Step | Hyp | Ref
| Expression |
1 | | uzssz 11583 |
. . . . . 6
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
2 | | zssre 11261 |
. . . . . 6
⊢ ℤ
⊆ ℝ |
3 | 1, 2 | sstri 3577 |
. . . . 5
⊢
(ℤ≥‘𝑀) ⊆ ℝ |
4 | 3 | a1i 11 |
. . . 4
⊢ (𝜑 →
(ℤ≥‘𝑀) ⊆ ℝ) |
5 | | ioodvbdlimc1lem2.m |
. . . . . . 7
⊢ 𝑀 = ((⌊‘(1 / (𝐵 − 𝐴))) + 1) |
6 | | ioodvbdlimc1lem2.b |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ∈ ℝ) |
7 | | ioodvbdlimc1lem2.a |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ ℝ) |
8 | 6, 7 | resubcld 10337 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℝ) |
9 | | ioodvbdlimc1lem2.altb |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 < 𝐵) |
10 | 7, 6 | posdifd 10493 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
11 | 9, 10 | mpbid 221 |
. . . . . . . . . . 11
⊢ (𝜑 → 0 < (𝐵 − 𝐴)) |
12 | 11 | gt0ne0d 10471 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐵 − 𝐴) ≠ 0) |
13 | 8, 12 | rereccld 10731 |
. . . . . . . . 9
⊢ (𝜑 → (1 / (𝐵 − 𝐴)) ∈ ℝ) |
14 | | 0red 9920 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ∈
ℝ) |
15 | 8, 11 | recgt0d 10837 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < (1 / (𝐵 − 𝐴))) |
16 | 14, 13, 15 | ltled 10064 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ (1 / (𝐵 − 𝐴))) |
17 | | flge0nn0 12483 |
. . . . . . . . 9
⊢ (((1 /
(𝐵 − 𝐴)) ∈ ℝ ∧ 0 ≤
(1 / (𝐵 − 𝐴))) → (⌊‘(1 /
(𝐵 − 𝐴))) ∈
ℕ0) |
18 | 13, 16, 17 | syl2anc 691 |
. . . . . . . 8
⊢ (𝜑 → (⌊‘(1 / (𝐵 − 𝐴))) ∈
ℕ0) |
19 | | peano2nn0 11210 |
. . . . . . . 8
⊢
((⌊‘(1 / (𝐵 − 𝐴))) ∈ ℕ0 →
((⌊‘(1 / (𝐵
− 𝐴))) + 1) ∈
ℕ0) |
20 | 18, 19 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((⌊‘(1 /
(𝐵 − 𝐴))) + 1) ∈
ℕ0) |
21 | 5, 20 | syl5eqel 2692 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈
ℕ0) |
22 | 21 | nn0zd 11356 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ ℤ) |
23 | | eqid 2610 |
. . . . . 6
⊢
(ℤ≥‘𝑀) = (ℤ≥‘𝑀) |
24 | 23 | uzsup 12524 |
. . . . 5
⊢ (𝑀 ∈ ℤ →
sup((ℤ≥‘𝑀), ℝ*, < ) =
+∞) |
25 | 22, 24 | syl 17 |
. . . 4
⊢ (𝜑 →
sup((ℤ≥‘𝑀), ℝ*, < ) =
+∞) |
26 | | ioodvbdlimc1lem2.f |
. . . . . . 7
⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ) |
27 | 26 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝐹:(𝐴(,)𝐵)⟶ℝ) |
28 | 7 | rexrd 9968 |
. . . . . . . 8
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
29 | 28 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈
ℝ*) |
30 | 6 | rexrd 9968 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
31 | 30 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝐵 ∈
ℝ*) |
32 | 7 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℝ) |
33 | | eluzelre 11574 |
. . . . . . . . . 10
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 𝑗 ∈ ℝ) |
34 | 33 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ ℝ) |
35 | | 0red 9920 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 0 ∈
ℝ) |
36 | | 0red 9920 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 0 ∈ ℝ) |
37 | | 1red 9934 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 1 ∈ ℝ) |
38 | 36, 37 | readdcld 9948 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → (0 + 1) ∈
