Step |
Hyp |
Ref |
Expression |
1 |
|
minveco.x |
⊢ 𝑋 = ( BaseSet ‘ 𝑈 ) |
2 |
|
minveco.m |
⊢ 𝑀 = ( −𝑣 ‘ 𝑈 ) |
3 |
|
minveco.n |
⊢ 𝑁 = ( normCV ‘ 𝑈 ) |
4 |
|
minveco.y |
⊢ 𝑌 = ( BaseSet ‘ 𝑊 ) |
5 |
|
minveco.u |
⊢ ( 𝜑 → 𝑈 ∈ CPreHilOLD ) |
6 |
|
minveco.w |
⊢ ( 𝜑 → 𝑊 ∈ ( ( SubSp ‘ 𝑈 ) ∩ CBan ) ) |
7 |
|
minveco.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑋 ) |
8 |
|
minveco.d |
⊢ 𝐷 = ( IndMet ‘ 𝑈 ) |
9 |
|
minveco.j |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
10 |
|
minveco.r |
⊢ 𝑅 = ran ( 𝑦 ∈ 𝑌 ↦ ( 𝑁 ‘ ( 𝐴 𝑀 𝑦 ) ) ) |
11 |
|
minveco.s |
⊢ 𝑆 = inf ( 𝑅 , ℝ , < ) |
12 |
|
minveco.f |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ 𝑌 ) |
13 |
|
minveco.1 |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) ≤ ( ( 𝑆 ↑ 2 ) + ( 1 / 𝑛 ) ) ) |
14 |
|
4re |
⊢ 4 ∈ ℝ |
15 |
|
4pos |
⊢ 0 < 4 |
16 |
14 15
|
elrpii |
⊢ 4 ∈ ℝ+ |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ+ ) |
18 |
|
2z |
⊢ 2 ∈ ℤ |
19 |
|
rpexpcl |
⊢ ( ( 𝑥 ∈ ℝ+ ∧ 2 ∈ ℤ ) → ( 𝑥 ↑ 2 ) ∈ ℝ+ ) |
20 |
17 18 19
|
sylancl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 𝑥 ↑ 2 ) ∈ ℝ+ ) |
21 |
|
rpdivcl |
⊢ ( ( 4 ∈ ℝ+ ∧ ( 𝑥 ↑ 2 ) ∈ ℝ+ ) → ( 4 / ( 𝑥 ↑ 2 ) ) ∈ ℝ+ ) |
22 |
16 20 21
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 4 / ( 𝑥 ↑ 2 ) ) ∈ ℝ+ ) |
23 |
|
rprege0 |
⊢ ( ( 4 / ( 𝑥 ↑ 2 ) ) ∈ ℝ+ → ( ( 4 / ( 𝑥 ↑ 2 ) ) ∈ ℝ ∧ 0 ≤ ( 4 / ( 𝑥 ↑ 2 ) ) ) ) |
24 |
|
flge0nn0 |
⊢ ( ( ( 4 / ( 𝑥 ↑ 2 ) ) ∈ ℝ ∧ 0 ≤ ( 4 / ( 𝑥 ↑ 2 ) ) ) → ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) ∈ ℕ0 ) |
25 |
|
nn0p1nn |
⊢ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) ∈ ℕ0 → ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ∈ ℕ ) |
26 |
22 23 24 25
|
4syl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ∈ ℕ ) |
27 |
|
phnv |
⊢ ( 𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec ) |
28 |
1 8
|
imsmet |
⊢ ( 𝑈 ∈ NrmCVec → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
29 |
5 27 28
|
3syl |
⊢ ( 𝜑 → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
30 |
29
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
31 |
5 27
|
syl |
⊢ ( 𝜑 → 𝑈 ∈ NrmCVec ) |
32 |
|
inss1 |
⊢ ( ( SubSp ‘ 𝑈 ) ∩ CBan ) ⊆ ( SubSp ‘ 𝑈 ) |
33 |
32 6
|
sselid |
⊢ ( 𝜑 → 𝑊 ∈ ( SubSp ‘ 𝑈 ) ) |
34 |
|
eqid |
⊢ ( SubSp ‘ 𝑈 ) = ( SubSp ‘ 𝑈 ) |
35 |
1 4 34
|
sspba |
⊢ ( ( 𝑈 ∈ NrmCVec ∧ 𝑊 ∈ ( SubSp ‘ 𝑈 ) ) → 𝑌 ⊆ 𝑋 ) |
36 |
31 33 35
|
syl2anc |
⊢ ( 𝜑 → 𝑌 ⊆ 𝑋 ) |
37 |
36
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → 𝑌 ⊆ 𝑋 ) |
38 |
12
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → 𝐹 : ℕ ⟶ 𝑌 ) |
39 |
26
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ∈ ℕ ) |
40 |
38 39
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ 𝑌 ) |
41 |
37 40
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ 𝑋 ) |
42 |
|
eluznn |
⊢ ( ( ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ∈ ℕ ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → 𝑛 ∈ ℕ ) |
43 |
26 42
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → 𝑛 ∈ ℕ ) |
44 |
38 43
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝑌 ) |
45 |
37 44
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝑋 ) |
46 |
|
metcl |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ 𝑋 ∧ ( 𝐹 ‘ 𝑛 ) ∈ 𝑋 ) → ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
