Step |
Hyp |
Ref |
Expression |
1 |
|
lhop.a |
|- ( ph -> A C_ RR ) |
2 |
|
lhop.f |
|- ( ph -> F : A --> RR ) |
3 |
|
lhop.g |
|- ( ph -> G : A --> RR ) |
4 |
|
lhop.i |
|- ( ph -> I e. ( topGen ` ran (,) ) ) |
5 |
|
lhop.b |
|- ( ph -> B e. I ) |
6 |
|
lhop.d |
|- D = ( I \ { B } ) |
7 |
|
lhop.if |
|- ( ph -> D C_ dom ( RR _D F ) ) |
8 |
|
lhop.ig |
|- ( ph -> D C_ dom ( RR _D G ) ) |
9 |
|
lhop.f0 |
|- ( ph -> 0 e. ( F limCC B ) ) |
10 |
|
lhop.g0 |
|- ( ph -> 0 e. ( G limCC B ) ) |
11 |
|
lhop.gn0 |
|- ( ph -> -. 0 e. ( G " D ) ) |
12 |
|
lhop.gd0 |
|- ( ph -> -. 0 e. ( ( RR _D G ) " D ) ) |
13 |
|
lhop.c |
|- ( ph -> C e. ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) limCC B ) ) |
14 |
|
eqid |
|- ( ( abs o. - ) |` ( RR X. RR ) ) = ( ( abs o. - ) |` ( RR X. RR ) ) |
15 |
14
|
rexmet |
|- ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) |
16 |
15
|
a1i |
|- ( ph -> ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) ) |
17 |
|
eqid |
|- ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) |
18 |
14 17
|
tgioo |
|- ( topGen ` ran (,) ) = ( MetOpen ` ( ( abs o. - ) |` ( RR X. RR ) ) ) |
19 |
18
|
mopni2 |
|- ( ( ( ( abs o. - ) |` ( RR X. RR ) ) e. ( *Met ` RR ) /\ I e. ( topGen ` ran (,) ) /\ B e. I ) -> E. r e. RR+ ( B ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ I ) |
20 |
16 4 5 19
|
syl3anc |
|- ( ph -> E. r e. RR+ ( B ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ I ) |
21 |
|
elssuni |
|- ( I e. ( topGen ` ran (,) ) -> I C_ U. ( topGen ` ran (,) ) ) |
22 |
|
uniretop |
|- RR = U. ( topGen ` ran (,) ) |
23 |
21 22
|
sseqtrrdi |
|- ( I e. ( topGen ` ran (,) ) -> I C_ RR ) |
24 |
4 23
|
syl |
|- ( ph -> I C_ RR ) |
25 |
24 5
|
sseldd |
|- ( ph -> B e. RR ) |
26 |
|
rpre |
|- ( r e. RR+ -> r e. RR ) |
27 |
14
|
bl2ioo |
|- ( ( B e. RR /\ r e. RR ) -> ( B ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) = ( ( B - r ) (,) ( B + r ) ) ) |
28 |
25 26 27
|
syl2an |
|- ( ( ph /\ r e. RR+ ) -> ( B ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) = ( ( B - r ) (,) ( B + r ) ) ) |
29 |
28
|
sseq1d |
|- ( ( ph /\ r e. RR+ ) -> ( ( B ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ I <-> ( ( B - r ) (,) ( B + r ) ) C_ I ) ) |
30 |
25
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> B e. RR ) |
31 |
|
simprl |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> r e. RR+ ) |
32 |
31
|
rpred |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> r e. RR ) |
33 |
30 32
|
resubcld |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B - r ) e. RR ) |
34 |
33
|
rexrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B - r ) e. RR* ) |
35 |
30 31
|
ltsubrpd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B - r ) < B ) |
36 |
2
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> F : A --> RR ) |
37 |
|
ssun1 |
|- ( ( B - r ) (,) B ) C_ ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) |
38 |
|
unass |
|- ( ( { B } u. ( ( B - r ) (,) B ) ) u. ( B (,) ( B + r ) ) ) = ( { B } u. ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) |
39 |
|
uncom |
|- ( { B } u. ( ( B - r ) (,) B ) ) = ( ( ( B - r ) (,) B ) u. { B } ) |
40 |
39
|
uneq1i |
|- ( ( { B } u. ( ( B - r ) (,) B ) ) u. ( B (,) ( B + r ) ) ) = ( ( ( ( B - r ) (,) B ) u. { B } ) u. ( B (,) ( B + r ) ) ) |
41 |
38 40
|
eqtr3i |
|- ( { B } u. ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) = ( ( ( ( B - r ) (,) B ) u. { B } ) u. ( B (,) ( B + r ) ) ) |
42 |
30
|
rexrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> B e. RR* ) |
43 |
30 32
|
readdcld |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B + r ) e. RR ) |
44 |
43
|
rexrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B + r ) e. RR* ) |
45 |
30 31
|
ltaddrpd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> B < ( B + r ) ) |
46 |
|
ioojoin |
|- ( ( ( ( B - r ) e. RR* /\ B e. RR* /\ ( B + r ) e. RR* ) /\ ( ( B - r ) < B /\ B < ( B + r ) ) ) -> ( ( ( ( B - r ) (,) B ) u. { B } ) u. ( B (,) ( B + r ) ) ) = ( ( B - r ) (,) ( B + r ) ) ) |
47 |
34 42 44 35 45 46
|
syl32anc |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( ( B - r ) (,) B ) u. { B } ) u. ( B (,) ( B + r ) ) ) = ( ( B - r ) (,) ( B + r ) ) ) |
48 |
41 47
|
syl5eq |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( { B } u. ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) = ( ( B - r ) (,) ( B + r ) ) ) |
49 |
|
elioo2 |
|- ( ( ( B - r ) e. RR* /\ ( B + r ) e. RR* ) -> ( B e. ( ( B - r ) (,) ( B + r ) ) <-> ( B e. RR /\ ( B - r ) < B /\ B < ( B + r ) ) ) ) |
50 |
34 44 49
|
syl2anc |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B e. ( ( B - r ) (,) ( B + r ) ) <-> ( B e. RR /\ ( B - r ) < B /\ B < ( B + r ) ) ) ) |
51 |
30 35 45 50
|
mpbir3and |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> B e. ( ( B - r ) (,) ( B + r ) ) ) |
52 |
51
|
snssd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> { B } C_ ( ( B - r ) (,) ( B + r ) ) ) |
53 |
|
incom |
|- ( { B } i^i ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) = ( ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) i^i { B } ) |
