Step |
Hyp |
Ref |
Expression |
1 |
|
rrndistlt.i |
⊢ ( 𝜑 → 𝐼 ∈ Fin ) |
2 |
|
rrndistlt.z |
⊢ ( 𝜑 → 𝐼 ≠ ∅ ) |
3 |
|
rrndistlt.n |
⊢ 𝑁 = ( ♯ ‘ 𝐼 ) |
4 |
|
rrndistlt.x |
⊢ ( 𝜑 → 𝑋 ∈ ( ℝ ↑m 𝐼 ) ) |
5 |
|
rrndistlt.y |
⊢ ( 𝜑 → 𝑌 ∈ ( ℝ ↑m 𝐼 ) ) |
6 |
|
rrndistlt.l |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) < 𝐸 ) |
7 |
|
rrndistlt.e |
⊢ ( 𝜑 → 𝐸 ∈ ℝ+ ) |
8 |
|
rrndistlt.d |
⊢ 𝐷 = ( dist ‘ ( ℝ^ ‘ 𝐼 ) ) |
9 |
|
elmapi |
⊢ ( 𝑋 ∈ ( ℝ ↑m 𝐼 ) → 𝑋 : 𝐼 ⟶ ℝ ) |
10 |
4 9
|
syl |
⊢ ( 𝜑 → 𝑋 : 𝐼 ⟶ ℝ ) |
11 |
|
ax-resscn |
⊢ ℝ ⊆ ℂ |
12 |
11
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℂ ) |
13 |
10 12
|
fssd |
⊢ ( 𝜑 → 𝑋 : 𝐼 ⟶ ℂ ) |
14 |
13
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑖 ) ∈ ℂ ) |
15 |
|
elmapi |
⊢ ( 𝑌 ∈ ( ℝ ↑m 𝐼 ) → 𝑌 : 𝐼 ⟶ ℝ ) |
16 |
5 15
|
syl |
⊢ ( 𝜑 → 𝑌 : 𝐼 ⟶ ℝ ) |
17 |
16 12
|
fssd |
⊢ ( 𝜑 → 𝑌 : 𝐼 ⟶ ℂ ) |
18 |
17
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑖 ) ∈ ℂ ) |
19 |
14 18
|
subcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ∈ ℂ ) |
20 |
19
|
abscld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ∈ ℝ ) |
21 |
20
|
resqcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ↑ 2 ) ∈ ℝ ) |
22 |
7
|
rpred |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
23 |
22
|
resqcld |
⊢ ( 𝜑 → ( 𝐸 ↑ 2 ) ∈ ℝ ) |
24 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( 𝐸 ↑ 2 ) ∈ ℝ ) |
25 |
19
|
absge0d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 0 ≤ ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ) |
26 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝐸 ∈ ℝ ) |
27 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 𝐸 ∈ ℝ+ ) |
28 |
27
|
rpge0d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 0 ≤ 𝐸 ) |
29 |
|
lt2sq |
⊢ ( ( ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ∈ ℝ ∧ 0 ≤ ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ) ∧ ( 𝐸 ∈ ℝ ∧ 0 ≤ 𝐸 ) ) → ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) < 𝐸 ↔ ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ↑ 2 ) < ( 𝐸 ↑ 2 ) ) ) |
30 |
20 25 26 28 29
|
syl22anc |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) < 𝐸 ↔ ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ↑ 2 ) < ( 𝐸 ↑ 2 ) ) ) |
31 |
6 30
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ↑ 2 ) < ( 𝐸 ↑ 2 ) ) |
32 |
1 2 21 24 31
|
fsumlt |
⊢ ( 𝜑 → Σ 𝑖 ∈ 𝐼 ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ↑ 2 ) < Σ 𝑖 ∈ 𝐼 ( 𝐸 ↑ 2 ) ) |
33 |
10
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( 𝑋 ‘ 𝑖 ) ∈ ℝ ) |
34 |
16
