Step |
Hyp |
Ref |
Expression |
1 |
|
ioorrnopn.x |
|- ( ph -> X e. Fin ) |
2 |
|
ioorrnopn.a |
|- ( ph -> A : X --> RR ) |
3 |
|
ioorrnopn.b |
|- ( ph -> B : X --> RR ) |
4 |
|
p0ex |
|- { (/) } e. _V |
5 |
4
|
prid2 |
|- { (/) } e. { (/) , { (/) } } |
6 |
5
|
a1i |
|- ( X = (/) -> { (/) } e. { (/) , { (/) } } ) |
7 |
|
ixpeq1 |
|- ( X = (/) -> X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) = X_ i e. (/) ( ( A ` i ) (,) ( B ` i ) ) ) |
8 |
|
ixp0x |
|- X_ i e. (/) ( ( A ` i ) (,) ( B ` i ) ) = { (/) } |
9 |
8
|
a1i |
|- ( X = (/) -> X_ i e. (/) ( ( A ` i ) (,) ( B ` i ) ) = { (/) } ) |
10 |
7 9
|
eqtrd |
|- ( X = (/) -> X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) = { (/) } ) |
11 |
|
2fveq3 |
|- ( X = (/) -> ( TopOpen ` ( RR^ ` X ) ) = ( TopOpen ` ( RR^ ` (/) ) ) ) |
12 |
|
rrxtopn0b |
|- ( TopOpen ` ( RR^ ` (/) ) ) = { (/) , { (/) } } |
13 |
12
|
a1i |
|- ( X = (/) -> ( TopOpen ` ( RR^ ` (/) ) ) = { (/) , { (/) } } ) |
14 |
11 13
|
eqtrd |
|- ( X = (/) -> ( TopOpen ` ( RR^ ` X ) ) = { (/) , { (/) } } ) |
15 |
10 14
|
eleq12d |
|- ( X = (/) -> ( X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) e. ( TopOpen ` ( RR^ ` X ) ) <-> { (/) } e. { (/) , { (/) } } ) ) |
16 |
6 15
|
mpbird |
|- ( X = (/) -> X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) e. ( TopOpen ` ( RR^ ` X ) ) ) |
17 |
16
|
adantl |
|- ( ( ph /\ X = (/) ) -> X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) e. ( TopOpen ` ( RR^ ` X ) ) ) |
18 |
|
neqne |
|- ( -. X = (/) -> X =/= (/) ) |
19 |
18
|
adantl |
|- ( ( ph /\ -. X = (/) ) -> X =/= (/) ) |
20 |
|
fveq2 |
|- ( i = j -> ( A ` i ) = ( A ` j ) ) |
21 |
|
fveq2 |
|- ( i = j -> ( B ` i ) = ( B ` j ) ) |
22 |
20 21
|
oveq12d |
|- ( i = j -> ( ( A ` i ) (,) ( B ` i ) ) = ( ( A ` j ) (,) ( B ` j ) ) ) |
23 |
22
|
cbvixpv |
|- X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) = X_ j e. X ( ( A ` j ) (,) ( B ` j ) ) |
24 |
23
|
eleq2i |
|- ( f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) <-> f e. X_ j e. X ( ( A ` j ) (,) ( B ` j ) ) ) |
25 |
24
|
biimpi |
|- ( f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) -> f e. X_ j e. X ( ( A ` j ) (,) ( B ` j ) ) ) |
26 |
25
|
adantl |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) -> f e. X_ j e. X ( ( A ` j ) (,) ( B ` j ) ) ) |
27 |
1
|
ad2antrr |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) -> X e. Fin ) |
28 |
24 27
|
sylan2br |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ j e. X ( ( A ` j ) (,) ( B ` j ) ) ) -> X e. Fin ) |
29 |
|
simplr |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) -> X =/= (/) ) |
30 |
24 29
|
sylan2br |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ j e. X ( ( A ` j ) (,) ( B ` j ) ) ) -> X =/= (/) ) |
31 |
2
|
ad2antrr |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) -> A : X --> RR ) |
32 |
24 31
|
sylan2br |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ j e. X ( ( A ` j ) (,) ( B ` j ) ) ) -> A : X --> RR ) |
33 |
3
|
ad2antrr |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) -> B : X --> RR ) |
34 |
24 33
|
sylan2br |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ j e. X ( ( A ` j ) (,) ( B ` j ) ) ) -> B : X --> RR ) |
