Step |
Hyp |
Ref |
Expression |
1 |
|
pserf.g |
|- G = ( x e. CC |-> ( n e. NN0 |-> ( ( A ` n ) x. ( x ^ n ) ) ) ) |
2 |
|
pserf.f |
|- F = ( y e. S |-> sum_ j e. NN0 ( ( G ` y ) ` j ) ) |
3 |
|
pserf.a |
|- ( ph -> A : NN0 --> CC ) |
4 |
|
pserf.r |
|- R = sup ( { r e. RR | seq 0 ( + , ( G ` r ) ) e. dom ~~> } , RR* , < ) |
5 |
|
psercn.s |
|- S = ( `' abs " ( 0 [,) R ) ) |
6 |
|
psercn.m |
|- M = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) |
7 |
|
sumex |
|- sum_ j e. NN0 ( ( G ` y ) ` j ) e. _V |
8 |
7
|
rgenw |
|- A. y e. S sum_ j e. NN0 ( ( G ` y ) ` j ) e. _V |
9 |
2
|
fnmpt |
|- ( A. y e. S sum_ j e. NN0 ( ( G ` y ) ` j ) e. _V -> F Fn S ) |
10 |
8 9
|
mp1i |
|- ( ph -> F Fn S ) |
11 |
|
cnvimass |
|- ( `' abs " ( 0 [,) R ) ) C_ dom abs |
12 |
|
absf |
|- abs : CC --> RR |
13 |
12
|
fdmi |
|- dom abs = CC |
14 |
11 13
|
sseqtri |
|- ( `' abs " ( 0 [,) R ) ) C_ CC |
15 |
5 14
|
eqsstri |
|- S C_ CC |
16 |
15
|
a1i |
|- ( ph -> S C_ CC ) |
17 |
16
|
sselda |
|- ( ( ph /\ a e. S ) -> a e. CC ) |
18 |
|
0cn |
|- 0 e. CC |
19 |
|
eqid |
|- ( abs o. - ) = ( abs o. - ) |
20 |
19
|
cnmetdval |
|- ( ( 0 e. CC /\ a e. CC ) -> ( 0 ( abs o. - ) a ) = ( abs ` ( 0 - a ) ) ) |
21 |
18 17 20
|
sylancr |
|- ( ( ph /\ a e. S ) -> ( 0 ( abs o. - ) a ) = ( abs ` ( 0 - a ) ) ) |
22 |
|
abssub |
|- ( ( 0 e. CC /\ a e. CC ) -> ( abs ` ( 0 - a ) ) = ( abs ` ( a - 0 ) ) ) |
23 |
18 17 22
|
sylancr |
|- ( ( ph /\ a e. S ) -> ( abs ` ( 0 - a ) ) = ( abs ` ( a - 0 ) ) ) |
24 |
17
|
subid1d |
|- ( ( ph /\ a e. S ) -> ( a - 0 ) = a ) |
25 |
24
|
fveq2d |
|- ( ( ph /\ a e. S ) -> ( abs ` ( a - 0 ) ) = ( abs ` a ) ) |
26 |
21 23 25
|
3eqtrd |
|- ( ( ph /\ a e. S ) -> ( 0 ( abs o. - ) a ) = ( abs ` a ) ) |
27 |
|
breq2 |
|- ( ( ( ( abs ` a ) + R ) / 2 ) = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) -> ( ( abs ` a ) < ( ( ( abs ` a ) + R ) / 2 ) <-> ( abs ` a ) < if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) ) ) |
28 |
|
breq2 |
|- ( ( ( abs ` a ) + 1 ) = if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) -> ( ( abs ` a ) < ( ( abs ` a ) + 1 ) <-> ( abs ` a ) < if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) ) ) |
29 |
|
simpr |
|- ( ( ph /\ a e. S ) -> a e. S ) |
30 |
29 5
|
eleqtrdi |
|- ( ( ph /\ a e. S ) -> a e. ( `' abs " ( 0 [,) R ) ) ) |
31 |
|
ffn |
|- ( abs : CC --> RR -> abs Fn CC ) |
32 |
|
elpreima |
|- ( abs Fn CC -> ( a e. ( `' abs " ( 0 [,) R ) ) <-> ( a e. CC /\ ( abs ` a ) e. ( 0 [,) R ) ) ) ) |
33 |
12 31 32
|
mp2b |
|- ( a e. ( `' abs " ( 0 [,) R ) ) <-> ( a e. CC /\ ( abs ` a ) e. ( 0 [,) R ) ) ) |
34 |
30 33
|
sylib |
|- ( ( ph /\ a e. S ) -> ( a e. CC /\ ( abs ` a ) e. ( 0 [,) R ) ) ) |
35 |
34
|
simprd |
|- ( ( ph /\ a e. S ) -> ( abs ` a ) e. ( 0 [,) R ) ) |
36 |
|
0re |
|- 0 e. RR |
37 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