47 |
30 41 45 46
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
48 |
47
|
resqcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) ∈ ℝ ) |
49 |
39
|
nnrpd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ∈ ℝ+ ) |
50 |
49
|
rpreccld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ ℝ+ ) |
51 |
|
rpmulcl |
⊢ ( ( 4 ∈ ℝ+ ∧ ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ ℝ+ ) → ( 4 · ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ∈ ℝ+ ) |
52 |
16 50 51
|
sylancr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 4 · ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ∈ ℝ+ ) |
53 |
52
|
rpred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 4 · ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ∈ ℝ ) |
54 |
20
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝑥 ↑ 2 ) ∈ ℝ+ ) |
55 |
54
|
rpred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝑥 ↑ 2 ) ∈ ℝ ) |
56 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → 𝑈 ∈ CPreHilOLD ) |
57 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → 𝑊 ∈ ( ( SubSp ‘ 𝑈 ) ∩ CBan ) ) |
58 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → 𝐴 ∈ 𝑋 ) |
59 |
26
|
nnrpd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ∈ ℝ+ ) |
60 |
59
|
rpreccld |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ ℝ+ ) |
61 |
60
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ ℝ+ ) |
62 |
61
|
rpred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ ℝ ) |
63 |
61
|
rpge0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → 0 ≤ ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) |
64 |
12
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → 𝐹 : ℕ ⟶ 𝑌 ) |
65 |
64
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝑌 ) |
66 |
43 65
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝑌 ) |
67 |
|
fveq2 |
⊢ ( 𝑛 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) |
68 |
67
|
oveq2d |
⊢ ( 𝑛 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) = ( 𝐴 𝐷 ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ) |
69 |
68
|
oveq1d |
⊢ ( 𝑛 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) = ( ( 𝐴 𝐷 ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ↑ 2 ) ) |
70 |
|
oveq2 |
⊢ ( 𝑛 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( 1 / 𝑛 ) = ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) |
71 |
70
|
oveq2d |
⊢ ( 𝑛 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( ( 𝑆 ↑ 2 ) + ( 1 / 𝑛 ) ) = ( ( 𝑆 ↑ 2 ) + ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ) |
72 |
69 71
|
breq12d |
⊢ ( 𝑛 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( ( ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) ≤ ( ( 𝑆 ↑ 2 ) + ( 1 / 𝑛 ) ) ↔ ( ( 𝐴 𝐷 ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ↑ 2 ) ≤ ( ( 𝑆 ↑ 2 ) + ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ) ) |
73 |
13
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ ( ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) ≤ ( ( 𝑆 ↑ 2 ) + ( 1 / 𝑛 ) ) ) |
74 |
73
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ∀ 𝑛 ∈ ℕ ( ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) ≤ ( ( 𝑆 ↑ 2 ) + ( 1 / 𝑛 ) ) ) |
75 |