ℝ) |
39 | 38 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (0 + 1) ∈
ℝ) |
40 | 36 | ltp1d 10833 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 0 < (0 + 1)) |
41 | 40 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 0 < (0 +
1)) |
42 | | eluzel2 11568 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
43 | 42 | zred 11358 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℝ) |
44 | 43 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℝ) |
45 | 13 | flcld 12461 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (⌊‘(1 / (𝐵 − 𝐴))) ∈ ℤ) |
46 | 45 | zred 11358 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (⌊‘(1 / (𝐵 − 𝐴))) ∈ ℝ) |
47 | | 1red 9934 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ∈
ℝ) |
48 | 18 | nn0ge0d 11231 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 0 ≤ (⌊‘(1 /
(𝐵 − 𝐴)))) |
49 | 14, 46, 47, 48 | leadd1dd 10520 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (0 + 1) ≤
((⌊‘(1 / (𝐵
− 𝐴))) +
1)) |
50 | 49, 5 | syl6breqr 4625 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (0 + 1) ≤ 𝑀) |
51 | 50 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (0 + 1) ≤ 𝑀) |
52 | | eluzle 11576 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈
(ℤ≥‘𝑀) → 𝑀 ≤ 𝑗) |
53 | 52 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑀 ≤ 𝑗) |
54 | 39, 44, 34, 51, 53 | letrd 10073 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (0 + 1) ≤ 𝑗) |
55 | 35, 39, 34, 41, 54 | ltletrd 10076 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 0 < 𝑗) |
56 | 55 | gt0ne0d 10471 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ≠ 0) |
57 | 34, 56 | rereccld 10731 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (1 / 𝑗) ∈
ℝ) |
58 | 32, 57 | readdcld 9948 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐴 + (1 / 𝑗)) ∈ ℝ) |
59 | 34, 55 | elrpd 11745 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ ℝ+) |
60 | 59 | rpreccld 11758 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (1 / 𝑗) ∈
ℝ+) |
61 | 32, 60 | ltaddrpd 11781 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝐴 < (𝐴 + (1 / 𝑗))) |
62 | 21 | nn0red 11229 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℝ) |
63 | 14, 47 | readdcld 9948 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (0 + 1) ∈
ℝ) |
64 | 46, 47 | readdcld 9948 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((⌊‘(1 /
(𝐵 − 𝐴))) + 1) ∈
ℝ) |
65 | 14 | ltp1d 10833 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < (0 +
1)) |
66 | 14, 63, 64, 65, 49 | ltletrd 10076 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 0 < ((⌊‘(1
/ (𝐵 − 𝐴))) + 1)) |
67 | 66, 5 | syl6breqr 4625 |
. . . . . . . . . . . 12
⊢ (𝜑 → 0 < 𝑀) |
68 | 67 | gt0ne0d 10471 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ≠ 0) |
69 | 62, 68 | rereccld 10731 |
. . . . . . . . . 10
⊢ (𝜑 → (1 / 𝑀) ∈ ℝ) |
70 | 69 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (1 / 𝑀) ∈ ℝ) |
71 | 32, 70 | readdcld 9948 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐴 + (1 / 𝑀)) ∈ ℝ) |
72 | 6 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝐵 ∈ ℝ) |