35 |
|
simpr |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) -> f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) |
36 |
24 35
|
sylan2br |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ j e. X ( ( A ` j ) (,) ( B ` j ) ) ) -> f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) |
37 |
|
eqid |
|- ran ( i e. X |-> if ( ( ( B ` i ) - ( f ` i ) ) <_ ( ( f ` i ) - ( A ` i ) ) , ( ( B ` i ) - ( f ` i ) ) , ( ( f ` i ) - ( A ` i ) ) ) ) = ran ( i e. X |-> if ( ( ( B ` i ) - ( f ` i ) ) <_ ( ( f ` i ) - ( A ` i ) ) , ( ( B ` i ) - ( f ` i ) ) , ( ( f ` i ) - ( A ` i ) ) ) ) |
38 |
|
fveq2 |
|- ( j = i -> ( B ` j ) = ( B ` i ) ) |
39 |
|
fveq2 |
|- ( j = i -> ( f ` j ) = ( f ` i ) ) |
40 |
38 39
|
oveq12d |
|- ( j = i -> ( ( B ` j ) - ( f ` j ) ) = ( ( B ` i ) - ( f ` i ) ) ) |
41 |
|
fveq2 |
|- ( j = i -> ( A ` j ) = ( A ` i ) ) |
42 |
39 41
|
oveq12d |
|- ( j = i -> ( ( f ` j ) - ( A ` j ) ) = ( ( f ` i ) - ( A ` i ) ) ) |
43 |
40 42
|
breq12d |
|- ( j = i -> ( ( ( B ` j ) - ( f ` j ) ) <_ ( ( f ` j ) - ( A ` j ) ) <-> ( ( B ` i ) - ( f ` i ) ) <_ ( ( f ` i ) - ( A ` i ) ) ) ) |
44 |
43 40 42
|
ifbieq12d |
|- ( j = i -> if ( ( ( B ` j ) - ( f ` j ) ) <_ ( ( f ` j ) - ( A ` j ) ) , ( ( B ` j ) - ( f ` j ) ) , ( ( f ` j ) - ( A ` j ) ) ) = if ( ( ( B ` i ) - ( f ` i ) ) <_ ( ( f ` i ) - ( A ` i ) ) , ( ( B ` i ) - ( f ` i ) ) , ( ( f ` i ) - ( A ` i ) ) ) ) |
45 |
44
|
cbvmptv |
|- ( j e. X |-> if ( ( ( B ` j ) - ( f ` j ) ) <_ ( ( f ` j ) - ( A ` j ) ) , ( ( B ` j ) - ( f ` j ) ) , ( ( f ` j ) - ( A ` j ) ) ) ) = ( i e. X |-> if ( ( ( B ` i ) - ( f ` i ) ) <_ ( ( f ` i ) - ( A ` i ) ) , ( ( B ` i ) - ( f ` i ) ) , ( ( f ` i ) - ( A ` i ) ) ) ) |
46 |
45
|
rneqi |
|- ran ( j e. X |-> if ( ( ( B ` j ) - ( f ` j ) ) <_ ( ( f ` j ) - ( A ` j ) ) , ( ( B ` j ) - ( f ` j ) ) , ( ( f ` j ) - ( A ` j ) ) ) ) = ran ( i e. X |-> if ( ( ( B ` i ) - ( f ` i ) ) <_ ( ( f ` i ) - ( A ` i ) ) , ( ( B ` i ) - ( f ` i ) ) , ( ( f ` i ) - ( A ` i ) ) ) ) |
47 |
46
|
infeq1i |
|- inf ( ran ( j e. X |-> if ( ( ( B ` j ) - ( f ` j ) ) <_ ( ( f ` j ) - ( A ` j ) ) , ( ( B ` j ) - ( f ` j ) ) , ( ( f ` j ) - ( A ` j ) ) ) ) , RR , < ) = inf ( ran ( i e. X |-> if ( ( ( B ` i ) - ( f ` i ) ) <_ ( ( f ` i ) - ( A ` i ) ) , ( ( B ` i ) - ( f ` i ) ) , ( ( f ` i ) - ( A ` i ) ) ) ) , RR , < ) |
48 |
|
eqid |
|- ( f ( ball ` ( a e. ( RR ^m X ) , b e. ( RR ^m X ) |-> ( sqrt ` sum_ k e. X ( ( ( a ` k ) - ( b ` k ) ) ^ 2 ) ) ) ) inf ( ran ( j e. X |-> if ( ( ( B ` j ) - ( f ` j ) ) <_ ( ( f ` j ) - ( A ` j ) ) , ( ( B ` j ) - ( f ` j ) ) , ( ( f ` j ) - ( A ` j ) ) ) ) , RR , < ) ) = ( f ( ball ` ( a e. ( RR ^m X ) , b e. ( RR ^m X ) |-> ( sqrt ` sum_ k e. X ( ( ( a ` k ) - ( b ` k ) ) ^ 2 ) ) ) ) inf ( ran ( j e. X |-> if ( ( ( B ` j ) - ( f ` j ) ) <_ ( ( f ` j ) - ( A ` j ) ) , ( ( B ` j ) - ( f ` j ) ) , ( ( f ` j ) - ( A ` j ) ) ) ) , RR , < ) ) |
49 |
|
fveq1 |
|- ( a = g -> ( a ` k ) = ( g ` k ) ) |
50 |
49
|
oveq1d |
|- ( a = g -> ( ( a ` k ) - ( b ` k ) ) = ( ( g ` k ) - ( b ` k ) ) ) |
51 |
50
|
oveq1d |
