Step |
Hyp |
Ref |
Expression |
1 |
|
divsqrtsum.2 |
⊢ 𝐹 = ( 𝑥 ∈ ℝ+ ↦ ( Σ 𝑛 ∈ ( 1 ... ( ⌊ ‘ 𝑥 ) ) ( 1 / ( √ ‘ 𝑛 ) ) − ( 2 · ( √ ‘ 𝑥 ) ) ) ) |
2 |
|
ioorp |
⊢ ( 0 (,) +∞ ) = ℝ+ |
3 |
2
|
eqcomi |
⊢ ℝ+ = ( 0 (,) +∞ ) |
4 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
5 |
|
1zzd |
⊢ ( ⊤ → 1 ∈ ℤ ) |
6 |
|
0red |
⊢ ( ⊤ → 0 ∈ ℝ ) |
7 |
|
1re |
⊢ 1 ∈ ℝ |
8 |
|
0nn0 |
⊢ 0 ∈ ℕ0 |
9 |
7 8
|
nn0addge2i |
⊢ 1 ≤ ( 0 + 1 ) |
10 |
9
|
a1i |
⊢ ( ⊤ → 1 ≤ ( 0 + 1 ) ) |
11 |
|
2re |
⊢ 2 ∈ ℝ |
12 |
|
rpsqrtcl |
⊢ ( 𝑥 ∈ ℝ+ → ( √ ‘ 𝑥 ) ∈ ℝ+ ) |
13 |
12
|
adantl |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( √ ‘ 𝑥 ) ∈ ℝ+ ) |
14 |
13
|
rpred |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( √ ‘ 𝑥 ) ∈ ℝ ) |
15 |
|
remulcl |
⊢ ( ( 2 ∈ ℝ ∧ ( √ ‘ 𝑥 ) ∈ ℝ ) → ( 2 · ( √ ‘ 𝑥 ) ) ∈ ℝ ) |
16 |
11 14 15
|
sylancr |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( 2 · ( √ ‘ 𝑥 ) ) ∈ ℝ ) |
17 |
13
|
rprecred |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( 1 / ( √ ‘ 𝑥 ) ) ∈ ℝ ) |
18 |
|
nnrp |
⊢ ( 𝑥 ∈ ℕ → 𝑥 ∈ ℝ+ ) |
19 |
18 17
|
sylan2 |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℕ ) → ( 1 / ( √ ‘ 𝑥 ) ) ∈ ℝ ) |
20 |
|
reelprrecn |
⊢ ℝ ∈ { ℝ , ℂ } |
21 |
20
|
a1i |
⊢ ( ⊤ → ℝ ∈ { ℝ , ℂ } ) |
22 |
13
|
rpcnd |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( √ ‘ 𝑥 ) ∈ ℂ ) |
23 |
|
2rp |
⊢ 2 ∈ ℝ+ |
24 |
|
rpmulcl |
⊢ ( ( 2 ∈ ℝ+ ∧ ( √ ‘ 𝑥 ) ∈ ℝ+ ) → ( 2 · ( √ ‘ 𝑥 ) ) ∈ ℝ+ ) |
25 |
23 13 24
|
sylancr |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( 2 · ( √ ‘ 𝑥 ) ) ∈ ℝ+ ) |
26 |
25
|
rpreccld |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( 1 / ( 2 · ( √ ‘ 𝑥 ) ) ) ∈ ℝ+ ) |
27 |
|
dvsqrt |
⊢ ( ℝ D ( 𝑥 ∈ ℝ+ ↦ ( √ ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ+ ↦ ( 1 / ( 2 · ( √ ‘ 𝑥 ) ) ) ) |
28 |
27
|
a1i |
⊢ ( ⊤ → ( ℝ D ( 𝑥 ∈ ℝ+ ↦ ( √ ‘ 𝑥 ) ) ) = ( 𝑥 ∈ ℝ+ ↦ ( 1 / ( 2 · ( √ ‘ 𝑥 ) ) ) ) ) |
29 |
|
2cnd |
⊢ ( ⊤ → 2 ∈ ℂ ) |
30 |
21 22 26 28 29
|
dvmptcmul |
⊢ ( ⊤ → ( ℝ D ( 𝑥 ∈ ℝ+ ↦ ( 2 · ( √ ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ℝ+ ↦ ( 2 · ( 1 / ( 2 · ( √ ‘ 𝑥 ) ) ) ) ) ) |
31 |
|
2cnd |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → 2 ∈ ℂ ) |