38 |
1 3 4
|
radcnvcl |
|- ( ph -> R e. ( 0 [,] +oo ) ) |
39 |
38
|
adantr |
|- ( ( ph /\ a e. S ) -> R e. ( 0 [,] +oo ) ) |
40 |
37 39
|
sselid |
|- ( ( ph /\ a e. S ) -> R e. RR* ) |
41 |
|
elico2 |
|- ( ( 0 e. RR /\ R e. RR* ) -> ( ( abs ` a ) e. ( 0 [,) R ) <-> ( ( abs ` a ) e. RR /\ 0 <_ ( abs ` a ) /\ ( abs ` a ) < R ) ) ) |
42 |
36 40 41
|
sylancr |
|- ( ( ph /\ a e. S ) -> ( ( abs ` a ) e. ( 0 [,) R ) <-> ( ( abs ` a ) e. RR /\ 0 <_ ( abs ` a ) /\ ( abs ` a ) < R ) ) ) |
43 |
35 42
|
mpbid |
|- ( ( ph /\ a e. S ) -> ( ( abs ` a ) e. RR /\ 0 <_ ( abs ` a ) /\ ( abs ` a ) < R ) ) |
44 |
43
|
simp3d |
|- ( ( ph /\ a e. S ) -> ( abs ` a ) < R ) |
45 |
44
|
adantr |
|- ( ( ( ph /\ a e. S ) /\ R e. RR ) -> ( abs ` a ) < R ) |
46 |
17
|
abscld |
|- ( ( ph /\ a e. S ) -> ( abs ` a ) e. RR ) |
47 |
|
avglt1 |
|- ( ( ( abs ` a ) e. RR /\ R e. RR ) -> ( ( abs ` a ) < R <-> ( abs ` a ) < ( ( ( abs ` a ) + R ) / 2 ) ) ) |
48 |
46 47
|
sylan |
|- ( ( ( ph /\ a e. S ) /\ R e. RR ) -> ( ( abs ` a ) < R <-> ( abs ` a ) < ( ( ( abs ` a ) + R ) / 2 ) ) ) |
49 |
45 48
|
mpbid |
|- ( ( ( ph /\ a e. S ) /\ R e. RR ) -> ( abs ` a ) < ( ( ( abs ` a ) + R ) / 2 ) ) |
50 |
46
|
ltp1d |
|- ( ( ph /\ a e. S ) -> ( abs ` a ) < ( ( abs ` a ) + 1 ) ) |
51 |
50
|
adantr |
|- ( ( ( ph /\ a e. S ) /\ -. R e. RR ) -> ( abs ` a ) < ( ( abs ` a ) + 1 ) ) |
52 |
27 28 49 51
|
ifbothda |
|- ( ( ph /\ a e. S ) -> ( abs ` a ) < if ( R e. RR , ( ( ( abs ` a ) + R ) / 2 ) , ( ( abs ` a ) + 1 ) ) ) |
53 |
52 6
|
breqtrrdi |
|- ( ( ph /\ a e. S ) -> ( abs ` a ) < M ) |
54 |
26 53
|
eqbrtrd |
|- ( ( ph /\ a e. S ) -> ( 0 ( abs o. - ) a ) < M ) |
55 |
|
cnxmet |
|- ( abs o. - ) e. ( *Met ` CC ) |
56 |
1 2 3 4 5 6
|
psercnlem1 |
|- ( ( ph /\ a e. S ) -> ( M e. RR+ /\ ( abs ` a ) < M /\ M < R ) ) |
57 |
56
|
simp1d |
|- ( ( ph /\ a e. S ) -> M e. RR+ ) |
58 |
57
|
rpxrd |
|- ( ( ph /\ a e. S ) -> M e. RR* ) |
59 |
|
elbl |
|- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ M e. RR* ) -> ( a e. ( 0 ( ball ` ( abs o. - ) ) M ) <-> ( a e. CC /\ ( 0 ( abs o. - ) a ) < M ) ) ) |
60 |
55 18 58 59
|
mp3an12i |
|- ( ( ph /\ a e. S ) -> ( a e. ( 0 ( ball ` ( abs o. - ) ) M ) <-> ( a e. CC /\ ( 0 ( abs o. - ) a ) < M ) ) ) |
61 |
17 54 60
|
mpbir2and |
|- ( ( ph /\ a e. S ) -> a e. ( 0 ( ball ` ( abs o. - ) ) M ) ) |
62 |
61
|
fvresd |
|- ( ( ph /\ a e. S ) -> ( ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) ` a ) = ( F ` a ) ) |
63 |
2
|
reseq1i |
|- ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) = ( ( y e. S |-> sum_ j e. NN0 ( ( G ` y ) ` j ) ) |` ( 0 ( ball ` ( abs o. - ) ) M ) ) |
64 |
1 2 3 4 5 56
|
psercnlem2 |
|- ( ( ph /\ a e. S ) -> ( a e. ( 0 ( ball ` ( abs o. - ) ) M ) /\ ( 0 ( ball ` ( abs o. - ) ) M ) C_ ( `' abs " ( 0 [,] M ) ) /\ ( `' abs " ( 0 [,] M ) ) C_ S ) ) |
65 |
64
