Step |
Hyp |
Ref |
Expression |
1 |
|
prdstmdd.y |
|- Y = ( S Xs_ R ) |
2 |
|
prdstmdd.i |
|- ( ph -> I e. W ) |
3 |
|
prdstmdd.s |
|- ( ph -> S e. V ) |
4 |
|
prdstmdd.r |
|- ( ph -> R : I --> TopMnd ) |
5 |
|
tmdmnd |
|- ( x e. TopMnd -> x e. Mnd ) |
6 |
5
|
ssriv |
|- TopMnd C_ Mnd |
7 |
|
fss |
|- ( ( R : I --> TopMnd /\ TopMnd C_ Mnd ) -> R : I --> Mnd ) |
8 |
4 6 7
|
sylancl |
|- ( ph -> R : I --> Mnd ) |
9 |
1 2 3 8
|
prdsmndd |
|- ( ph -> Y e. Mnd ) |
10 |
|
tmdtps |
|- ( x e. TopMnd -> x e. TopSp ) |
11 |
10
|
ssriv |
|- TopMnd C_ TopSp |
12 |
|
fss |
|- ( ( R : I --> TopMnd /\ TopMnd C_ TopSp ) -> R : I --> TopSp ) |
13 |
4 11 12
|
sylancl |
|- ( ph -> R : I --> TopSp ) |
14 |
1 3 2 13
|
prdstps |
|- ( ph -> Y e. TopSp ) |
15 |
|
eqid |
|- ( Base ` Y ) = ( Base ` Y ) |
16 |
3
|
3ad2ant1 |
|- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> S e. V ) |
17 |
2
|
3ad2ant1 |
|- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> I e. W ) |
18 |
4
|
ffnd |
|- ( ph -> R Fn I ) |
19 |
18
|
3ad2ant1 |
|- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> R Fn I ) |
20 |
|
simp2 |
|- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> f e. ( Base ` Y ) ) |
21 |
|
simp3 |
|- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> g e. ( Base ` Y ) ) |
22 |
|
eqid |
|- ( +g ` Y ) = ( +g ` Y ) |
23 |
1 15 16 17 19 20 21 22
|
prdsplusgval |
|- ( ( ph /\ f e. ( Base ` Y ) /\ g e. ( Base ` Y ) ) -> ( f ( +g ` Y ) g ) = ( k e. I |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) ) |
24 |
23
|
mpoeq3dva |
|- ( ph -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( f ( +g ` Y ) g ) ) = ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( k e. I |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) ) ) |
25 |
|
eqid |
|- ( +f ` Y ) = ( +f ` Y ) |
26 |
15 22 25
|
plusffval |
|- ( +f ` Y ) = ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( f ( +g ` Y ) g ) ) |
27 |
|
vex |
|- f e. _V |
28 |
|
vex |
|- g e. _V |
29 |
27 28
|
op1std |
|- ( z = <. f , g >. -> ( 1st ` z ) = f ) |
30 |
29
|
fveq1d |
|- ( z = <. f , g >. -> ( ( 1st ` z ) ` k ) = ( f ` k ) ) |
31 |
27 28
|
op2ndd |
|- ( z = <. f , g >. -> ( 2nd ` z ) = g ) |
32 |
31
|
fveq1d |
|- ( z = <. f , g >. -> ( ( 2nd ` z ) ` k ) = ( g ` k ) ) |
33 |
30 32
|
oveq12d |
|- ( z = <. f , g >. -> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) = ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) |
34 |
33
|
mpteq2dv |
|- ( z = <. f , g >. -> ( k e. I |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) = ( k e. I |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) ) |
35 |
34
|
mpompt |
|- ( z e. ( ( Base ` Y ) X. ( Base ` Y ) ) |-> ( k e. I |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) ) = ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( k e. I |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) ) |
36 |
24 26 35
|
3eqtr4g |
|- ( ph -> ( +f ` Y ) = ( z e. ( ( Base ` Y ) X. ( Base ` Y ) ) |-> ( k e. I |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) ) ) |
37 |
|
eqid |
|- ( Xt_ ` ( TopOpen o. R ) ) = ( Xt_ ` ( TopOpen o. R ) ) |
38 |
|
eqid |
|- ( TopOpen ` Y ) = ( TopOpen ` Y ) |
39 |
15 38