|- ( a = g -> ( ( ( a ` k ) - ( b ` k ) ) ^ 2 ) = ( ( ( g ` k ) - ( b ` k ) ) ^ 2 ) ) |
52 |
51
|
sumeq2sdv |
|- ( a = g -> sum_ k e. X ( ( ( a ` k ) - ( b ` k ) ) ^ 2 ) = sum_ k e. X ( ( ( g ` k ) - ( b ` k ) ) ^ 2 ) ) |
53 |
52
|
fveq2d |
|- ( a = g -> ( sqrt ` sum_ k e. X ( ( ( a ` k ) - ( b ` k ) ) ^ 2 ) ) = ( sqrt ` sum_ k e. X ( ( ( g ` k ) - ( b ` k ) ) ^ 2 ) ) ) |
54 |
|
fveq1 |
|- ( b = h -> ( b ` k ) = ( h ` k ) ) |
55 |
54
|
oveq2d |
|- ( b = h -> ( ( g ` k ) - ( b ` k ) ) = ( ( g ` k ) - ( h ` k ) ) ) |
56 |
55
|
oveq1d |
|- ( b = h -> ( ( ( g ` k ) - ( b ` k ) ) ^ 2 ) = ( ( ( g ` k ) - ( h ` k ) ) ^ 2 ) ) |
57 |
56
|
sumeq2sdv |
|- ( b = h -> sum_ k e. X ( ( ( g ` k ) - ( b ` k ) ) ^ 2 ) = sum_ k e. X ( ( ( g ` k ) - ( h ` k ) ) ^ 2 ) ) |
58 |
57
|
fveq2d |
|- ( b = h -> ( sqrt ` sum_ k e. X ( ( ( g ` k ) - ( b ` k ) ) ^ 2 ) ) = ( sqrt ` sum_ k e. X ( ( ( g ` k ) - ( h ` k ) ) ^ 2 ) ) ) |
59 |
53 58
|
cbvmpov |
|- ( a e. ( RR ^m X ) , b e. ( RR ^m X ) |-> ( sqrt ` sum_ k e. X ( ( ( a ` k ) - ( b ` k ) ) ^ 2 ) ) ) = ( g e. ( RR ^m X ) , h e. ( RR ^m X ) |-> ( sqrt ` sum_ k e. X ( ( ( g ` k ) - ( h ` k ) ) ^ 2 ) ) ) |
60 |
28 30 32 34 36 37 47 48 59
|
ioorrnopnlem |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ j e. X ( ( A ` j ) (,) ( B ` j ) ) ) -> E. v e. ( TopOpen ` ( RR^ ` X ) ) ( f e. v /\ v C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) ) |
61 |
26 60
|
syldan |
|- ( ( ( ph /\ X =/= (/) ) /\ f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) -> E. v e. ( TopOpen ` ( RR^ ` X ) ) ( f e. v /\ v C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) ) |
62 |
61
|
ralrimiva |
|- ( ( ph /\ X =/= (/) ) -> A. f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) E. v e. ( TopOpen ` ( RR^ ` X ) ) ( f e. v /\ v C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) ) |
63 |
|
eqid |
|- ( TopOpen ` ( RR^ ` X ) ) = ( TopOpen ` ( RR^ ` X ) ) |
64 |
63
|
rrxtop |
|- ( X e. Fin -> ( TopOpen ` ( RR^ ` X ) ) e. Top ) |
65 |
1 64
|
syl |
|- ( ph -> ( TopOpen ` ( RR^ ` X ) ) e. Top ) |
66 |
65
|
adantr |
|- ( ( ph /\ X =/= (/) ) -> ( TopOpen ` ( RR^ ` X ) ) e. Top ) |
67 |
|
eltop2 |
|- ( ( TopOpen ` ( RR^ ` X ) ) e. Top -> ( X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) e. ( TopOpen ` ( RR^ ` X ) ) <-> A. f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) E. v e. ( TopOpen ` ( RR^ ` X ) ) ( f e. v /\ v C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) ) ) |
68 |
66 67
|
syl |
|- ( ( ph /\ X =/= (/) ) -> ( X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) e. ( TopOpen ` ( RR^ ` X ) ) <-> A. f e. X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) E. v e. ( TopOpen ` ( RR^ ` X ) ) ( f e. v /\ v C_ X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) ) ) ) |
69 |
62 68
|
mpbird |
|- ( ( ph /\ X =/= (/) ) -> X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) e. ( TopOpen ` ( RR^ ` X ) ) ) |
70 |
19 69
|
syldan |
|- ( ( ph /\ -. X = (/) ) -> X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) e. ( TopOpen ` ( RR^ ` X ) ) ) |
71 |
17 70
|
pm2.61dan |
|- ( ph -> X_ i e. X ( ( A ` i ) (,) ( B ` i ) ) e. ( TopOpen ` ( RR^ ` X ) ) ) |