54 |
|
ubioo |
|- -. B e. ( ( B - r ) (,) B ) |
55 |
|
lbioo |
|- -. B e. ( B (,) ( B + r ) ) |
56 |
54 55
|
pm3.2ni |
|- -. ( B e. ( ( B - r ) (,) B ) \/ B e. ( B (,) ( B + r ) ) ) |
57 |
|
elun |
|- ( B e. ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) <-> ( B e. ( ( B - r ) (,) B ) \/ B e. ( B (,) ( B + r ) ) ) ) |
58 |
56 57
|
mtbir |
|- -. B e. ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) |
59 |
|
disjsn |
|- ( ( ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) i^i { B } ) = (/) <-> -. B e. ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) |
60 |
58 59
|
mpbir |
|- ( ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) i^i { B } ) = (/) |
61 |
53 60
|
eqtri |
|- ( { B } i^i ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) = (/) |
62 |
|
uneqdifeq |
|- ( ( { B } C_ ( ( B - r ) (,) ( B + r ) ) /\ ( { B } i^i ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) = (/) ) -> ( ( { B } u. ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) = ( ( B - r ) (,) ( B + r ) ) <-> ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) = ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) ) |
63 |
52 61 62
|
sylancl |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( { B } u. ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) = ( ( B - r ) (,) ( B + r ) ) <-> ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) = ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) ) |
64 |
48 63
|
mpbid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) = ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) ) |
65 |
37 64
|
sseqtrrid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( B - r ) (,) B ) C_ ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) ) |
66 |
|
ssdif |
|- ( ( ( B - r ) (,) ( B + r ) ) C_ I -> ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) C_ ( I \ { B } ) ) |
67 |
66
|
ad2antll |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) C_ ( I \ { B } ) ) |
68 |
67 6
|
sseqtrrdi |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) C_ D ) |
69 |
|
ax-resscn |
|- RR C_ CC |
70 |
69
|
a1i |
|- ( ph -> RR C_ CC ) |
71 |
|
fss |
|- ( ( F : A --> RR /\ RR C_ CC ) -> F : A --> CC ) |
72 |
2 69 71
|
sylancl |
|- ( ph -> F : A --> CC ) |
73 |
70 72 1
|
dvbss |
|- ( ph -> dom ( RR _D F ) C_ A ) |
74 |
7 73
|
sstrd |
|- ( ph -> D C_ A ) |
75 |
74
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> D C_ A ) |
76 |
68 75
|
sstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) C_ A ) |
77 |
65 76
|
sstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( B - r ) (,) B ) C_ A ) |
78 |
36 77
|
fssresd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( F |` ( ( B - r ) (,) B ) ) : ( ( B - r ) (,) B ) --> RR ) |
79 |
3
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> G : A --> RR ) |
80 |
79 77
|
fssresd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( G |` ( ( B - r ) (,) B ) ) : ( ( B - r ) (,) B ) --> RR ) |
81 |
69
|
a1i |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> RR C_ CC ) |
82 |
72
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> F : A --> CC ) |
83 |
1
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> A C_ RR ) |
84 |
|
ioossre |
|- ( ( B - r ) (,) B ) C_ RR |
85 |
84
|
a1i |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( B - r ) (,) B ) C_ RR ) |
86 |
|
eqid |
|- ( TopOpen ` CCfld ) = ( TopOpen ` CCfld ) |
87 |
86
|
tgioo2 |
|- ( topGen ` ran (,) ) = ( ( TopOpen ` CCfld ) |`t RR ) |
88 |
86 87
|
dvres |
|- ( ( ( RR C_ CC /\ F : A --> CC ) /\ ( A C_ RR /\ ( ( B - r ) (,) B ) C_ RR ) ) -> ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( B - r ) (,) B ) ) ) ) |
89 |
81 82 83 85 88
|
syl22anc |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( B - r ) (,) B ) ) ) ) |
90 |
|
retop |
|- ( topGen ` ran (,) ) e. Top |
91 |
|
iooretop |
|- ( ( B - r ) (,) B ) e. ( topGen ` ran (,) ) |
92 |
|
isopn3i |
|- ( ( ( topGen ` ran (,) ) e. Top /\ ( ( B - r ) (,) B ) e. ( topGen ` ran (,) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( ( B - r ) (,) B ) ) = ( ( B - r ) (,) B ) ) |
93 |
90 91 92
|
mp2an |
|- ( ( int ` ( topGen ` ran (,) ) ) ` ( ( B - r ) (,) B ) ) = ( ( B - r ) (,) B ) |
94 |
93
|
reseq2i |
|- ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( B - r ) (,) B ) ) ) = ( ( RR _D F ) |` ( ( B - r ) (,) B ) ) |
95 |
89 94
|
eqtrdi |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) = ( ( RR _D F ) |` ( ( B - r ) (,) B ) ) ) |
96 |
95
|
dmeqd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) = dom ( ( RR _D F ) |` ( ( B - r ) (,) B ) ) ) |
97 |
65 68
|
sstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( B - r ) (,) B ) C_ D ) |
98 |
7
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> D C_ dom ( RR _D F ) ) |
99 |
97 98
|
sstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( B - r ) (,) B ) C_ dom ( RR _D F ) ) |
100 |
|
ssdmres |
|- ( ( ( B - r ) (,) B ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( ( B - r ) (,) B ) ) = ( ( B - r ) (,) B ) ) |