72 74 39
|
rspcdva |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝐴 𝐷 ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ↑ 2 ) ≤ ( ( 𝑆 ↑ 2 ) + ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ) |
76 |
37 66
|
sseldd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝐹 ‘ 𝑛 ) ∈ 𝑋 ) |
77 |
|
metcl |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ ( 𝐹 ‘ 𝑛 ) ∈ 𝑋 ) → ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
78 |
30 58 76 77
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ) |
79 |
78
|
resqcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) ∈ ℝ ) |
80 |
1 2 3 4 5 6 7 8 9 10
|
minvecolem1 |
⊢ ( 𝜑 → ( 𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀ 𝑤 ∈ 𝑅 0 ≤ 𝑤 ) ) |
81 |
|
0re |
⊢ 0 ∈ ℝ |
82 |
|
breq1 |
⊢ ( 𝑥 = 0 → ( 𝑥 ≤ 𝑤 ↔ 0 ≤ 𝑤 ) ) |
83 |
82
|
ralbidv |
⊢ ( 𝑥 = 0 → ( ∀ 𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ↔ ∀ 𝑤 ∈ 𝑅 0 ≤ 𝑤 ) ) |
84 |
83
|
rspcev |
⊢ ( ( 0 ∈ ℝ ∧ ∀ 𝑤 ∈ 𝑅 0 ≤ 𝑤 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ) |
85 |
81 84
|
mpan |
⊢ ( ∀ 𝑤 ∈ 𝑅 0 ≤ 𝑤 → ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ) |
86 |
85
|
3anim3i |
⊢ ( ( 𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∀ 𝑤 ∈ 𝑅 0 ≤ 𝑤 ) → ( 𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ) ) |
87 |
|
infrecl |
⊢ ( ( 𝑅 ⊆ ℝ ∧ 𝑅 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑤 ∈ 𝑅 𝑥 ≤ 𝑤 ) → inf ( 𝑅 , ℝ , < ) ∈ ℝ ) |
88 |
80 86 87
|
3syl |
⊢ ( 𝜑 → inf ( 𝑅 , ℝ , < ) ∈ ℝ ) |
89 |
11 88
|
eqeltrid |
⊢ ( 𝜑 → 𝑆 ∈ ℝ ) |
90 |
89
|
resqcld |
⊢ ( 𝜑 → ( 𝑆 ↑ 2 ) ∈ ℝ ) |
91 |
90
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝑆 ↑ 2 ) ∈ ℝ ) |
92 |
43
|
nnrecred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 1 / 𝑛 ) ∈ ℝ ) |
93 |
91 92
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝑆 ↑ 2 ) + ( 1 / 𝑛 ) ) ∈ ℝ ) |
94 |
91 62
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝑆 ↑ 2 ) + ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ∈ ℝ ) |
95 |
13
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ℕ ) → ( ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) ≤ ( ( 𝑆 ↑ 2 ) + ( 1 / 𝑛 ) ) ) |
96 |
43 95
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) ≤ ( ( 𝑆 ↑ 2 ) + ( 1 / 𝑛 ) ) ) |
97 |
|
eluzle |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) → ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ≤ 𝑛 ) |
98 |
97
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ≤ 𝑛 ) |
99 |
49
|
rpregt0d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ∈ ℝ ∧ 0 < ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) |
100 |
|
nnre |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ ) |
101 |
|
nngt0 |
⊢ ( 𝑛 ∈ ℕ → 0 < 𝑛 ) |
102 |
100 101
|
jca |
⊢ ( 𝑛 ∈ ℕ → ( 𝑛 ∈ ℝ ∧ 0 < 𝑛 ) ) |
103 |
43 102
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝑛 ∈ ℝ ∧ 0 < 𝑛 ) ) |
104 |
|
lerec |
⊢ ( ( ( ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ∈ ℝ ∧ 0 < ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∧ ( 𝑛 ∈ ℝ ∧ 0 < 𝑛 ) ) → ( ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ≤ 𝑛 ↔ ( 1 / 𝑛 ) ≤ ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ) |
105 |
99 103 104
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ≤ 𝑛 ↔ ( 1 / 𝑛 ) ≤ ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ) |