|
simp2d |
|- ( ( ph /\ a e. S ) -> ( 0 ( ball ` ( abs o. - ) ) M ) C_ ( `' abs " ( 0 [,] M ) ) ) |
66 |
64
|
simp3d |
|- ( ( ph /\ a e. S ) -> ( `' abs " ( 0 [,] M ) ) C_ S ) |
67 |
65 66
|
sstrd |
|- ( ( ph /\ a e. S ) -> ( 0 ( ball ` ( abs o. - ) ) M ) C_ S ) |
68 |
67
|
resmptd |
|- ( ( ph /\ a e. S ) -> ( ( y e. S |-> sum_ j e. NN0 ( ( G ` y ) ` j ) ) |` ( 0 ( ball ` ( abs o. - ) ) M ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> sum_ j e. NN0 ( ( G ` y ) ` j ) ) ) |
69 |
63 68
|
eqtrid |
|- ( ( ph /\ a e. S ) -> ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> sum_ j e. NN0 ( ( G ` y ) ` j ) ) ) |
70 |
|
eqid |
|- ( y e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> sum_ j e. NN0 ( ( G ` y ) ` j ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> sum_ j e. NN0 ( ( G ` y ) ` j ) ) |
71 |
3
|
adantr |
|- ( ( ph /\ a e. S ) -> A : NN0 --> CC ) |
72 |
|
fveq2 |
|- ( k = y -> ( G ` k ) = ( G ` y ) ) |
73 |
72
|
seqeq3d |
|- ( k = y -> seq 0 ( + , ( G ` k ) ) = seq 0 ( + , ( G ` y ) ) ) |
74 |
73
|
fveq1d |
|- ( k = y -> ( seq 0 ( + , ( G ` k ) ) ` s ) = ( seq 0 ( + , ( G ` y ) ) ` s ) ) |
75 |
74
|
cbvmptv |
|- ( k e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> ( seq 0 ( + , ( G ` k ) ) ` s ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> ( seq 0 ( + , ( G ` y ) ) ` s ) ) |
76 |
|
fveq2 |
|- ( s = i -> ( seq 0 ( + , ( G ` y ) ) ` s ) = ( seq 0 ( + , ( G ` y ) ) ` i ) ) |
77 |
76
|
mpteq2dv |
|- ( s = i -> ( y e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> ( seq 0 ( + , ( G ` y ) ) ` s ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> ( seq 0 ( + , ( G ` y ) ) ` i ) ) ) |
78 |
75 77
|
eqtrid |
|- ( s = i -> ( k e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> ( seq 0 ( + , ( G ` k ) ) ` s ) ) = ( y e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> ( seq 0 ( + , ( G ` y ) ) ` i ) ) ) |
79 |
78
|
cbvmptv |
|- ( s e. NN0 |-> ( k e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> ( seq 0 ( + , ( G ` k ) ) ` s ) ) ) = ( i e. NN0 |-> ( y e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> ( seq 0 ( + , ( G ` y ) ) ` i ) ) ) |
80 |
57
|
rpred |
|- ( ( ph /\ a e. S ) -> M e. RR ) |
81 |
56
|
simp3d |
|- ( ( ph /\ a e. S ) -> M < R ) |
82 |
1 70 71 4 79 80 81 65
|
psercn2 |
|- ( ( ph /\ a e. S ) -> ( y e. ( 0 ( ball ` ( abs o. - ) ) M ) |-> sum_ j e. NN0 ( ( G ` y ) ` j ) ) e. ( ( 0 ( ball ` ( abs o. - ) ) M ) -cn-> CC ) ) |
83 |
69 82
|
eqeltrd |
|- ( ( ph /\ a e. S ) -> ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) e. ( ( 0 ( ball ` ( abs o. - ) ) M ) -cn-> CC ) ) |
84 |
|
cncff |
|- ( ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) e. ( ( 0 ( ball ` ( abs o. - ) ) M ) -cn-> CC ) -> ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) : ( 0 ( ball ` ( abs o. - ) ) M ) --> CC ) |
85 |