73 | 62, 67 | elrpd 11745 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈
ℝ+) |
74 | 73 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈
ℝ+) |
75 | | 1red 9934 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 1 ∈
ℝ) |
76 | | 0le1 10430 |
. . . . . . . . . . 11
⊢ 0 ≤
1 |
77 | 76 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 0 ≤
1) |
78 | 74, 59, 75, 77, 53 | lediv2ad 11770 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (1 / 𝑗) ≤ (1 / 𝑀)) |
79 | 57, 70, 32, 78 | leadd2dd 10521 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐴 + (1 / 𝑗)) ≤ (𝐴 + (1 / 𝑀))) |
80 | 5 | eqcomi 2619 |
. . . . . . . . . . . . 13
⊢
((⌊‘(1 / (𝐵 − 𝐴))) + 1) = 𝑀 |
81 | 80 | oveq2i 6560 |
. . . . . . . . . . . 12
⊢ (1 /
((⌊‘(1 / (𝐵
− 𝐴))) + 1)) = (1 /
𝑀) |
82 | 81, 69 | syl5eqel 2692 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 / ((⌊‘(1 /
(𝐵 − 𝐴))) + 1)) ∈
ℝ) |
83 | 13, 15 | elrpd 11745 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1 / (𝐵 − 𝐴)) ∈
ℝ+) |
84 | 64, 66 | elrpd 11745 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((⌊‘(1 /
(𝐵 − 𝐴))) + 1) ∈
ℝ+) |
85 | | 1rp 11712 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℝ+ |
86 | 85 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ∈
ℝ+) |
87 | | fllelt 12460 |
. . . . . . . . . . . . . . 15
⊢ ((1 /
(𝐵 − 𝐴)) ∈ ℝ →
((⌊‘(1 / (𝐵
− 𝐴))) ≤ (1 /
(𝐵 − 𝐴)) ∧ (1 / (𝐵 − 𝐴)) < ((⌊‘(1 / (𝐵 − 𝐴))) + 1))) |
88 | 13, 87 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((⌊‘(1 /
(𝐵 − 𝐴))) ≤ (1 / (𝐵 − 𝐴)) ∧ (1 / (𝐵 − 𝐴)) < ((⌊‘(1 / (𝐵 − 𝐴))) + 1))) |
89 | 88 | simprd 478 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1 / (𝐵 − 𝐴)) < ((⌊‘(1 / (𝐵 − 𝐴))) + 1)) |
90 | 83, 84, 86, 89 | ltdiv2dd 38448 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1 / ((⌊‘(1 /
(𝐵 − 𝐴))) + 1)) < (1 / (1 / (𝐵 − 𝐴)))) |
91 | 8 | recnd 9947 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐵 − 𝐴) ∈ ℂ) |
92 | 91, 12 | recrecd 10677 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1 / (1 / (𝐵 − 𝐴))) = (𝐵 − 𝐴)) |
93 | 90, 92 | breqtrd 4609 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 / ((⌊‘(1 /
(𝐵 − 𝐴))) + 1)) < (𝐵 − 𝐴)) |
94 | 82, 8, 7, 93 | ltadd2dd 10075 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 + (1 / ((⌊‘(1 / (𝐵 − 𝐴))) + 1))) < (𝐴 + (𝐵 − 𝐴))) |
95 | 5 | oveq2i 6560 |
. . . . . . . . . . . 12
⊢ (1 /
𝑀) = (1 /
((⌊‘(1 / (𝐵
− 𝐴))) +
1)) |
96 | 95 | oveq2i 6560 |
. . . . . . . . . . 11
⊢ (𝐴 + (1 / 𝑀)) = (𝐴 + (1 / ((⌊‘(1 / (𝐵 − 𝐴))) + 1))) |
97 | 96 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 + (1 / 𝑀)) = (𝐴 + (1 / ((⌊‘(1 / (𝐵 − 𝐴))) + 1)))) |
98 | 7 | recnd 9947 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ ℂ) |
99 | 6 | recnd 9947 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ∈ ℂ) |
100 | 98, 99 | pncan3d 10274 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
101 | 100 | eqcomd 2616 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 = (𝐴 + (𝐵 − 𝐴))) |
102 | 94, 97, 101 | 3brtr4d 4615 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 + (1 / 𝑀)) < 𝐵) |