|
istps |
|- ( Y e. TopSp <-> ( TopOpen ` Y ) e. ( TopOn ` ( Base ` Y ) ) ) |
40 |
14 39
|
sylib |
|- ( ph -> ( TopOpen ` Y ) e. ( TopOn ` ( Base ` Y ) ) ) |
41 |
|
txtopon |
|- ( ( ( TopOpen ` Y ) e. ( TopOn ` ( Base ` Y ) ) /\ ( TopOpen ` Y ) e. ( TopOn ` ( Base ` Y ) ) ) -> ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) e. ( TopOn ` ( ( Base ` Y ) X. ( Base ` Y ) ) ) ) |
42 |
40 40 41
|
syl2anc |
|- ( ph -> ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) e. ( TopOn ` ( ( Base ` Y ) X. ( Base ` Y ) ) ) ) |
43 |
|
topnfn |
|- TopOpen Fn _V |
44 |
|
ssv |
|- TopSp C_ _V |
45 |
|
fnssres |
|- ( ( TopOpen Fn _V /\ TopSp C_ _V ) -> ( TopOpen |` TopSp ) Fn TopSp ) |
46 |
43 44 45
|
mp2an |
|- ( TopOpen |` TopSp ) Fn TopSp |
47 |
|
fvres |
|- ( x e. TopSp -> ( ( TopOpen |` TopSp ) ` x ) = ( TopOpen ` x ) ) |
48 |
|
eqid |
|- ( TopOpen ` x ) = ( TopOpen ` x ) |
49 |
48
|
tpstop |
|- ( x e. TopSp -> ( TopOpen ` x ) e. Top ) |
50 |
47 49
|
eqeltrd |
|- ( x e. TopSp -> ( ( TopOpen |` TopSp ) ` x ) e. Top ) |
51 |
50
|
rgen |
|- A. x e. TopSp ( ( TopOpen |` TopSp ) ` x ) e. Top |
52 |
|
ffnfv |
|- ( ( TopOpen |` TopSp ) : TopSp --> Top <-> ( ( TopOpen |` TopSp ) Fn TopSp /\ A. x e. TopSp ( ( TopOpen |` TopSp ) ` x ) e. Top ) ) |
53 |
46 51 52
|
mpbir2an |
|- ( TopOpen |` TopSp ) : TopSp --> Top |
54 |
|
fco2 |
|- ( ( ( TopOpen |` TopSp ) : TopSp --> Top /\ R : I --> TopSp ) -> ( TopOpen o. R ) : I --> Top ) |
55 |
53 13 54
|
sylancr |
|- ( ph -> ( TopOpen o. R ) : I --> Top ) |
56 |
33
|
mpompt |
|- ( z e. ( ( Base ` Y ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) = ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) |
57 |
|
eqid |
|- ( TopOpen ` ( R ` k ) ) = ( TopOpen ` ( R ` k ) ) |
58 |
|
eqid |
|- ( +g ` ( R ` k ) ) = ( +g ` ( R ` k ) ) |
59 |
4
|
ffvelrnda |
|- ( ( ph /\ k e. I ) -> ( R ` k ) e. TopMnd ) |
60 |
40
|
adantr |
|- ( ( ph /\ k e. I ) -> ( TopOpen ` Y ) e. ( TopOn ` ( Base ` Y ) ) ) |
61 |
60 60
|
cnmpt1st |
|- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> f ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` Y ) ) ) |
62 |
1 3 2 18 38
|
prdstopn |
|- ( ph -> ( TopOpen ` Y ) = ( Xt_ ` ( TopOpen o. R ) ) ) |
63 |
62
|
adantr |
|- ( ( ph /\ k e. I ) -> ( TopOpen ` Y ) = ( Xt_ ` ( TopOpen o. R ) ) ) |
64 |
63 60
|
eqeltrrd |
|- ( ( ph /\ k e. I ) -> ( Xt_ ` ( TopOpen o. R ) ) e. ( TopOn ` ( Base ` Y ) ) ) |
65 |
|
toponuni |
|- ( ( Xt_ ` ( TopOpen o. R ) ) e. ( TopOn ` ( Base ` Y ) ) -> ( Base ` Y ) = U. ( Xt_ ` ( TopOpen o. R ) ) ) |
66 |
64 65
|
syl |
|- ( ( ph /\ k e. I ) -> ( Base ` Y ) = U. ( Xt_ ` ( TopOpen o. R ) ) ) |
67 |
66
|
mpteq1d |
|- ( ( ph /\ k e. I ) -> ( x e. ( Base ` Y ) |-> ( x ` k ) ) = ( x e. U. ( Xt_ ` ( TopOpen o. R ) ) |-> ( x ` k ) ) ) |
68 |
2
|
adantr |
|- ( ( ph /\ k e. I ) -> I e. W ) |
69 |
55
|
adantr |
|- ( ( ph /\ k e. I ) -> ( TopOpen o. R ) : I --> Top ) |
70 |
|
simpr |
|- ( ( ph /\ k e. I ) -> k e. I ) |
71 |
|
eqid |
|- U. ( Xt_ ` ( TopOpen o. R ) ) = U. ( Xt_ ` ( TopOpen o. R ) ) |
72 |
71 37
|
ptpjcn |
|- ( ( I e. W /\ ( TopOpen o. R ) : I --> Top /\ k e. I ) -> ( x e. U. ( Xt_ ` ( TopOpen o. R ) ) |-> ( x ` k ) ) e. ( ( Xt_ ` ( TopOpen o. R ) ) Cn ( ( TopOpen o. R ) ` k ) ) ) |