83 84
|
syl |
|- ( ( ph /\ a e. S ) -> ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) : ( 0 ( ball ` ( abs o. - ) ) M ) --> CC ) |
86 |
85 61
|
ffvelrnd |
|- ( ( ph /\ a e. S ) -> ( ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) ` a ) e. CC ) |
87 |
62 86
|
eqeltrrd |
|- ( ( ph /\ a e. S ) -> ( F ` a ) e. CC ) |
88 |
87
|
ralrimiva |
|- ( ph -> A. a e. S ( F ` a ) e. CC ) |
89 |
|
ffnfv |
|- ( F : S --> CC <-> ( F Fn S /\ A. a e. S ( F ` a ) e. CC ) ) |
90 |
10 88 89
|
sylanbrc |
|- ( ph -> F : S --> CC ) |
91 |
67 15
|
sstrdi |
|- ( ( ph /\ a e. S ) -> ( 0 ( ball ` ( abs o. - ) ) M ) C_ CC ) |
92 |
|
ssid |
|- CC C_ CC |
93 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
94 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) = ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) |
95 |
93
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
96 |
95
|
toponrestid |
|- ( TopOpen ` CCfld ) = ( ( TopOpen ` CCfld ) |`t CC ) |
97 |
93 94 96
|
cncfcn |
|- ( ( ( 0 ( ball ` ( abs o. - ) ) M ) C_ CC /\ CC C_ CC ) -> ( ( 0 ( ball ` ( abs o. - ) ) M ) -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) Cn ( TopOpen ` CCfld ) ) ) |
98 |
91 92 97
|
sylancl |
|- ( ( ph /\ a e. S ) -> ( ( 0 ( ball ` ( abs o. - ) ) M ) -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) Cn ( TopOpen ` CCfld ) ) ) |
99 |
83 98
|
eleqtrd |
|- ( ( ph /\ a e. S ) -> ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) Cn ( TopOpen ` CCfld ) ) ) |
100 |
93
|
cnfldtop |
|- ( TopOpen ` CCfld ) e. Top |
101 |
|
unicntop |
|- CC = U. ( TopOpen ` CCfld ) |
102 |
101
|
restuni |
|- ( ( ( TopOpen ` CCfld ) e. Top /\ ( 0 ( ball ` ( abs o. - ) ) M ) C_ CC ) -> ( 0 ( ball ` ( abs o. - ) ) M ) = U. ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) ) |
103 |
100 91 102
|
sylancr |
|- ( ( ph /\ a e. S ) -> ( 0 ( ball ` ( abs o. - ) ) M ) = U. ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) ) |
104 |
61 103
|
eleqtrd |
|- ( ( ph /\ a e. S ) -> a e. U. ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) ) |
105 |
|
eqid |
|- U. ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) = U. ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) |
106 |
105
|
cncnpi |
|- ( ( ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) e. ( ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) Cn ( TopOpen ` CCfld ) ) /\ a e. U. ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) ) -> ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) CnP ( TopOpen ` CCfld ) ) ` a ) ) |
107 |
99 104 106
|
syl2anc |
|- ( ( ph /\ a e. S ) -> ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) e. ( ( ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) CnP ( TopOpen ` CCfld ) ) ` a ) ) |
108 |
|
cnex |
|- CC e. _V |
109 |
108 15
|
ssexi |
|- S e. _V |
110 |
109
|
a1i |
|- ( ( ph /\ a e. S ) -> S e. _V ) |
