Step |
Hyp |
Ref |
Expression |
1 |
|
rrndistlt.i |
|- ( ph -> I e. Fin ) |
2 |
|
rrndistlt.z |
|- ( ph -> I =/= (/) ) |
3 |
|
rrndistlt.n |
|- N = ( # ` I ) |
4 |
|
rrndistlt.x |
|- ( ph -> X e. ( RR ^m I ) ) |
5 |
|
rrndistlt.y |
|- ( ph -> Y e. ( RR ^m I ) ) |
6 |
|
rrndistlt.l |
|- ( ( ph /\ i e. I ) -> ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) < E ) |
7 |
|
rrndistlt.e |
|- ( ph -> E e. RR+ ) |
8 |
|
rrndistlt.d |
|- D = ( dist ` ( RR^ ` I ) ) |
9 |
|
elmapi |
|- ( X e. ( RR ^m I ) -> X : I --> RR ) |
10 |
4 9
|
syl |
|- ( ph -> X : I --> RR ) |
11 |
|
ax-resscn |
|- RR C_ CC |
12 |
11
|
a1i |
|- ( ph -> RR C_ CC ) |
13 |
10 12
|
fssd |
|- ( ph -> X : I --> CC ) |
14 |
13
|
ffvelrnda |
|- ( ( ph /\ i e. I ) -> ( X ` i ) e. CC ) |
15 |
|
elmapi |
|- ( Y e. ( RR ^m I ) -> Y : I --> RR ) |
16 |
5 15
|
syl |
|- ( ph -> Y : I --> RR ) |
17 |
16 12
|
fssd |
|- ( ph -> Y : I --> CC ) |
18 |
17
|
ffvelrnda |
|- ( ( ph /\ i e. I ) -> ( Y ` i ) e. CC ) |
19 |
14 18
|
subcld |
|- ( ( ph /\ i e. I ) -> ( ( X ` i ) - ( Y ` i ) ) e. CC ) |
20 |
19
|
abscld |
|- ( ( ph /\ i e. I ) -> ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) e. RR ) |
21 |
20
|
resqcld |
|- ( ( ph /\ i e. I ) -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) e. RR ) |
22 |
7
|
rpred |
|- ( ph -> E e. RR ) |
23 |
22
|
resqcld |
|- ( ph -> ( E ^ 2 ) e. RR ) |
24 |
23
|
adantr |
|- ( ( ph /\ i e. I ) -> ( E ^ 2 ) e. RR ) |
25 |
19
|
absge0d |
|- ( ( ph /\ i e. I ) -> 0 <_ ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ) |
26 |
22
|
adantr |
|- ( ( ph /\ i e. I ) -> E e. RR ) |
27 |
7
|
adantr |
|- ( ( ph /\ i e. I ) -> E e. RR+ ) |
28 |
27
|
rpge0d |
|- ( ( ph /\ i e. I ) -> 0 <_ E ) |
29 |
|
lt2sq |
|- ( ( ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) e. RR /\ 0 <_ ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ) /\ ( E e. RR /\ 0 <_ E ) ) -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) < E <-> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) < ( E ^ 2 ) ) ) |
30 |
20 25 26 28 29
|
syl22anc |
|- ( ( ph /\ i e. I ) -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) < E <-> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) < ( E ^ 2 ) ) ) |
31 |
6 30
|
mpbid |
|- ( ( ph /\ i e. I ) -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) < ( E ^ 2 ) ) |
32 |
1 2 21 24 31
|
fsumlt |
|- ( ph -> sum_ i e. I ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) < sum_ i e. I ( E ^ 2 ) ) |
33 |
10
|
ffvelrnda |
|- ( ( ph /\ i e. I ) -> ( X ` i ) e. RR ) |
34 |
16
|
ffvelrnda |
|- ( ( ph /\ i e. I ) -> ( Y ` i ) e. RR ) |
35 |
33 34
|
resubcld |
|- ( ( ph /\ i e. I ) -> ( ( X ` i ) - ( Y ` i ) ) e. RR ) |
36 |
|
absresq |