32 |
|
1cnd |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → 1 ∈ ℂ ) |
33 |
25
|
rpcnne0d |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( ( 2 · ( √ ‘ 𝑥 ) ) ∈ ℂ ∧ ( 2 · ( √ ‘ 𝑥 ) ) ≠ 0 ) ) |
34 |
|
divass |
⊢ ( ( 2 ∈ ℂ ∧ 1 ∈ ℂ ∧ ( ( 2 · ( √ ‘ 𝑥 ) ) ∈ ℂ ∧ ( 2 · ( √ ‘ 𝑥 ) ) ≠ 0 ) ) → ( ( 2 · 1 ) / ( 2 · ( √ ‘ 𝑥 ) ) ) = ( 2 · ( 1 / ( 2 · ( √ ‘ 𝑥 ) ) ) ) ) |
35 |
31 32 33 34
|
syl3anc |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( ( 2 · 1 ) / ( 2 · ( √ ‘ 𝑥 ) ) ) = ( 2 · ( 1 / ( 2 · ( √ ‘ 𝑥 ) ) ) ) ) |
36 |
13
|
rpcnne0d |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( ( √ ‘ 𝑥 ) ∈ ℂ ∧ ( √ ‘ 𝑥 ) ≠ 0 ) ) |
37 |
|
rpcnne0 |
⊢ ( 2 ∈ ℝ+ → ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) |
38 |
23 37
|
mp1i |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) |
39 |
|
divcan5 |
⊢ ( ( 1 ∈ ℂ ∧ ( ( √ ‘ 𝑥 ) ∈ ℂ ∧ ( √ ‘ 𝑥 ) ≠ 0 ) ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) → ( ( 2 · 1 ) / ( 2 · ( √ ‘ 𝑥 ) ) ) = ( 1 / ( √ ‘ 𝑥 ) ) ) |
40 |
32 36 38 39
|
syl3anc |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( ( 2 · 1 ) / ( 2 · ( √ ‘ 𝑥 ) ) ) = ( 1 / ( √ ‘ 𝑥 ) ) ) |
41 |
35 40
|
eqtr3d |
⊢ ( ( ⊤ ∧ 𝑥 ∈ ℝ+ ) → ( 2 · ( 1 / ( 2 · ( √ ‘ 𝑥 ) ) ) ) = ( 1 / ( √ ‘ 𝑥 ) ) ) |
42 |
41
|
mpteq2dva |
⊢ ( ⊤ → ( 𝑥 ∈ ℝ+ ↦ ( 2 · ( 1 / ( 2 · ( √ ‘ 𝑥 ) ) ) ) ) = ( 𝑥 ∈ ℝ+ ↦ ( 1 / ( √ ‘ 𝑥 ) ) ) ) |
43 |
30 42
|
eqtrd |
⊢ ( ⊤ → ( ℝ D ( 𝑥 ∈ ℝ+ ↦ ( 2 · ( √ ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ℝ+ ↦ ( 1 / ( √ ‘ 𝑥 ) ) ) ) |
44 |
|
fveq2 |
⊢ ( 𝑥 = 𝑛 → ( √ ‘ 𝑥 ) = ( √ ‘ 𝑛 ) ) |
45 |
44
|
oveq2d |
⊢ ( 𝑥 = 𝑛 → ( 1 / ( √ ‘ 𝑥 ) ) = ( 1 / ( √ ‘ 𝑛 ) ) ) |
46 |
|
simp3r |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → 𝑥 ≤ 𝑛 ) |
47 |
|
simp2l |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → 𝑥 ∈ ℝ+ ) |
48 |
47
|
rprege0d |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ) |
49 |
|
simp2r |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → 𝑛 ∈ ℝ+ ) |
50 |
49
|
rprege0d |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → ( 𝑛 ∈ ℝ ∧ 0 ≤ 𝑛 ) ) |
51 |
|
sqrtle |