101 |
99 100
|
sylib |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( ( RR _D F ) |` ( ( B - r ) (,) B ) ) = ( ( B - r ) (,) B ) ) |
102 |
96 101
|
eqtrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) = ( ( B - r ) (,) B ) ) |
103 |
|
fss |
|- ( ( G : A --> RR /\ RR C_ CC ) -> G : A --> CC ) |
104 |
3 69 103
|
sylancl |
|- ( ph -> G : A --> CC ) |
105 |
104
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> G : A --> CC ) |
106 |
86 87
|
dvres |
|- ( ( ( RR C_ CC /\ G : A --> CC ) /\ ( A C_ RR /\ ( ( B - r ) (,) B ) C_ RR ) ) -> ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) = ( ( RR _D G ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( B - r ) (,) B ) ) ) ) |
107 |
81 105 83 85 106
|
syl22anc |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) = ( ( RR _D G ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( B - r ) (,) B ) ) ) ) |
108 |
93
|
reseq2i |
|- ( ( RR _D G ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( ( B - r ) (,) B ) ) ) = ( ( RR _D G ) |` ( ( B - r ) (,) B ) ) |
109 |
107 108
|
eqtrdi |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) = ( ( RR _D G ) |` ( ( B - r ) (,) B ) ) ) |
110 |
109
|
dmeqd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) = dom ( ( RR _D G ) |` ( ( B - r ) (,) B ) ) ) |
111 |
8
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> D C_ dom ( RR _D G ) ) |
112 |
97 111
|
sstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( B - r ) (,) B ) C_ dom ( RR _D G ) ) |
113 |
|
ssdmres |
|- ( ( ( B - r ) (,) B ) C_ dom ( RR _D G ) <-> dom ( ( RR _D G ) |` ( ( B - r ) (,) B ) ) = ( ( B - r ) (,) B ) ) |
114 |
112 113
|
sylib |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( ( RR _D G ) |` ( ( B - r ) (,) B ) ) = ( ( B - r ) (,) B ) ) |
115 |
110 114
|
eqtrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) = ( ( B - r ) (,) B ) ) |
116 |
|
limcresi |
|- ( F limCC B ) C_ ( ( F |` ( ( B - r ) (,) B ) ) limCC B ) |
117 |
9
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> 0 e. ( F limCC B ) ) |
118 |
116 117
|
sselid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> 0 e. ( ( F |` ( ( B - r ) (,) B ) ) limCC B ) ) |
119 |
|
limcresi |
|- ( G limCC B ) C_ ( ( G |` ( ( B - r ) (,) B ) ) limCC B ) |
120 |
10
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> 0 e. ( G limCC B ) ) |
121 |
119 120
|
sselid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> 0 e. ( ( G |` ( ( B - r ) (,) B ) ) limCC B ) ) |
122 |
|
df-ima |
|- ( G " ( ( B - r ) (,) B ) ) = ran ( G |` ( ( B - r ) (,) B ) ) |
123 |
|
imass2 |
|- ( ( ( B - r ) (,) B ) C_ D -> ( G " ( ( B - r ) (,) B ) ) C_ ( G " D ) ) |
124 |
97 123
|
syl |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( G " ( ( B - r ) (,) B ) ) C_ ( G " D ) ) |
125 |
122 124
|
eqsstrrid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ran ( G |` ( ( B - r ) (,) B ) ) C_ ( G " D ) ) |
126 |
11
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> -. 0 e. ( G " D ) ) |
127 |
125 126
|
ssneldd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> -. 0 e. ran ( G |` ( ( B - r ) (,) B ) ) ) |
128 |
109
|
rneqd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ran ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) = ran ( ( RR _D G ) |` ( ( B - r ) (,) B ) ) ) |
129 |
|
df-ima |
|- ( ( RR _D G ) " ( ( B - r ) (,) B ) ) = ran ( ( RR _D G ) |` ( ( B - r ) (,) B ) ) |
130 |
128 129
|
eqtr4di |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ran ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) = ( ( RR _D G ) " ( ( B - r ) (,) B ) ) ) |
131 |
|
imass2 |
|- ( ( ( B - r ) (,) B ) C_ D -> ( ( RR _D G ) " ( ( B - r ) (,) B ) ) C_ ( ( RR _D G ) " D ) ) |
132 |
97 131
|
syl |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( RR _D G ) " ( ( B - r ) (,) B ) ) C_ ( ( RR _D G ) " D ) ) |
133 |
130 132
|
eqsstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ran ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) C_ ( ( RR _D G ) " D ) ) |
134 |
12
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> -. 0 e. ( ( RR _D G ) " D ) ) |
135 |
133 134
|
ssneldd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> -. 0 e. ran ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) ) |
136 |
|
limcresi |
|- ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) limCC B ) C_ ( ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) |` ( ( B - r ) (,) B ) ) limCC B ) |
137 |
97
|
resmptd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) |` ( ( B - r ) (,) B ) ) = ( z e. ( ( B - r ) (,) B ) |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) ) |
138 |
95
|
fveq1d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) ` z ) = ( ( ( RR _D F ) |` ( ( B - r ) (,) B ) ) ` z ) ) |
139 |
|
fvres |
|- ( z e. ( ( B - r ) (,) B ) -> ( ( ( RR _D F ) |` ( ( B - r ) (,) B ) ) ` z ) = ( ( RR _D F ) ` z ) ) |
140 |
138 139
|
sylan9eq |
|- ( ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) /\ z e. ( ( B - r ) (,) B ) ) -> ( ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) ` z ) = ( ( RR _D F ) ` z ) ) |