111 |
|
restabs |
|- ( ( ( TopOpen ` CCfld ) e. Top /\ ( 0 ( ball ` ( abs o. - ) ) M ) C_ S /\ S e. _V ) -> ( ( ( TopOpen ` CCfld ) |`t S ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) = ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) ) |
112 |
100 67 110 111
|
mp3an2i |
|- ( ( ph /\ a e. S ) -> ( ( ( TopOpen ` CCfld ) |`t S ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) = ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) ) |
113 |
112
|
oveq1d |
|- ( ( ph /\ a e. S ) -> ( ( ( ( TopOpen ` CCfld ) |`t S ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) CnP ( TopOpen ` CCfld ) ) = ( ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) CnP ( TopOpen ` CCfld ) ) ) |
114 |
113
|
fveq1d |
|- ( ( ph /\ a e. S ) -> ( ( ( ( ( TopOpen ` CCfld ) |`t S ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) CnP ( TopOpen ` CCfld ) ) ` a ) = ( ( ( ( TopOpen ` CCfld ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) CnP ( TopOpen ` CCfld ) ) ` a ) ) |
115 |
107 114
|
eleqtrrd |
|- ( ( ph /\ a e. S ) -> ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) e. ( ( ( ( ( TopOpen ` CCfld ) |`t S ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) CnP ( TopOpen ` CCfld ) ) ` a ) ) |
116 |
|
resttop |
|- ( ( ( TopOpen ` CCfld ) e. Top /\ S e. _V ) -> ( ( TopOpen ` CCfld ) |`t S ) e. Top ) |
117 |
100 109 116
|
mp2an |
|- ( ( TopOpen ` CCfld ) |`t S ) e. Top |
118 |
117
|
a1i |
|- ( ( ph /\ a e. S ) -> ( ( TopOpen ` CCfld ) |`t S ) e. Top ) |
119 |
|
df-ss |
|- ( ( 0 ( ball ` ( abs o. - ) ) M ) C_ S <-> ( ( 0 ( ball ` ( abs o. - ) ) M ) i^i S ) = ( 0 ( ball ` ( abs o. - ) ) M ) ) |
120 |
67 119
|
sylib |
|- ( ( ph /\ a e. S ) -> ( ( 0 ( ball ` ( abs o. - ) ) M ) i^i S ) = ( 0 ( ball ` ( abs o. - ) ) M ) ) |
121 |
93
|
cnfldtopn |
|- ( TopOpen ` CCfld ) = ( MetOpen ` ( abs o. - ) ) |
122 |
121
|
blopn |
|- ( ( ( abs o. - ) e. ( *Met ` CC ) /\ 0 e. CC /\ M e. RR* ) -> ( 0 ( ball ` ( abs o. - ) ) M ) e. ( TopOpen ` CCfld ) ) |
123 |
55 18 58 122
|
mp3an12i |
|- ( ( ph /\ a e. S ) -> ( 0 ( ball ` ( abs o. - ) ) M ) e. ( TopOpen ` CCfld ) ) |
124 |
|
elrestr |
|- ( ( ( TopOpen ` CCfld ) e. Top /\ S e. _V /\ ( 0 ( ball ` ( abs o. - ) ) M ) e. ( TopOpen ` CCfld ) ) -> ( ( 0 ( ball ` ( abs o. - ) ) M ) i^i S ) e. ( ( TopOpen ` CCfld ) |`t S ) ) |
125 |
100 109 123 124
|
mp3an12i |
|- ( ( ph /\ a e. S ) -> ( ( 0 ( ball ` ( abs o. - ) ) M ) i^i S ) e. ( ( TopOpen ` CCfld ) |`t S ) ) |
126 |
120 125
|
eqeltrrd |
|- ( ( ph /\ a e. S ) -> ( 0 ( ball ` ( abs o. - ) ) M ) e. ( ( TopOpen ` CCfld ) |`t S ) ) |
127 |
|
isopn3i |
|- ( ( ( ( TopOpen ` CCfld ) |`t S ) e. Top /\ ( 0 ( ball ` ( abs o. - ) ) M ) e. ( ( TopOpen ` CCfld ) |`t S ) ) -> ( ( int ` ( ( TopOpen ` CCfld ) |`t S ) ) ` ( 0 ( ball ` ( abs o. - ) ) M ) ) = ( 0 ( ball ` ( abs o. - ) ) M ) ) |