⊢ ( ( ( 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥 ) ∧ ( 𝑛 ∈ ℝ ∧ 0 ≤ 𝑛 ) ) → ( 𝑥 ≤ 𝑛 ↔ ( √ ‘ 𝑥 ) ≤ ( √ ‘ 𝑛 ) ) ) |
52 |
48 50 51
|
syl2anc |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → ( 𝑥 ≤ 𝑛 ↔ ( √ ‘ 𝑥 ) ≤ ( √ ‘ 𝑛 ) ) ) |
53 |
46 52
|
mpbid |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → ( √ ‘ 𝑥 ) ≤ ( √ ‘ 𝑛 ) ) |
54 |
47
|
rpsqrtcld |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → ( √ ‘ 𝑥 ) ∈ ℝ+ ) |
55 |
49
|
rpsqrtcld |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → ( √ ‘ 𝑛 ) ∈ ℝ+ ) |
56 |
54 55
|
lerecd |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → ( ( √ ‘ 𝑥 ) ≤ ( √ ‘ 𝑛 ) ↔ ( 1 / ( √ ‘ 𝑛 ) ) ≤ ( 1 / ( √ ‘ 𝑥 ) ) ) ) |
57 |
53 56
|
mpbid |
⊢ ( ( ⊤ ∧ ( 𝑥 ∈ ℝ+ ∧ 𝑛 ∈ ℝ+ ) ∧ ( 0 ≤ 𝑥 ∧ 𝑥 ≤ 𝑛 ) ) → ( 1 / ( √ ‘ 𝑛 ) ) ≤ ( 1 / ( √ ‘ 𝑥 ) ) ) |
58 |
|
sqrtlim |
⊢ ( 𝑥 ∈ ℝ+ ↦ ( 1 / ( √ ‘ 𝑥 ) ) ) ⇝𝑟 0 |
59 |
58
|
a1i |
⊢ ( ⊤ → ( 𝑥 ∈ ℝ+ ↦ ( 1 / ( √ ‘ 𝑥 ) ) ) ⇝𝑟 0 ) |
60 |
|
fveq2 |
⊢ ( 𝑥 = 𝐴 → ( √ ‘ 𝑥 ) = ( √ ‘ 𝐴 ) ) |
61 |
60
|
oveq2d |
⊢ ( 𝑥 = 𝐴 → ( 1 / ( √ ‘ 𝑥 ) ) = ( 1 / ( √ ‘ 𝐴 ) ) ) |
62 |
3 4 5 6 10 6 16 17 19 43 45 57 1 59 61
|
dvfsumrlim3 |
⊢ ( ⊤ → ( 𝐹 : ℝ+ ⟶ ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ( ( 𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ∧ 0 ≤ 𝐴 ) → ( abs ‘ ( ( 𝐹 ‘ 𝐴 ) − 𝐿 ) ) ≤ ( 1 / ( √ ‘ 𝐴 ) ) ) ) ) |
63 |
62
|
simp1d |
⊢ ( ⊤ → 𝐹 : ℝ+ ⟶ ℝ ) |
64 |
63
|
mptru |
⊢ 𝐹 : ℝ+ ⟶ ℝ |
65 |
62
|
simp2d |
⊢ ( ⊤ → 𝐹 ∈ dom ⇝𝑟 ) |
66 |
65
|
mptru |
⊢ 𝐹 ∈ dom ⇝𝑟 |
67 |
|
rpge0 |
⊢ ( 𝐴 ∈ ℝ+ → 0 ≤ 𝐴 ) |
68 |
67
|
adantl |
⊢ ( ( 𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ) → 0 ≤ 𝐴 ) |
69 |
62
|
simp3d |
⊢ ( ⊤ → ( ( 𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ∧ 0 ≤ 𝐴 ) → ( abs ‘ ( ( 𝐹 ‘ 𝐴 ) − 𝐿 ) ) ≤ ( 1 / ( √ ‘ 𝐴 ) ) ) ) |
70 |
69
|
mptru |
⊢ ( ( 𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ∧ 0 ≤ 𝐴 ) → ( abs ‘ ( ( 𝐹 ‘ 𝐴 ) − 𝐿 ) ) ≤ ( 1 / ( √ ‘ 𝐴 ) ) ) |
71 |
68 70
|
mpd3an3 |
⊢ ( ( 𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ) → ( abs ‘ ( ( 𝐹 ‘ 𝐴 ) − 𝐿 ) ) ≤ ( 1 / ( √ ‘ 𝐴 ) ) ) |
72 |
64 66 71
|
3pm3.2i |
⊢ ( 𝐹 : ℝ+ ⟶ ℝ ∧ 𝐹 ∈ dom ⇝𝑟 ∧ ( ( 𝐹 ⇝𝑟 𝐿 ∧ 𝐴 ∈ ℝ+ ) → ( abs ‘ ( ( 𝐹 ‘ 𝐴 ) − 𝐿 ) ) ≤ ( 1 / ( √ ‘ 𝐴 ) ) ) ) |