
164 167
|
unex |
|- ( ( { <. ( 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 |
169 |
168
|
a1i |
|- ( 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 ) |
170 |
17 159 160 161 169
|
ovmpod |
|- ( ph -> ( S Xs_ 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 >. } ) ) ) |
171 |
1 170
|
eqtrid |
|- ( 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 >. } ) ) ) |