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( 𝑌 ‘ 𝑖 ) ∈ ℝ ) |
35 |
33 34
|
resubcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ∈ ℝ ) |
36 |
|
absresq |
⊢ ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ∈ ℝ → ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ↑ 2 ) = ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) |
37 |
35 36
|
syl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ↑ 2 ) = ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) |
38 |
37
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) = ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ↑ 2 ) ) |
39 |
38
|
sumeq2dv |
⊢ ( 𝜑 → Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) = Σ 𝑖 ∈ 𝐼 ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ↑ 2 ) ) |
40 |
11 23
|
sselid |
⊢ ( 𝜑 → ( 𝐸 ↑ 2 ) ∈ ℂ ) |
41 |
|
fsumconst |
⊢ ( ( 𝐼 ∈ Fin ∧ ( 𝐸 ↑ 2 ) ∈ ℂ ) → Σ 𝑖 ∈ 𝐼 ( 𝐸 ↑ 2 ) = ( ( ♯ ‘ 𝐼 ) · ( 𝐸 ↑ 2 ) ) ) |
42 |
1 40 41
|
syl2anc |
⊢ ( 𝜑 → Σ 𝑖 ∈ 𝐼 ( 𝐸 ↑ 2 ) = ( ( ♯ ‘ 𝐼 ) · ( 𝐸 ↑ 2 ) ) ) |
43 |
|
eqcom |
⊢ ( 𝑁 = ( ♯ ‘ 𝐼 ) ↔ ( ♯ ‘ 𝐼 ) = 𝑁 ) |
44 |
3 43
|
mpbi |
⊢ ( ♯ ‘ 𝐼 ) = 𝑁 |
45 |
44
|
oveq1i |
⊢ ( ( ♯ ‘ 𝐼 ) · ( 𝐸 ↑ 2 ) ) = ( 𝑁 · ( 𝐸 ↑ 2 ) ) |
46 |
45
|
a1i |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝐼 ) · ( 𝐸 ↑ 2 ) ) = ( 𝑁 · ( 𝐸 ↑ 2 ) ) ) |
47 |
42 46
|
eqtr2d |
⊢ ( 𝜑 → ( 𝑁 · ( 𝐸 ↑ 2 ) ) = Σ 𝑖 ∈ 𝐼 ( 𝐸 ↑ 2 ) ) |
48 |
39 47
|
breq12d |
⊢ ( 𝜑 → ( Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) < ( 𝑁 · ( 𝐸 ↑ 2 ) ) ↔ Σ 𝑖 ∈ 𝐼 ( ( abs ‘ ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ) ↑ 2 ) < Σ 𝑖 ∈ 𝐼 ( 𝐸 ↑ 2 ) ) ) |
49 |
32 48
|
mpbird |
⊢ ( 𝜑 → Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) < ( 𝑁 · ( 𝐸 ↑ 2 ) ) ) |
50 |
|
nfv |
⊢ Ⅎ 𝑖 𝜑 |
51 |
35
|
resqcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ∈ ℝ ) |
52 |
50 1 51
|
fsumreclf |
⊢ ( 𝜑 → Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ∈ ℝ ) |
53 |
35
|
sqge0d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → 0 ≤ ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) |
54 |
1 51 53
|
fsumge0 |
⊢ ( 𝜑 → 0 ≤ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) |
55 |
|
hashcl |
⊢ ( 𝐼 ∈ Fin → ( ♯ ‘ 𝐼 ) ∈ ℕ0 ) |
56 |
1 55
|
syl |
⊢ ( 𝜑 → ( ♯ ‘ 𝐼 ) ∈ ℕ0 ) |
57 |
3 56
|
eqeltrid |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
58 |
57
|
nn0red |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
59 |
58 23
|
remulcld |
⊢ ( 𝜑 → ( 𝑁 · ( 𝐸 ↑ 2 ) ) ∈ ℝ ) |
60 |
57
|
nn0ge0d |
⊢ ( 𝜑 → 0 ≤ 𝑁 ) |
61 |
22
|
sqge0d |
⊢ ( 𝜑 → 0 ≤ ( 𝐸 ↑ 2 ) ) |