141 |
109
|
fveq1d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) ` z ) = ( ( ( RR _D G ) |` ( ( B - r ) (,) B ) ) ` z ) ) |
142 |
|
fvres |
|- ( z e. ( ( B - r ) (,) B ) -> ( ( ( RR _D G ) |` ( ( B - r ) (,) B ) ) ` z ) = ( ( RR _D G ) ` z ) ) |
143 |
141 142
|
sylan9eq |
|- ( ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) /\ z e. ( ( B - r ) (,) B ) ) -> ( ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) ` z ) = ( ( RR _D G ) ` z ) ) |
144 |
140 143
|
oveq12d |
|- ( ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) /\ z e. ( ( B - r ) (,) B ) ) -> ( ( ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) ` z ) / ( ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) ` z ) ) = ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) |
145 |
144
|
mpteq2dva |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( z e. ( ( B - r ) (,) B ) |-> ( ( ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) ` z ) / ( ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) ` z ) ) ) = ( z e. ( ( B - r ) (,) B ) |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) ) |
146 |
137 145
|
eqtr4d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) |` ( ( B - r ) (,) B ) ) = ( z e. ( ( B - r ) (,) B ) |-> ( ( ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) ` z ) / ( ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) ` z ) ) ) ) |
147 |
146
|
oveq1d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) |` ( ( B - r ) (,) B ) ) limCC B ) = ( ( z e. ( ( B - r ) (,) B ) |-> ( ( ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) ` z ) / ( ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) ` z ) ) ) limCC B ) ) |
148 |
136 147
|
sseqtrid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) limCC B ) C_ ( ( z e. ( ( B - r ) (,) B ) |-> ( ( ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) ` z ) / ( ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) ` z ) ) ) limCC B ) ) |
149 |
13
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> C e. ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) limCC B ) ) |
150 |
148 149
|
sseldd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> C e. ( ( z e. ( ( B - r ) (,) B ) |-> ( ( ( RR _D ( F |` ( ( B - r ) (,) B ) ) ) ` z ) / ( ( RR _D ( G |` ( ( B - r ) (,) B ) ) ) ` z ) ) ) limCC B ) ) |
151 |
34 30 35 78 80 102 115 118 121 127 135 150
|
lhop2 |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> C e. ( ( z e. ( ( B - r ) (,) B ) |-> ( ( ( F |` ( ( B - r ) (,) B ) ) ` z ) / ( ( G |` ( ( B - r ) (,) B ) ) ` z ) ) ) limCC B ) ) |
152 |
65
|
resmptd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( ( B - r ) (,) B ) ) = ( z e. ( ( B - r ) (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) ) |
153 |
|
fvres |
|- ( z e. ( ( B - r ) (,) B ) -> ( ( F |` ( ( B - r ) (,) B ) ) ` z ) = ( F ` z ) ) |
154 |
|
fvres |
|- ( z e. ( ( B - r ) (,) B ) -> ( ( G |` ( ( B - r ) (,) B ) ) ` z ) = ( G ` z ) ) |
155 |
153 154
|
oveq12d |
|- ( z e. ( ( B - r ) (,) B ) -> ( ( ( F |` ( ( B - r ) (,) B ) ) ` z ) / ( ( G |` ( ( B - r ) (,) B ) ) ` z ) ) = ( ( F ` z ) / ( G ` z ) ) ) |
156 |
155
|
mpteq2ia |
|- ( z e. ( ( B - r ) (,) B ) |-> ( ( ( F |` ( ( B - r ) (,) B ) ) ` z ) / ( ( G |` ( ( B - r ) (,) B ) ) ` z ) ) ) = ( z e. ( ( B - r ) (,) B ) |-> ( ( F ` z ) / ( G ` z ) ) ) |
157 |
152 156
|
eqtr4di |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( ( B - r ) (,) B ) ) = ( z e. ( ( B - r ) (,) B ) |-> ( ( ( F |` ( ( B - r ) (,) B ) ) ` z ) / ( ( G |` ( ( B - r ) (,) B ) ) ` z ) ) ) ) |
158 |
157
|
oveq1d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( ( B - r ) (,) B ) ) limCC B ) = ( ( z e. ( ( B - r ) (,) B ) |-> ( ( ( F |` ( ( B - r ) (,) B ) ) ` z ) / ( ( G |` ( ( B - r ) (,) B ) ) ` z ) ) ) limCC B ) ) |
159 |
151 158
|
eleqtrrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> C e. ( ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( ( B - r ) (,) B ) ) limCC B ) ) |
160 |
|
ssun2 |
|- ( B (,) ( B + r ) ) C_ ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) |
161 |
160 64
|
sseqtrrid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B (,) ( B + r ) ) C_ ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) ) |
162 |
161 76
|
sstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B (,) ( B + r ) ) C_ A ) |
163 |
36 162
|
fssresd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( F |` ( B (,) ( B + r ) ) ) : ( B (,) ( B + r ) ) --> RR ) |
164 |
79 162
|
fssresd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( G |` ( B (,) ( B + r ) ) ) : ( B (,) ( B + r ) ) --> RR ) |
165 |
|
ioossre |
|- ( B (,) ( B + r ) ) C_ RR |
166 |
165
|
a1i |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B (,) ( B + r ) ) C_ RR ) |
167 |
86 87
|
dvres |
|- ( ( ( RR C_ CC /\ F : A --> CC ) /\ ( A C_ RR /\ ( B (,) ( B + r ) ) C_ RR ) ) -> ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( B (,) ( B + r ) ) ) ) ) |
168 |
81 82 83 166 167
|