|- ( ( ( X ` i ) - ( Y ` i ) ) e. RR -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) = ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) |
37 |
35 36
|
syl |
|- ( ( ph /\ i e. I ) -> ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) = ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) |
38 |
37
|
eqcomd |
|- ( ( ph /\ i e. I ) -> ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) = ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) ) |
39 |
38
|
sumeq2dv |
|- ( ph -> sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) = sum_ i e. I ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) ) |
40 |
11 23
|
sselid |
|- ( ph -> ( E ^ 2 ) e. CC ) |
41 |
|
fsumconst |
|- ( ( I e. Fin /\ ( E ^ 2 ) e. CC ) -> sum_ i e. I ( E ^ 2 ) = ( ( # ` I ) x. ( E ^ 2 ) ) ) |
42 |
1 40 41
|
syl2anc |
|- ( ph -> sum_ i e. I ( E ^ 2 ) = ( ( # ` I ) x. ( E ^ 2 ) ) ) |
43 |
|
eqcom |
|- ( N = ( # ` I ) <-> ( # ` I ) = N ) |
44 |
3 43
|
mpbi |
|- ( # ` I ) = N |
45 |
44
|
oveq1i |
|- ( ( # ` I ) x. ( E ^ 2 ) ) = ( N x. ( E ^ 2 ) ) |
46 |
45
|
a1i |
|- ( ph -> ( ( # ` I ) x. ( E ^ 2 ) ) = ( N x. ( E ^ 2 ) ) ) |
47 |
42 46
|
eqtr2d |
|- ( ph -> ( N x. ( E ^ 2 ) ) = sum_ i e. I ( E ^ 2 ) ) |
48 |
39 47
|
breq12d |
|- ( ph -> ( sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) < ( N x. ( E ^ 2 ) ) <-> sum_ i e. I ( ( abs ` ( ( X ` i ) - ( Y ` i ) ) ) ^ 2 ) < sum_ i e. I ( E ^ 2 ) ) ) |
49 |
32 48
|
mpbird |
|- ( ph -> sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) < ( N x. ( E ^ 2 ) ) ) |
50 |
|
nfv |
|- F/ i ph |
51 |
35
|
resqcld |
|- ( ( ph /\ i e. I ) -> ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) e. RR ) |
52 |
50 1 51
|
fsumreclf |
|- ( ph -> sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) e. RR ) |
53 |
35
|
sqge0d |
|- ( ( ph /\ i e. I ) -> 0 <_ ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) |
54 |
1 51 53
|
fsumge0 |
|- ( ph -> 0 <_ sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) |
55 |
|
hashcl |
|- ( I e. Fin -> ( # ` I ) e. NN0 ) |
56 |
1 55
|
syl |
|- ( ph -> ( # ` I ) e. NN0 ) |
57 |
3 56
|
eqeltrid |
|- ( ph -> N e. NN0 ) |
58 |
57
|
nn0red |
|- ( ph -> N e. RR ) |
59 |
58 23
|
remulcld |
|- ( ph -> ( N x. ( E ^ 2 ) ) e. RR ) |
60 |
57
|
nn0ge0d |
|- ( ph -> 0 <_ N ) |
61 |
22
|
sqge0d |
|- ( ph -> 0 <_ ( E ^ 2 ) ) |
62 |
58 23 60 61
|
mulge0d |
|- ( ph -> 0 <_ ( N x. ( E ^ 2 ) ) ) |
63 |
52 54 59 62
|
sqrtltd |
|- ( ph -> ( sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) < ( N x. ( E ^ 2 ) ) <-> ( sqrt ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) < ( sqrt ` ( N x. ( E ^ 2 ) ) ) ) ) |
64 |
49 63
|
mpbid |
|- ( ph -> ( sqrt ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) < ( sqrt ` ( N x. ( E ^ 2 ) ) ) ) |
65 |
8
|
a1i |
