Step |
Hyp |
Ref |
Expression |
1 |
|
metdscn.f |
|- F = ( x e. X |-> inf ( ran ( y e. S |-> ( x D y ) ) , RR* , < ) ) |
2 |
|
metdscn.j |
|- J = ( MetOpen ` D ) |
3 |
|
simpll1 |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) -> D e. ( *Met ` X ) ) |
4 |
|
simprl |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) -> z e. J ) |
5 |
|
simprr |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) -> A e. z ) |
6 |
2
|
mopni2 |
|- ( ( D e. ( *Met ` X ) /\ z e. J /\ A e. z ) -> E. r e. RR+ ( A ( ball ` D ) r ) C_ z ) |
7 |
3 4 5 6
|
syl3anc |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) -> E. r e. RR+ ( A ( ball ` D ) r ) C_ z ) |
8 |
|
simprr |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> ( A ( ball ` D ) r ) C_ z ) |
9 |
8
|
ssrind |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> ( ( A ( ball ` D ) r ) i^i S ) C_ ( z i^i S ) ) |
10 |
|
rpgt0 |
|- ( r e. RR+ -> 0 < r ) |
11 |
|
0re |
|- 0 e. RR |
12 |
|
rpre |
|- ( r e. RR+ -> r e. RR ) |
13 |
|
ltnle |
|- ( ( 0 e. RR /\ r e. RR ) -> ( 0 < r <-> -. r <_ 0 ) ) |
14 |
11 12 13
|
sylancr |
|- ( r e. RR+ -> ( 0 < r <-> -. r <_ 0 ) ) |
15 |
10 14
|
mpbid |
|- ( r e. RR+ -> -. r <_ 0 ) |
16 |
15
|
ad2antrl |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> -. r <_ 0 ) |
17 |
|
simpllr |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> ( F ` A ) = 0 ) |
18 |
17
|
breq2d |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> ( r <_ ( F ` A ) <-> r <_ 0 ) ) |
19 |
3
|
adantr |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> D e. ( *Met ` X ) ) |
20 |
|
simpl2 |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) -> S C_ X ) |
21 |
20
|
ad2antrr |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> S C_ X ) |
22 |
|
simpl3 |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) -> A e. X ) |
23 |
22
|
ad2antrr |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> A e. X ) |
24 |
|
rpxr |
|- ( r e. RR+ -> r e. RR* ) |
25 |
24
|
ad2antrl |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> r e. RR* ) |
26 |
1
|
metdsge |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ r e. RR* ) -> ( r <_ ( F ` A ) <-> ( S i^i ( A ( ball ` D ) r ) ) = (/) ) ) |
27 |
19 21 23 25 26
|
syl31anc |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> ( r <_ ( F ` A ) <-> ( S i^i ( A ( ball ` D ) r ) ) = (/) ) ) |
28 |
18 27
|
bitr3d |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> ( r <_ 0 <-> ( S i^i ( A ( ball ` D ) r ) ) = (/) ) ) |
29 |
|
incom |
|- ( S i^i ( A ( ball ` D ) r ) ) = ( ( A ( ball ` D ) r ) i^i S ) |
30 |
29
|
eqeq1i |
|- ( ( S i^i ( A ( ball ` D ) r ) ) = (/) <-> ( ( A ( ball ` D ) r ) i^i S ) = (/) ) |
31 |
28 30
|
bitrdi |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> ( r <_ 0 <-> ( ( A ( ball ` D ) r ) i^i S ) = (/) ) ) |
32 |
31
|
necon3bbid |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> ( -. r <_ 0 <-> ( ( A ( ball ` D ) r ) i^i S ) =/= (/) ) ) |
33 |
16 32
|
mpbid |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> ( ( A ( ball ` D ) r ) i^i S ) =/= (/) ) |
34 |
|
ssn0 |
|- ( ( ( ( A ( ball ` D ) r ) i^i S ) C_ ( z i^i S ) /\ ( ( A ( ball ` D ) r ) i^i S ) =/= (/) ) -> ( z i^i S ) =/= (/) ) |
35 |
9 33 34
|
syl2anc |
|- ( ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) /\ ( r e. RR+ /\ ( A ( ball ` D ) r ) C_ z ) ) -> ( z i^i S ) =/= (/) ) |
36 |
7 35
|
rexlimddv |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ ( z e. J /\ A e. z ) ) -> ( z i^i S ) =/= (/) ) |
37 |
36
|
expr |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) /\ z e. J ) -> ( A e. z -> ( z i^i S ) =/= (/) ) ) |
38 |
37
|
ralrimiva |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) -> A. z e. J ( A e. z -> ( z i^i S ) =/= (/) ) ) |
39 |
2
|
mopntopon |
|- ( D e. ( *Met ` X ) -> J e. ( TopOn ` X ) ) |
40 |
39
|
3ad2ant1 |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> J e. ( TopOn ` X ) ) |
41 |
40
|
adantr |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) -> J e. ( TopOn ` X ) ) |
42 |
|
topontop |
|- ( J e. ( TopOn ` X ) -> J e. Top ) |
43 |
41 42
|
syl |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) -> J e. Top ) |
44 |
|
toponuni |
|- ( J e. ( TopOn ` X ) -> X = U. J ) |
45 |
41 44
|
syl |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) -> X = U. J ) |
46 |
20 45
|
sseqtrd |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) -> S C_ U. J ) |
47 |
22 45
|
eleqtrd |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) -> A e. U. J ) |
48 |
|
eqid |
|- U. J = U. J |
49 |
48
|
elcls |
|- ( ( J e. Top /\ S C_ U. J /\ A e. U. J ) -> ( A e. ( ( cls ` J ) ` S ) <-> A. z e. J ( A e. z -> ( z i^i S ) =/= (/) ) ) ) |
50 |
43 46 47 49
|
syl3anc |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) -> ( A e. ( ( cls ` J ) ` S ) <-> A. z e. J ( A e. z -> ( z i^i S ) =/= (/) ) ) ) |
51 |
38 50
|
mpbird |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) = 0 ) -> A e. ( ( cls ` J ) ` S ) ) |
52 |
|
incom |
|- ( ( A ( ball ` D ) ( F ` A ) ) i^i S ) = ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) |
53 |
1
|
metdsf |
|- ( ( D e. ( *Met ` X ) /\ S C_ X ) -> F : X --> ( 0 [,] +oo ) ) |
54 |
53
|
ffvelrnda |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X ) /\ A e. X ) -> ( F ` A ) e. ( 0 [,] +oo ) ) |
55 |
54
|
3impa |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> ( F ` A ) e. ( 0 [,] +oo ) ) |
56 |
|
eliccxr |
|- ( ( F ` A ) e. ( 0 [,] +oo ) -> ( F ` A ) e. RR* ) |
57 |
55 56
|
syl |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> ( F ` A ) e. RR* ) |
58 |
57
|
xrleidd |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> ( F ` A ) <_ ( F ` A ) ) |
59 |
1
|
metdsge |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ ( F ` A ) e. RR* ) -> ( ( F ` A ) <_ ( F ` A ) <-> ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) = (/) ) ) |
60 |
57 59
|
mpdan |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> ( ( F ` A ) <_ ( F ` A ) <-> ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) = (/) ) ) |
61 |
58 60
|
mpbid |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> ( S i^i ( A ( ball ` D ) ( F ` A ) ) ) = (/) ) |
62 |
52 61
|
syl5eq |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> ( ( A ( ball ` D ) ( F ` A ) ) i^i S ) = (/) ) |
63 |
62
|
adantr |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) -> ( ( A ( ball ` D ) ( F ` A ) ) i^i S ) = (/) ) |
64 |
40
|
ad2antrr |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> J e. ( TopOn ` X ) ) |
65 |
64 42
|
syl |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> J e. Top ) |