syl22anc |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) = ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( B (,) ( B + r ) ) ) ) ) |
169 |
|
iooretop |
|- ( B (,) ( B + r ) ) e. ( topGen ` ran (,) ) |
170 |
|
isopn3i |
|- ( ( ( topGen ` ran (,) ) e. Top /\ ( B (,) ( B + r ) ) e. ( topGen ` ran (,) ) ) -> ( ( int ` ( topGen ` ran (,) ) ) ` ( B (,) ( B + r ) ) ) = ( B (,) ( B + r ) ) ) |
171 |
90 169 170
|
mp2an |
|- ( ( int ` ( topGen ` ran (,) ) ) ` ( B (,) ( B + r ) ) ) = ( B (,) ( B + r ) ) |
172 |
171
|
reseq2i |
|- ( ( RR _D F ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( B (,) ( B + r ) ) ) ) = ( ( RR _D F ) |` ( B (,) ( B + r ) ) ) |
173 |
168 172
|
eqtrdi |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) = ( ( RR _D F ) |` ( B (,) ( B + r ) ) ) ) |
174 |
173
|
dmeqd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) = dom ( ( RR _D F ) |` ( B (,) ( B + r ) ) ) ) |
175 |
161 68
|
sstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B (,) ( B + r ) ) C_ D ) |
176 |
175 98
|
sstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B (,) ( B + r ) ) C_ dom ( RR _D F ) ) |
177 |
|
ssdmres |
|- ( ( B (,) ( B + r ) ) C_ dom ( RR _D F ) <-> dom ( ( RR _D F ) |` ( B (,) ( B + r ) ) ) = ( B (,) ( B + r ) ) ) |
178 |
176 177
|
sylib |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( ( RR _D F ) |` ( B (,) ( B + r ) ) ) = ( B (,) ( B + r ) ) ) |
179 |
174 178
|
eqtrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) = ( B (,) ( B + r ) ) ) |
180 |
86 87
|
dvres |
|- ( ( ( RR C_ CC /\ G : A --> CC ) /\ ( A C_ RR /\ ( B (,) ( B + r ) ) C_ RR ) ) -> ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) = ( ( RR _D G ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( B (,) ( B + r ) ) ) ) ) |
181 |
81 105 83 166 180
|
syl22anc |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) = ( ( RR _D G ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( B (,) ( B + r ) ) ) ) ) |
182 |
171
|
reseq2i |
|- ( ( RR _D G ) |` ( ( int ` ( topGen ` ran (,) ) ) ` ( B (,) ( B + r ) ) ) ) = ( ( RR _D G ) |` ( B (,) ( B + r ) ) ) |
183 |
181 182
|
eqtrdi |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) = ( ( RR _D G ) |` ( B (,) ( B + r ) ) ) ) |
184 |
183
|
dmeqd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) = dom ( ( RR _D G ) |` ( B (,) ( B + r ) ) ) ) |
185 |
175 111
|
sstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B (,) ( B + r ) ) C_ dom ( RR _D G ) ) |
186 |
|
ssdmres |
|- ( ( B (,) ( B + r ) ) C_ dom ( RR _D G ) <-> dom ( ( RR _D G ) |` ( B (,) ( B + r ) ) ) = ( B (,) ( B + r ) ) ) |
187 |
185 186
|
sylib |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( ( RR _D G ) |` ( B (,) ( B + r ) ) ) = ( B (,) ( B + r ) ) ) |
188 |
184 187
|
eqtrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> dom ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) = ( B (,) ( B + r ) ) ) |
189 |
|
limcresi |
|- ( F limCC B ) C_ ( ( F |` ( B (,) ( B + r ) ) ) limCC B ) |
190 |
189 117
|
sselid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> 0 e. ( ( F |` ( B (,) ( B + r ) ) ) limCC B ) ) |
191 |
|
limcresi |
|- ( G limCC B ) C_ ( ( G |` ( B (,) ( B + r ) ) ) limCC B ) |
192 |
191 120
|
sselid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> 0 e. ( ( G |` ( B (,) ( B + r ) ) ) limCC B ) ) |
193 |
|
df-ima |
|- ( G " ( B (,) ( B + r ) ) ) = ran ( G |` ( B (,) ( B + r ) ) ) |
194 |
|
imass2 |
|- ( ( B (,) ( B + r ) ) C_ D -> ( G " ( B (,) ( B + r ) ) ) C_ ( G " D ) ) |
195 |
175 194
|
syl |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( G " ( B (,) ( B + r ) ) ) C_ ( G " D ) ) |
196 |
193 195
|
eqsstrrid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ran ( G |` ( B (,) ( B + r ) ) ) C_ ( G " D ) ) |
197 |
196 126
|
ssneldd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> -. 0 e. ran ( G |` ( B (,) ( B + r ) ) ) ) |
198 |
183
|
rneqd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ran ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) = ran ( ( RR _D G ) |` ( B (,) ( B + r ) ) ) ) |
199 |
|
df-ima |
|- ( ( RR _D G ) " ( B (,) ( B + r ) ) ) = ran ( ( RR _D G ) |` ( B (,) ( B + r ) ) ) |
200 |
198 199
|
eqtr4di |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ran ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) = ( ( RR _D G ) " ( B (,) ( B + r ) ) ) ) |
201 |
|
imass2 |
|- ( ( B (,) ( B + r ) ) C_ D -> ( ( RR _D G ) " ( B (,) ( B + r ) ) ) C_ ( ( RR _D G ) " D ) ) |
202 |
175 201
|
syl |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( RR _D G ) " ( B (,) ( B + r ) ) ) C_ ( ( RR _D G ) " D ) ) |
203 |
200 202
|
eqsstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ran ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) C_ ( ( RR _D G ) " D ) ) |
204 |
203 134
|
ssneldd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> -. 0 e. ran ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) ) |
205 |
|
limcresi |
|- ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) limCC B ) C_ ( ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) |` ( B (,) ( B + r ) ) ) limCC B ) |
206 |
175
|