|- ( ph -> D = ( dist ` ( RR^ ` I ) ) ) |
66 |
|
eqid |
|- ( RR^ ` I ) = ( RR^ ` I ) |
67 |
|
eqid |
|- ( RR ^m I ) = ( RR ^m I ) |
68 |
66 67
|
rrxdsfi |
|- ( I e. Fin -> ( dist ` ( RR^ ` I ) ) = ( f e. ( RR ^m I ) , g e. ( RR ^m I ) |-> ( sqrt ` sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) ) ) ) |
69 |
1 68
|
syl |
|- ( ph -> ( dist ` ( RR^ ` I ) ) = ( f e. ( RR ^m I ) , g e. ( RR ^m I ) |-> ( sqrt ` sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) ) ) ) |
70 |
65 69
|
eqtrd |
|- ( ph -> D = ( f e. ( RR ^m I ) , g e. ( RR ^m I ) |-> ( sqrt ` sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) ) ) ) |
71 |
|
fveq1 |
|- ( f = X -> ( f ` i ) = ( X ` i ) ) |
72 |
71
|
adantr |
|- ( ( f = X /\ g = Y ) -> ( f ` i ) = ( X ` i ) ) |
73 |
|
fveq1 |
|- ( g = Y -> ( g ` i ) = ( Y ` i ) ) |
74 |
73
|
adantl |
|- ( ( f = X /\ g = Y ) -> ( g ` i ) = ( Y ` i ) ) |
75 |
72 74
|
oveq12d |
|- ( ( f = X /\ g = Y ) -> ( ( f ` i ) - ( g ` i ) ) = ( ( X ` i ) - ( Y ` i ) ) ) |
76 |
75
|
oveq1d |
|- ( ( f = X /\ g = Y ) -> ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) = ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) |
77 |
76
|
sumeq2sdv |
|- ( ( f = X /\ g = Y ) -> sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) = sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) |
78 |
77
|
fveq2d |
|- ( ( f = X /\ g = Y ) -> ( sqrt ` sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) ) = ( sqrt ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) ) |
79 |
78
|
adantl |
|- ( ( ph /\ ( f = X /\ g = Y ) ) -> ( sqrt ` sum_ i e. I ( ( ( f ` i ) - ( g ` i ) ) ^ 2 ) ) = ( sqrt ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) ) |
80 |
52 54
|
resqrtcld |
|- ( ph -> ( sqrt ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) e. RR ) |
81 |
70 79 4 5 80
|
ovmpod |
|- ( ph -> ( X D Y ) = ( sqrt ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) ) |
82 |
|
sqrtmul |
|- ( ( ( N e. RR /\ 0 <_ N ) /\ ( ( E ^ 2 ) e. RR /\ 0 <_ ( E ^ 2 ) ) ) -> ( sqrt ` ( N x. ( E ^ 2 ) ) ) = ( ( sqrt ` N ) x. ( sqrt ` ( E ^ 2 ) ) ) ) |
83 |
58 60 23 61 82
|
syl22anc |
|- ( ph -> ( sqrt ` ( N x. ( E ^ 2 ) ) ) = ( ( sqrt ` N ) x. ( sqrt ` ( E ^ 2 ) ) ) ) |
84 |
7
|
rpge0d |
|- ( ph -> 0 <_ E ) |
85 |
22 84
|
sqrtsqd |
|- ( ph -> ( sqrt ` ( E ^ 2 ) ) = E ) |
86 |
85
|
oveq2d |
|- ( ph -> ( ( sqrt ` N ) x. ( sqrt ` ( E ^ 2 ) ) ) = ( ( sqrt ` N ) x. E ) ) |
87 |
83 86
|
eqtr2d |
|- ( ph -> ( ( sqrt ` N ) x. E ) = ( sqrt ` ( N x. ( E ^ 2 ) ) ) ) |
88 |
81 87
|
breq12d |
|- ( ph -> ( ( X D Y ) < ( ( sqrt ` N ) x. E ) <-> ( sqrt ` sum_ i e. I ( ( ( X ` i ) - ( Y ` i ) ) ^ 2 ) ) < ( sqrt ` ( N x. ( E ^ 2 ) ) ) ) ) |
89 |
64 88
|
mpbird |
|- ( ph -> ( X D Y ) < ( ( sqrt ` N ) x. E ) ) |