128 |
117 126 127
|
sylancr |
|- ( ( ph /\ a e. S ) -> ( ( int ` ( ( TopOpen ` CCfld ) |`t S ) ) ` ( 0 ( ball ` ( abs o. - ) ) M ) ) = ( 0 ( ball ` ( abs o. - ) ) M ) ) |
129 |
61 128
|
eleqtrrd |
|- ( ( ph /\ a e. S ) -> a e. ( ( int ` ( ( TopOpen ` CCfld ) |`t S ) ) ` ( 0 ( ball ` ( abs o. - ) ) M ) ) ) |
130 |
90
|
adantr |
|- ( ( ph /\ a e. S ) -> F : S --> CC ) |
131 |
101
|
restuni |
|- ( ( ( TopOpen ` CCfld ) e. Top /\ S C_ CC ) -> S = U. ( ( TopOpen ` CCfld ) |`t S ) ) |
132 |
100 15 131
|
mp2an |
|- S = U. ( ( TopOpen ` CCfld ) |`t S ) |
133 |
132 101
|
cnprest |
|- ( ( ( ( ( TopOpen ` CCfld ) |`t S ) e. Top /\ ( 0 ( ball ` ( abs o. - ) ) M ) C_ S ) /\ ( a e. ( ( int ` ( ( TopOpen ` CCfld ) |`t S ) ) ` ( 0 ( ball ` ( abs o. - ) ) M ) ) /\ F : S --> CC ) ) -> ( F e. ( ( ( ( TopOpen ` CCfld ) |`t S ) CnP ( TopOpen ` CCfld ) ) ` a ) <-> ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) e. ( ( ( ( ( TopOpen ` CCfld ) |`t S ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) CnP ( TopOpen ` CCfld ) ) ` a ) ) ) |
134 |
118 67 129 130 133
|
syl22anc |
|- ( ( ph /\ a e. S ) -> ( F e. ( ( ( ( TopOpen ` CCfld ) |`t S ) CnP ( TopOpen ` CCfld ) ) ` a ) <-> ( F |` ( 0 ( ball ` ( abs o. - ) ) M ) ) e. ( ( ( ( ( TopOpen ` CCfld ) |`t S ) |`t ( 0 ( ball ` ( abs o. - ) ) M ) ) CnP ( TopOpen ` CCfld ) ) ` a ) ) ) |
135 |
115 134
|
mpbird |
|- ( ( ph /\ a e. S ) -> F e. ( ( ( ( TopOpen ` CCfld ) |`t S ) CnP ( TopOpen ` CCfld ) ) ` a ) ) |
136 |
135
|
ralrimiva |
|- ( ph -> A. a e. S F e. ( ( ( ( TopOpen ` CCfld ) |`t S ) CnP ( TopOpen ` CCfld ) ) ` a ) ) |
137 |
|
resttopon |
|- ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ S C_ CC ) -> ( ( TopOpen ` CCfld ) |`t S ) e. ( TopOn ` S ) ) |
138 |
95 15 137
|
mp2an |
|- ( ( TopOpen ` CCfld ) |`t S ) e. ( TopOn ` S ) |
139 |
|
cncnp |
|- ( ( ( ( TopOpen ` CCfld ) |`t S ) e. ( TopOn ` S ) /\ ( TopOpen ` CCfld ) e. ( TopOn ` CC ) ) -> ( F e. ( ( ( TopOpen ` CCfld ) |`t S ) Cn ( TopOpen ` CCfld ) ) <-> ( F : S --> CC /\ A. a e. S F e. ( ( ( ( TopOpen ` CCfld ) |`t S ) CnP ( TopOpen ` CCfld ) ) ` a ) ) ) ) |
140 |
138 95 139
|
mp2an |
|- ( F e. ( ( ( TopOpen ` CCfld ) |`t S ) Cn ( TopOpen ` CCfld ) ) <-> ( F : S --> CC /\ A. a e. S F e. ( ( ( ( TopOpen ` CCfld ) |`t S ) CnP ( TopOpen ` CCfld ) ) ` a ) ) ) |
141 |
90 136 140
|
sylanbrc |
|- ( ph -> F e. ( ( ( TopOpen ` CCfld ) |`t S ) Cn ( TopOpen ` CCfld ) ) ) |
142 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t S ) = ( ( TopOpen ` CCfld ) |`t S ) |
143 |
93 142 96
|
cncfcn |
|- ( ( S C_ CC /\ CC C_ CC ) -> ( S -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t S ) Cn ( TopOpen ` CCfld ) ) ) |
144 |
15 92 143
|
mp2an |
|- ( S -cn-> CC ) = ( ( ( TopOpen ` CCfld ) |`t S ) Cn ( TopOpen ` CCfld ) ) |
145 |
141 144
|
eleqtrrdi |
|- ( ph -> F e. ( S -cn-> CC ) ) |