66 |
|
simpll2 |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> S C_ X ) |
67 |
64 44
|
syl |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> X = U. J ) |
68 |
66 67
|
sseqtrd |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> S C_ U. J ) |
69 |
|
simplr |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> A e. ( ( cls ` J ) ` S ) ) |
70 |
|
simpll1 |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> D e. ( *Met ` X ) ) |
71 |
|
simpll3 |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> A e. X ) |
72 |
57
|
ad2antrr |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> ( F ` A ) e. RR* ) |
73 |
2
|
blopn |
|- ( ( D e. ( *Met ` X ) /\ A e. X /\ ( F ` A ) e. RR* ) -> ( A ( ball ` D ) ( F ` A ) ) e. J ) |
74 |
70 71 72 73
|
syl3anc |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> ( A ( ball ` D ) ( F ` A ) ) e. J ) |
75 |
|
simpr |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> 0 < ( F ` A ) ) |
76 |
|
xblcntr |
|- ( ( D e. ( *Met ` X ) /\ A e. X /\ ( ( F ` A ) e. RR* /\ 0 < ( F ` A ) ) ) -> A e. ( A ( ball ` D ) ( F ` A ) ) ) |
77 |
70 71 72 75 76
|
syl112anc |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> A e. ( A ( ball ` D ) ( F ` A ) ) ) |
78 |
48
|
clsndisj |
|- ( ( ( J e. Top /\ S C_ U. J /\ A e. ( ( cls ` J ) ` S ) ) /\ ( ( A ( ball ` D ) ( F ` A ) ) e. J /\ A e. ( A ( ball ` D ) ( F ` A ) ) ) ) -> ( ( A ( ball ` D ) ( F ` A ) ) i^i S ) =/= (/) ) |
79 |
65 68 69 74 77 78
|
syl32anc |
|- ( ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) /\ 0 < ( F ` A ) ) -> ( ( A ( ball ` D ) ( F ` A ) ) i^i S ) =/= (/) ) |
80 |
79
|
ex |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) -> ( 0 < ( F ` A ) -> ( ( A ( ball ` D ) ( F ` A ) ) i^i S ) =/= (/) ) ) |
81 |
80
|
necon2bd |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) -> ( ( ( A ( ball ` D ) ( F ` A ) ) i^i S ) = (/) -> -. 0 < ( F ` A ) ) ) |
82 |
63 81
|
mpd |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) -> -. 0 < ( F ` A ) ) |
83 |
|
elxrge0 |
|- ( ( F ` A ) e. ( 0 [,] +oo ) <-> ( ( F ` A ) e. RR* /\ 0 <_ ( F ` A ) ) ) |
84 |
83
|
simprbi |
|- ( ( F ` A ) e. ( 0 [,] +oo ) -> 0 <_ ( F ` A ) ) |
85 |
55 84
|
syl |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> 0 <_ ( F ` A ) ) |
86 |
|
0xr |
|- 0 e. RR* |
87 |
|
xrleloe |
|- ( ( 0 e. RR* /\ ( F ` A ) e. RR* ) -> ( 0 <_ ( F ` A ) <-> ( 0 < ( F ` A ) \/ 0 = ( F ` A ) ) ) ) |
88 |
86 57 87
|
sylancr |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> ( 0 <_ ( F ` A ) <-> ( 0 < ( F ` A ) \/ 0 = ( F ` A ) ) ) ) |
89 |
85 88
|
mpbid |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> ( 0 < ( F ` A ) \/ 0 = ( F ` A ) ) ) |
90 |
89
|
adantr |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) -> ( 0 < ( F ` A ) \/ 0 = ( F ` A ) ) ) |
91 |
90
|
ord |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) -> ( -. 0 < ( F ` A ) -> 0 = ( F ` A ) ) ) |
92 |
82 91
|
mpd |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) -> 0 = ( F ` A ) ) |
93 |
92
|
eqcomd |
|- ( ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) /\ A e. ( ( cls ` J ) ` S ) ) -> ( F ` A ) = 0 ) |
94 |
51 93
|
impbida |
|- ( ( D e. ( *Met ` X ) /\ S C_ X /\ A e. X ) -> ( ( F ` A ) = 0 <-> A e. ( ( cls ` J ) ` S ) ) ) |