62 |
58 23 60 61
|
mulge0d |
⊢ ( 𝜑 → 0 ≤ ( 𝑁 · ( 𝐸 ↑ 2 ) ) ) |
63 |
52 54 59 62
|
sqrtltd |
⊢ ( 𝜑 → ( Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) < ( 𝑁 · ( 𝐸 ↑ 2 ) ) ↔ ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) < ( √ ‘ ( 𝑁 · ( 𝐸 ↑ 2 ) ) ) ) ) |
64 |
49 63
|
mpbid |
⊢ ( 𝜑 → ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) < ( √ ‘ ( 𝑁 · ( 𝐸 ↑ 2 ) ) ) ) |
65 |
8
|
a1i |
⊢ ( 𝜑 → 𝐷 = ( dist ‘ ( ℝ^ ‘ 𝐼 ) ) ) |
66 |
|
eqid |
⊢ ( ℝ^ ‘ 𝐼 ) = ( ℝ^ ‘ 𝐼 ) |
67 |
|
eqid |
⊢ ( ℝ ↑m 𝐼 ) = ( ℝ ↑m 𝐼 ) |
68 |
66 67
|
rrxdsfi |
⊢ ( 𝐼 ∈ Fin → ( dist ‘ ( ℝ^ ‘ 𝐼 ) ) = ( 𝑓 ∈ ( ℝ ↑m 𝐼 ) , 𝑔 ∈ ( ℝ ↑m 𝐼 ) ↦ ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑓 ‘ 𝑖 ) − ( 𝑔 ‘ 𝑖 ) ) ↑ 2 ) ) ) ) |
69 |
1 68
|
syl |
⊢ ( 𝜑 → ( dist ‘ ( ℝ^ ‘ 𝐼 ) ) = ( 𝑓 ∈ ( ℝ ↑m 𝐼 ) , 𝑔 ∈ ( ℝ ↑m 𝐼 ) ↦ ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑓 ‘ 𝑖 ) − ( 𝑔 ‘ 𝑖 ) ) ↑ 2 ) ) ) ) |
70 |
65 69
|
eqtrd |
⊢ ( 𝜑 → 𝐷 = ( 𝑓 ∈ ( ℝ ↑m 𝐼 ) , 𝑔 ∈ ( ℝ ↑m 𝐼 ) ↦ ( √ ‘ Σ 𝑖 ∈ 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 ) → Σ 𝑖 ∈ 𝐼 ( ( ( 𝑓 ‘ 𝑖 ) − ( 𝑔 ‘ 𝑖 ) ) ↑ 2 ) = Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) |
78 |
77
|
fveq2d |
⊢ ( ( 𝑓 = 𝑋 ∧ 𝑔 = 𝑌 ) → ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑓 ‘ 𝑖 ) − ( 𝑔 ‘ 𝑖 ) ) ↑ 2 ) ) = ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) ) |
79 |
78
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑓 = 𝑋 ∧ 𝑔 = 𝑌 ) ) → ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑓 ‘ 𝑖 ) − ( 𝑔 ‘ 𝑖 ) ) ↑ 2 ) ) = ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) ) |
80 |
52 54
|
resqrtcld |
⊢ ( 𝜑 → ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) ∈ ℝ ) |
81 |
70 79 4 5 80
|
ovmpod |
⊢ ( 𝜑 → ( 𝑋 𝐷 𝑌 ) = ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) ) |
82 |
|
sqrtmul |
⊢ ( ( ( 𝑁 ∈ ℝ ∧ 0 ≤ 𝑁 ) ∧ ( ( 𝐸 ↑ 2 ) ∈ ℝ ∧ 0 ≤ ( 𝐸 ↑ 2 ) ) ) → ( √ ‘ ( 𝑁 · ( 𝐸 ↑ 2 ) ) ) = ( ( √ ‘ 𝑁 ) · ( √ ‘ ( 𝐸 ↑ 2 ) ) ) ) |
83 |
58 60 23 61 82
|
syl22anc |
⊢ ( 𝜑 → ( √ ‘ ( 𝑁 · ( 𝐸 ↑ 2 ) ) ) = ( ( √ ‘ 𝑁 ) · ( √ ‘ ( 𝐸 ↑ 2 ) ) ) ) |
84 |
7
|
rpge0d |
⊢ ( 𝜑 → 0 ≤ 𝐸 ) |
85 |
22 84
|
sqrtsqd |
⊢ ( 𝜑 → ( √ ‘ ( 𝐸 ↑ 2 ) ) = 𝐸 ) |
86 |
85
|
oveq2d |
⊢ ( 𝜑 → ( ( √ ‘ 𝑁 ) · ( √ ‘ ( 𝐸 ↑ 2 ) ) ) = ( ( √ ‘ 𝑁 ) · 𝐸 ) ) |
87 |
83 86
|
eqtr2d |
⊢ ( 𝜑 → ( ( √ ‘ 𝑁 ) · 𝐸 ) = ( √ ‘ ( 𝑁 · ( 𝐸 ↑ 2 ) ) ) ) |
88 |
81 87
|
breq12d |
⊢ ( 𝜑 → ( ( 𝑋 𝐷 𝑌 ) < ( ( √ ‘ 𝑁 ) · 𝐸 ) ↔ ( √ ‘ Σ 𝑖 ∈ 𝐼 ( ( ( 𝑋 ‘ 𝑖 ) − ( 𝑌 ‘ 𝑖 ) ) ↑ 2 ) ) < ( √ ‘ ( 𝑁 · ( 𝐸 ↑ 2 ) ) ) ) ) |
89 |
64 88
|
mpbird |
⊢ ( 𝜑 → ( 𝑋 𝐷 𝑌 ) < ( ( √ ‘ 𝑁 ) · 𝐸 ) ) |