resmptd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) |` ( B (,) ( B + r ) ) ) = ( z e. ( B (,) ( B + r ) ) |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) ) |
207 |
173
|
fveq1d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) ` z ) = ( ( ( RR _D F ) |` ( B (,) ( B + r ) ) ) ` z ) ) |
208 |
|
fvres |
|- ( z e. ( B (,) ( B + r ) ) -> ( ( ( RR _D F ) |` ( B (,) ( B + r ) ) ) ` z ) = ( ( RR _D F ) ` z ) ) |
209 |
207 208
|
sylan9eq |
|- ( ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) /\ z e. ( B (,) ( B + r ) ) ) -> ( ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) ` z ) = ( ( RR _D F ) ` z ) ) |
210 |
183
|
fveq1d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) ` z ) = ( ( ( RR _D G ) |` ( B (,) ( B + r ) ) ) ` z ) ) |
211 |
|
fvres |
|- ( z e. ( B (,) ( B + r ) ) -> ( ( ( RR _D G ) |` ( B (,) ( B + r ) ) ) ` z ) = ( ( RR _D G ) ` z ) ) |
212 |
210 211
|
sylan9eq |
|- ( ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) /\ z e. ( B (,) ( B + r ) ) ) -> ( ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) ` z ) = ( ( RR _D G ) ` z ) ) |
213 |
209 212
|
oveq12d |
|- ( ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) /\ z e. ( B (,) ( B + r ) ) ) -> ( ( ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) ` z ) / ( ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) ` z ) ) = ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) |
214 |
213
|
mpteq2dva |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( z e. ( B (,) ( B + r ) ) |-> ( ( ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) ` z ) / ( ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) ` z ) ) ) = ( z e. ( B (,) ( B + r ) ) |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) ) |
215 |
206 214
|
eqtr4d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) |` ( B (,) ( B + r ) ) ) = ( z e. ( B (,) ( B + r ) ) |-> ( ( ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) ` z ) / ( ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) ` z ) ) ) ) |
216 |
215
|
oveq1d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) |` ( B (,) ( B + r ) ) ) limCC B ) = ( ( z e. ( B (,) ( B + r ) ) |-> ( ( ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) ` z ) / ( ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) ` z ) ) ) limCC B ) ) |
217 |
205 216
|
sseqtrid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. D |-> ( ( ( RR _D F ) ` z ) / ( ( RR _D G ) ` z ) ) ) limCC B ) C_ ( ( z e. ( B (,) ( B + r ) ) |-> ( ( ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) ` z ) / ( ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) ` z ) ) ) limCC B ) ) |
218 |
217 149
|
sseldd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> C e. ( ( z e. ( B (,) ( B + r ) ) |-> ( ( ( RR _D ( F |` ( B (,) ( B + r ) ) ) ) ` z ) / ( ( RR _D ( G |` ( B (,) ( B + r ) ) ) ) ` z ) ) ) limCC B ) ) |
219 |
30 44 45 163 164 179 188 190 192 197 204 218
|
lhop1 |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> C e. ( ( z e. ( B (,) ( B + r ) ) |-> ( ( ( F |` ( B (,) ( B + r ) ) ) ` z ) / ( ( G |` ( B (,) ( B + r ) ) ) ` z ) ) ) limCC B ) ) |
220 |
161
|
resmptd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( B (,) ( B + r ) ) ) = ( z e. ( B (,) ( B + r ) ) |-> ( ( F ` z ) / ( G ` z ) ) ) ) |
221 |
|
fvres |
|- ( z e. ( B (,) ( B + r ) ) -> ( ( F |` ( B (,) ( B + r ) ) ) ` z ) = ( F ` z ) ) |
222 |
|
fvres |
|- ( z e. ( B (,) ( B + r ) ) -> ( ( G |` ( B (,) ( B + r ) ) ) ` z ) = ( G ` z ) ) |
223 |
221 222
|
oveq12d |
|- ( z e. ( B (,) ( B + r ) ) -> ( ( ( F |` ( B (,) ( B + r ) ) ) ` z ) / ( ( G |` ( B (,) ( B + r ) ) ) ` z ) ) = ( ( F ` z ) / ( G ` z ) ) ) |
224 |
223
|
mpteq2ia |
|- ( z e. ( B (,) ( B + r ) ) |-> ( ( ( F |` ( B (,) ( B + r ) ) ) ` z ) / ( ( G |` ( B (,) ( B + r ) ) ) ` z ) ) ) = ( z e. ( B (,) ( B + r ) ) |-> ( ( F ` z ) / ( G ` z ) ) ) |
225 |
220 224
|
eqtr4di |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( B (,) ( B + r ) ) ) = ( z e. ( B (,) ( B + r ) ) |-> ( ( ( F |` ( B (,) ( B + r ) ) ) ` z ) / ( ( G |` ( B (,) ( B + r ) ) ) ` z ) ) ) ) |
226 |
225
|
oveq1d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( B (,) ( B + r ) ) ) limCC B ) = ( ( z e. ( B (,) ( B + r ) ) |-> ( ( ( F |` ( B (,) ( B + r ) ) ) ` z ) / ( ( G |` ( B (,) ( B + r ) ) ) ` z ) ) ) limCC B ) ) |
227 |
219 226
|
eleqtrrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> C e. ( ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( B (,) ( B + r ) ) ) limCC B ) ) |
228 |
159 227
|
elind |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> C e. ( ( ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( ( B - r ) (,) B ) ) limCC B ) i^i ( ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( B (,) ( B + r ) ) ) limCC B ) ) ) |
229 |
68
|
resmptd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) ) = ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) ) |
230 |
229
|
oveq1d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) ) limCC B ) = ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) limCC B ) ) |
231 |
74