106 |
98 105
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 1 / 𝑛 ) ≤ ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) |
107 |
92 62 91 106
|
leadd2dd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝑆 ↑ 2 ) + ( 1 / 𝑛 ) ) ≤ ( ( 𝑆 ↑ 2 ) + ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ) |
108 |
79 93 94 96 107
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝐴 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) ≤ ( ( 𝑆 ↑ 2 ) + ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ) |
109 |
1 2 3 4 56 57 58 8 9 10 11 62 63 40 66 75 108
|
minvecolem2 |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) ≤ ( 4 · ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) ) |
110 |
|
rpdivcl |
⊢ ( ( ( 𝑥 ↑ 2 ) ∈ ℝ+ ∧ 4 ∈ ℝ+ ) → ( ( 𝑥 ↑ 2 ) / 4 ) ∈ ℝ+ ) |
111 |
54 16 110
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝑥 ↑ 2 ) / 4 ) ∈ ℝ+ ) |
112 |
|
rpcnne0 |
⊢ ( ( 𝑥 ↑ 2 ) ∈ ℝ+ → ( ( 𝑥 ↑ 2 ) ∈ ℂ ∧ ( 𝑥 ↑ 2 ) ≠ 0 ) ) |
113 |
54 112
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝑥 ↑ 2 ) ∈ ℂ ∧ ( 𝑥 ↑ 2 ) ≠ 0 ) ) |
114 |
|
rpcnne0 |
⊢ ( 4 ∈ ℝ+ → ( 4 ∈ ℂ ∧ 4 ≠ 0 ) ) |
115 |
16 114
|
ax-mp |
⊢ ( 4 ∈ ℂ ∧ 4 ≠ 0 ) |
116 |
|
recdiv |
⊢ ( ( ( ( 𝑥 ↑ 2 ) ∈ ℂ ∧ ( 𝑥 ↑ 2 ) ≠ 0 ) ∧ ( 4 ∈ ℂ ∧ 4 ≠ 0 ) ) → ( 1 / ( ( 𝑥 ↑ 2 ) / 4 ) ) = ( 4 / ( 𝑥 ↑ 2 ) ) ) |
117 |
113 115 116
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 1 / ( ( 𝑥 ↑ 2 ) / 4 ) ) = ( 4 / ( 𝑥 ↑ 2 ) ) ) |
118 |
22
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 4 / ( 𝑥 ↑ 2 ) ) ∈ ℝ+ ) |
119 |
118
|
rpred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 4 / ( 𝑥 ↑ 2 ) ) ∈ ℝ ) |
120 |
|
flltp1 |
⊢ ( ( 4 / ( 𝑥 ↑ 2 ) ) ∈ ℝ → ( 4 / ( 𝑥 ↑ 2 ) ) < ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) |
121 |
119 120
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 4 / ( 𝑥 ↑ 2 ) ) < ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) |
122 |
117 121
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 1 / ( ( 𝑥 ↑ 2 ) / 4 ) ) < ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) |
123 |
111 49 122
|
ltrec1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) < ( ( 𝑥 ↑ 2 ) / 4 ) ) |
124 |
14 15
|
pm3.2i |
⊢ ( 4 ∈ ℝ ∧ 0 < 4 ) |
125 |
|
ltmuldiv2 |
⊢ ( ( ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ ℝ ∧ ( 𝑥 ↑ 2 ) ∈ ℝ ∧ ( 4 ∈ ℝ ∧ 0 < 4 ) ) → ( ( 4 · ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) < ( 𝑥 ↑ 2 ) ↔ ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) < ( ( 𝑥 ↑ 2 ) / 4 ) ) ) |
126 |
124 125
|
mp3an3 |
⊢ ( ( ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ ℝ ∧ ( 𝑥 ↑ 2 ) ∈ ℝ ) → ( ( 4 · ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) < ( 𝑥 ↑ 2 ) ↔ ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) < ( ( 𝑥 ↑ 2 ) / 4 ) ) ) |
127 |
62 55 126
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 4 · ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) < ( 𝑥 ↑ 2 ) ↔ ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) < ( ( 𝑥 ↑ 2 ) / 4 ) ) ) |
128 |
123 127
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 4 · ( 1 / ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) < ( 𝑥 ↑ 2 ) ) |
129 |
48 53 55 109 128
|
lelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) < ( 𝑥 ↑ 2 ) ) |
130 |
|
metge0 |
⊢ ( ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ∈ 𝑋 ∧ ( 𝐹 ‘ 𝑛 ) ∈ 𝑋 ) → 0 ≤ ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ) |
131 |
30 41 45 130
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → 0 ≤ ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ) |
132 |
|
rprege0 |
⊢ ( 𝑥 ∈ ℝ+ → ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ) |
133 |
132
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ) |
134 |
|
lt2sq |
⊢ ( ( ( ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ∈ ℝ ∧ 0 ≤ ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ) ∧ ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ) → ( ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ↔ ( ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) < ( 𝑥 ↑ 2 ) ) ) |
135 |
47 131 133 134
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ↔ ( ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ↑ 2 ) < ( 𝑥 ↑ 2 ) ) ) |
136 |
129 135
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) → ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) |
137 |
136
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∀ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) |
138 |
|
fveq2 |
⊢ ( 𝑗 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( ℤ≥ ‘ 𝑗 ) = ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) |
139 |
|
fveq2 |
⊢ ( 𝑗 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ) |
140 |
139
|
oveq1d |
⊢ ( 𝑗 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) = ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) ) |
141 |
140
|
breq1d |
⊢ ( 𝑗 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ↔ ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
142 |
138 141
|
raleqbidv |
⊢ ( 𝑗 = ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) → ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ↔ ∀ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
143 |
142
|
rspcev |
⊢ ( ( ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ∈ ℕ ∧ ∀ 𝑛 ∈ ( ℤ≥ ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) ( ( 𝐹 ‘ ( ( ⌊ ‘ ( 4 / ( 𝑥 ↑ 2 ) ) ) + 1 ) ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) → ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) |
144 |
26 137 143
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) |
145 |
144
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) |
146 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
147 |
1 8
|
imsxmet |
⊢ ( 𝑈 ∈ NrmCVec → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
148 |
5 27 147
|
3syl |
⊢ ( 𝜑 → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
149 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
150 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐹 ‘ 𝑛 ) = ( 𝐹 ‘ 𝑛 ) ) |
151 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℕ ) → ( 𝐹 ‘ 𝑗 ) = ( 𝐹 ‘ 𝑗 ) ) |
152 |
12 36
|
fssd |
⊢ ( 𝜑 → 𝐹 : ℕ ⟶ 𝑋 ) |
153 |
146 148 149 150 151 152
|
iscauf |
⊢ ( 𝜑 → ( 𝐹 ∈ ( Cau ‘ 𝐷 ) ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑗 ∈ ℕ ∀ 𝑛 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 𝐹 ‘ 𝑗 ) 𝐷 ( 𝐹 ‘ 𝑛 ) ) < 𝑥 ) ) |
154 |
145 153
|
mpbird |
⊢ ( 𝜑 → 𝐹 ∈ ( Cau ‘ 𝐷 ) ) |