73 |
68 69 70 72
|
syl3anc |
|- ( ( ph /\ k e. I ) -> ( x e. U. ( Xt_ ` ( TopOpen o. R ) ) |-> ( x ` k ) ) e. ( ( Xt_ ` ( TopOpen o. R ) ) Cn ( ( TopOpen o. R ) ` k ) ) ) |
74 |
67 73
|
eqeltrd |
|- ( ( ph /\ k e. I ) -> ( x e. ( Base ` Y ) |-> ( x ` k ) ) e. ( ( Xt_ ` ( TopOpen o. R ) ) Cn ( ( TopOpen o. R ) ` k ) ) ) |
75 |
63
|
eqcomd |
|- ( ( ph /\ k e. I ) -> ( Xt_ ` ( TopOpen o. R ) ) = ( TopOpen ` Y ) ) |
76 |
|
fvco3 |
|- ( ( R : I --> TopMnd /\ k e. I ) -> ( ( TopOpen o. R ) ` k ) = ( TopOpen ` ( R ` k ) ) ) |
77 |
4 76
|
sylan |
|- ( ( ph /\ k e. I ) -> ( ( TopOpen o. R ) ` k ) = ( TopOpen ` ( R ` k ) ) ) |
78 |
75 77
|
oveq12d |
|- ( ( ph /\ k e. I ) -> ( ( Xt_ ` ( TopOpen o. R ) ) Cn ( ( TopOpen o. R ) ` k ) ) = ( ( TopOpen ` Y ) Cn ( TopOpen ` ( R ` k ) ) ) ) |
79 |
74 78
|
eleqtrd |
|- ( ( ph /\ k e. I ) -> ( x e. ( Base ` Y ) |-> ( x ` k ) ) e. ( ( TopOpen ` Y ) Cn ( TopOpen ` ( R ` k ) ) ) ) |
80 |
|
fveq1 |
|- ( x = f -> ( x ` k ) = ( f ` k ) ) |
81 |
60 60 61 60 79 80
|
cnmpt21 |
|- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( f ` k ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` ( R ` k ) ) ) ) |
82 |
60 60
|
cnmpt2nd |
|- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> g ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` Y ) ) ) |
83 |
|
fveq1 |
|- ( x = g -> ( x ` k ) = ( g ` k ) ) |
84 |
60 60 82 60 79 83
|
cnmpt21 |
|- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( g ` k ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` ( R ` k ) ) ) ) |
85 |
57 58 59 60 60 81 84
|
cnmpt2plusg |
|- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` ( R ` k ) ) ) ) |
86 |
77
|
oveq2d |
|- ( ( ph /\ k e. I ) -> ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( ( TopOpen o. R ) ` k ) ) = ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` ( R ` k ) ) ) ) |
87 |
85 86
|
eleqtrrd |
|- ( ( ph /\ k e. I ) -> ( f e. ( Base ` Y ) , g e. ( Base ` Y ) |-> ( ( f ` k ) ( +g ` ( R ` k ) ) ( g ` k ) ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( ( TopOpen o. R ) ` k ) ) ) |
88 |
56 87
|
eqeltrid |
|- ( ( ph /\ k e. I ) -> ( z e. ( ( Base ` Y ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( ( TopOpen o. R ) ` k ) ) ) |
89 |
37 42 2 55 88
|
ptcn |
|- ( ph -> ( z e. ( ( Base ` Y ) X. ( Base ` Y ) ) |-> ( k e. I |-> ( ( ( 1st ` z ) ` k ) ( +g ` ( R ` k ) ) ( ( 2nd ` z ) ` k ) ) ) ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( Xt_ ` ( TopOpen o. R ) ) ) ) |
90 |
36 89
|
eqeltrd |
|- ( ph -> ( +f ` Y ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( Xt_ ` ( TopOpen o. R ) ) ) ) |
91 |
62
|
oveq2d |
|- ( ph -> ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` Y ) ) = ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( Xt_ ` ( TopOpen o. R ) ) ) ) |
92 |
90 91
|
eleqtrrd |
|- ( ph -> ( +f ` Y ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` Y ) ) ) |
93 |
25 38
|
istmd |
|- ( Y e. TopMnd <-> ( Y e. Mnd /\ Y e. TopSp /\ ( +f ` Y ) e. ( ( ( TopOpen ` Y ) tX ( TopOpen ` Y ) ) Cn ( TopOpen ` Y ) ) ) ) |
94 |
9 14 92 93
|
syl3anbrc |
|- ( ph -> Y e. TopMnd ) |