|
sselda |
|- ( ( ph /\ z e. D ) -> z e. A ) |
232 |
2
|
ffvelrnda |
|- ( ( ph /\ z e. A ) -> ( F ` z ) e. RR ) |
233 |
231 232
|
syldan |
|- ( ( ph /\ z e. D ) -> ( F ` z ) e. RR ) |
234 |
233
|
recnd |
|- ( ( ph /\ z e. D ) -> ( F ` z ) e. CC ) |
235 |
3
|
ffvelrnda |
|- ( ( ph /\ z e. A ) -> ( G ` z ) e. RR ) |
236 |
231 235
|
syldan |
|- ( ( ph /\ z e. D ) -> ( G ` z ) e. RR ) |
237 |
236
|
recnd |
|- ( ( ph /\ z e. D ) -> ( G ` z ) e. CC ) |
238 |
11
|
adantr |
|- ( ( ph /\ z e. D ) -> -. 0 e. ( G " D ) ) |
239 |
3
|
ffnd |
|- ( ph -> G Fn A ) |
240 |
239
|
adantr |
|- ( ( ph /\ z e. D ) -> G Fn A ) |
241 |
74
|
adantr |
|- ( ( ph /\ z e. D ) -> D C_ A ) |
242 |
|
simpr |
|- ( ( ph /\ z e. D ) -> z e. D ) |
243 |
|
fnfvima |
|- ( ( G Fn A /\ D C_ A /\ z e. D ) -> ( G ` z ) e. ( G " D ) ) |
244 |
240 241 242 243
|
syl3anc |
|- ( ( ph /\ z e. D ) -> ( G ` z ) e. ( G " D ) ) |
245 |
|
eleq1 |
|- ( ( G ` z ) = 0 -> ( ( G ` z ) e. ( G " D ) <-> 0 e. ( G " D ) ) ) |
246 |
244 245
|
syl5ibcom |
|- ( ( ph /\ z e. D ) -> ( ( G ` z ) = 0 -> 0 e. ( G " D ) ) ) |
247 |
246
|
necon3bd |
|- ( ( ph /\ z e. D ) -> ( -. 0 e. ( G " D ) -> ( G ` z ) =/= 0 ) ) |
248 |
238 247
|
mpd |
|- ( ( ph /\ z e. D ) -> ( G ` z ) =/= 0 ) |
249 |
234 237 248
|
divcld |
|- ( ( ph /\ z e. D ) -> ( ( F ` z ) / ( G ` z ) ) e. CC ) |
250 |
249
|
adantlr |
|- ( ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) /\ z e. D ) -> ( ( F ` z ) / ( G ` z ) ) e. CC ) |
251 |
250
|
fmpttd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) : D --> CC ) |
252 |
|
difss |
|- ( I \ { B } ) C_ I |
253 |
6 252
|
eqsstri |
|- D C_ I |
254 |
24 69
|
sstrdi |
|- ( ph -> I C_ CC ) |
255 |
253 254
|
sstrid |
|- ( ph -> D C_ CC ) |
256 |
255
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> D C_ CC ) |
257 |
|
eqid |
|- ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) = ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) |
258 |
6
|
uneq1i |
|- ( D u. { B } ) = ( ( I \ { B } ) u. { B } ) |
259 |
|
undif1 |
|- ( ( I \ { B } ) u. { B } ) = ( I u. { B } ) |
260 |
258 259
|
eqtri |
|- ( D u. { B } ) = ( I u. { B } ) |
261 |
|
simprr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( B - r ) (,) ( B + r ) ) C_ I ) |
262 |
52 261
|
sstrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> { B } C_ I ) |
263 |
|
ssequn2 |
|- ( { B } C_ I <-> ( I u. { B } ) = I ) |
264 |
262 263
|
sylib |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( I u. { B } ) = I ) |
265 |
260 264
|
syl5eq |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( D u. { B } ) = I ) |
266 |
265
|
oveq2d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) = ( ( TopOpen ` CCfld ) |`t I ) ) |
267 |
24
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> I C_ RR ) |
268 |
|
eqid |
|- ( topGen ` ran (,) ) = ( topGen ` ran (,) ) |
269 |
86 268
|
rerest |
|- ( I C_ RR -> ( ( TopOpen ` CCfld ) |`t I ) = ( ( topGen ` ran (,) ) |`t I ) ) |
270 |
267 269
|
syl |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( TopOpen ` CCfld ) |`t I ) = ( ( topGen ` ran (,) ) |`t I ) ) |
271 |
266 270
|
eqtrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) = ( ( topGen ` ran (,) ) |`t I ) ) |
272 |
271
|
fveq2d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( int ` ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) ) = ( int ` ( ( topGen ` ran (,) ) |`t I ) ) ) |
273 |
272
|
fveq1d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( int ` ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) ) ` ( ( B - r ) (,) ( B + r ) ) ) = ( ( int ` ( ( topGen ` ran (,) ) |`t I ) ) ` ( ( B - r ) (,) ( B + r ) ) ) ) |
274 |
86
|
cnfldtopon |
|- ( TopOpen ` CCfld ) e. ( TopOn ` CC ) |
275 |
254
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> I C_ CC ) |
276 |
|
resttopon |
|- ( ( ( TopOpen ` CCfld ) e. ( TopOn ` CC ) /\ I C_ CC ) -> ( ( TopOpen ` CCfld ) |`t I ) e. ( TopOn ` I ) ) |
277 |
274 275 276
|
sylancr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( TopOpen ` CCfld ) |`t I ) e. ( TopOn ` I ) ) |
278 |
|
topontop |
|- ( ( ( TopOpen ` CCfld ) |`t I ) e. ( TopOn ` I ) -> ( ( TopOpen ` CCfld ) |`t I ) e. Top ) |
279 |
277 278
|
syl |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( TopOpen ` CCfld ) |`t I ) e. Top ) |
280 |
270 279
|
eqeltrrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( topGen ` ran (,) ) |`t I ) e. Top ) |
281 |
|
iooretop |
|- ( ( B - r ) (,) ( B + r ) ) e. ( topGen ` ran (,) ) |
282 |
281
|
a1i |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( B - r ) (,) ( B + r ) ) e. ( topGen ` ran (,) ) ) |
283 |
4
|
adantr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> I e. ( topGen ` ran (,) ) ) |
284 |
|
restopn2 |
|- ( ( ( topGen ` ran (,) ) e. Top /\ I e. ( topGen ` ran (,) ) ) -> ( ( ( B - r ) (,) ( B + r ) ) e. ( ( topGen ` ran (,) ) |`t I ) <-> ( ( ( B - r ) (,) ( B + r ) ) e. ( topGen ` ran (,) ) /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) ) |
285 |
90 283 284
|
sylancr |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( B - r ) (,) ( B + r ) ) e. ( ( topGen ` ran (,) ) |`t I ) <-> ( ( ( B - r ) (,) ( B + r ) ) e. ( topGen ` ran (,) ) /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) ) |
286 |
282 261 285
|
mpbir2and |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( B - r ) (,) ( B + r ) ) e. ( ( topGen ` ran (,) ) |`t I ) ) |
287 |
|
isopn3i |
|- ( ( ( ( topGen ` ran (,) ) |`t I ) e. Top /\ ( ( B - r ) (,) ( B + r ) ) e. ( ( topGen ` ran (,) ) |`t I ) ) -> ( ( int ` ( ( topGen ` ran (,) ) |`t I ) ) ` ( ( B - r ) (,) ( B + r ) ) ) = ( ( B - r ) (,) ( B + r ) ) ) |
288 |
280 286 287
|
syl2anc |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( int ` ( ( topGen ` ran (,) ) |`t I ) ) ` ( ( B - r ) (,) ( B + r ) ) ) = ( ( B - r ) (,) ( B + r ) ) ) |
289 |
273 288
|
eqtrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( int ` ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) ) ` ( ( B - r ) (,) ( B + r ) ) ) = ( ( B - r ) (,) ( B + r ) ) ) |
290 |
51 289
|
eleqtrrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> B e. ( ( int ` ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) ) ` ( ( B - r ) (,) ( B + r ) ) ) ) |
291 |
|
undif1 |
|- ( ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) u. { B } ) = ( ( ( B - r ) (,) ( B + r ) ) u. { B } ) |
292 |
|
ssequn2 |
|- ( { B } C_ ( ( B - r ) (,) ( B + r ) ) <-> ( ( ( B - r ) (,) ( B + r ) ) u. { B } ) = ( ( B - r ) (,) ( B + r ) ) ) |
293 |
52 292
|
sylib |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( B - r ) (,) ( B + r ) ) u. { B } ) = ( ( B - r ) (,) ( B + r ) ) ) |
294 |
291 293
|
syl5eq |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) u. { B } ) = ( ( B - r ) (,) ( B + r ) ) ) |
295 |
294
|
fveq2d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( int ` ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) ) ` ( ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) u. { B } ) ) = ( ( int ` ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) ) ` ( ( B - r ) (,) ( B + r ) ) ) ) |
296 |
290 295
|
eleqtrrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> B e. ( ( int ` ( ( TopOpen ` CCfld ) |`t ( D u. { B } ) ) ) ` ( ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) u. { B } ) ) ) |
297 |
251 68 256 86 257 296
|
limcres |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) ) limCC B ) = ( ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) limCC B ) ) |
298 |
84 69
|
sstri |
|- ( ( B - r ) (,) B ) C_ CC |
299 |
298
|
a1i |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( B - r ) (,) B ) C_ CC ) |
300 |
165 69
|
sstri |
|- ( B (,) ( B + r ) ) C_ CC |
301 |
300
|
a1i |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( B (,) ( B + r ) ) C_ CC ) |
302 |
68
|
sselda |
|- ( ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) /\ z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) ) -> z e. D ) |
303 |
302 250
|
syldan |
|- ( ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) /\ z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) ) -> ( ( F ` z ) / ( G ` z ) ) e. CC ) |
304 |
303
|
fmpttd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) : ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) --> CC ) |
305 |
64
|
feq2d |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) : ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) --> CC <-> ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) : ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) --> CC ) ) |
306 |
304 305
|
mpbid |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) : ( ( ( B - r ) (,) B ) u. ( B (,) ( B + r ) ) ) --> CC ) |
307 |
299 301 306
|
limcun |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) limCC B ) = ( ( ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( ( B - r ) (,) B ) ) limCC B ) i^i ( ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( B (,) ( B + r ) ) ) limCC B ) ) ) |
308 |
230 297 307
|
3eqtr3rd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> ( ( ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( ( B - r ) (,) B ) ) limCC B ) i^i ( ( ( z e. ( ( ( B - r ) (,) ( B + r ) ) \ { B } ) |-> ( ( F ` z ) / ( G ` z ) ) ) |` ( B (,) ( B + r ) ) ) limCC B ) ) = ( ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) limCC B ) ) |
309 |
228 308
|
eleqtrd |
|- ( ( ph /\ ( r e. RR+ /\ ( ( B - r ) (,) ( B + r ) ) C_ I ) ) -> C e. ( ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) limCC B ) ) |
310 |
309
|
expr |
|- ( ( ph /\ r e. RR+ ) -> ( ( ( B - r ) (,) ( B + r ) ) C_ I -> C e. ( ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) limCC B ) ) ) |
311 |
29 310
|
sylbid |
|- ( ( ph /\ r e. RR+ ) -> ( ( B ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ I -> C e. ( ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) limCC B ) ) ) |
312 |
311
|
rexlimdva |
|- ( ph -> ( E. r e. RR+ ( B ( ball ` ( ( abs o. - ) |` ( RR X. RR ) ) ) r ) C_ I -> C e. ( ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) limCC B ) ) ) |
313 |
20 312
|
mpd |
|- ( ph -> C e. ( ( z e. D |-> ( ( F ` z ) / ( G ` z ) ) ) limCC B ) ) |