103 | 102 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐴 + (1 / 𝑀)) < 𝐵) |
104 | 58, 71, 72, 79, 103 | lelttrd 10074 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐴 + (1 / 𝑗)) < 𝐵) |
105 | 29, 31, 58, 61, 104 | eliood 38567 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐴 + (1 / 𝑗)) ∈ (𝐴(,)𝐵)) |
106 | 27, 105 | ffvelrnd 6268 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐹‘(𝐴 + (1 / 𝑗))) ∈ ℝ) |
107 | | ioodvbdlimc1lem2.s |
. . . . 5
⊢ 𝑆 = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐹‘(𝐴 + (1 / 𝑗)))) |
108 | 106, 107 | fmptd 6292 |
. . . 4
⊢ (𝜑 → 𝑆:(ℤ≥‘𝑀)⟶ℝ) |
109 | | ioodvbdlimc1lem2.dmdv |
. . . . . 6
⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) |
110 | | ioodvbdlimc1lem2.dvbd |
. . . . . 6
⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑦) |
111 | 7, 6, 9, 26, 109, 110 | dvbdfbdioo 38820 |
. . . . 5
⊢ (𝜑 → ∃𝑏 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) |
112 | 62 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) → 𝑀 ∈ ℝ) |
113 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → 𝑗 ∈ (ℤ≥‘𝑀)) |
114 | 107 | fvmpt2 6200 |
. . . . . . . . . . . . . 14
⊢ ((𝑗 ∈
(ℤ≥‘𝑀) ∧ (𝐹‘(𝐴 + (1 / 𝑗))) ∈ ℝ) → (𝑆‘𝑗) = (𝐹‘(𝐴 + (1 / 𝑗)))) |
115 | 113, 106,
114 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝑆‘𝑗) = (𝐹‘(𝐴 + (1 / 𝑗)))) |
116 | 115 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (abs‘(𝑆‘𝑗)) = (abs‘(𝐹‘(𝐴 + (1 / 𝑗))))) |
117 | 116 | adantlr 747 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (abs‘(𝑆‘𝑗)) = (abs‘(𝐹‘(𝐴 + (1 / 𝑗))))) |
118 | | simplr 788 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) |
119 | 105 | adantlr 747 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐴 + (1 / 𝑗)) ∈ (𝐴(,)𝐵)) |
120 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝐴 + (1 / 𝑗)) → (𝐹‘𝑥) = (𝐹‘(𝐴 + (1 / 𝑗)))) |
121 | 120 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝐴 + (1 / 𝑗)) → (abs‘(𝐹‘𝑥)) = (abs‘(𝐹‘(𝐴 + (1 / 𝑗))))) |
122 | 121 | breq1d 4593 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝐴 + (1 / 𝑗)) → ((abs‘(𝐹‘𝑥)) ≤ 𝑏 ↔ (abs‘(𝐹‘(𝐴 + (1 / 𝑗)))) ≤ 𝑏)) |
123 | 122 | rspccva 3281 |
. . . . . . . . . . . 12
⊢
((∀𝑥 ∈
(𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏 ∧ (𝐴 + (1 / 𝑗)) ∈ (𝐴(,)𝐵)) → (abs‘(𝐹‘(𝐴 + (1 / 𝑗)))) ≤ 𝑏) |
124 | 118, 119,
123 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘(𝐴 + (1 / 𝑗)))) ≤ 𝑏) |
125 | 117, 124 | eqbrtrd 4605 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (abs‘(𝑆‘𝑗)) ≤ 𝑏) |
126 | 125 | a1d 25 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝑀 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏)) |
127 | 126 | ralrimiva 2949 |
. . . . . . . 8
⊢ ((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) → ∀𝑗 ∈ (ℤ≥‘𝑀)(𝑀 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏)) |
128 | | breq1 4586 |
. . . . . . . . . . 11
⊢ (𝑘 = 𝑀 → (𝑘 ≤ 𝑗 ↔ 𝑀 ≤ 𝑗)) |
129 | 128 | imbi1d 330 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑀 → ((𝑘 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏) ↔ (𝑀 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏))) |
130 | 129 | ralbidv 2969 |
. . . . . . . . 9
⊢ (𝑘 = 𝑀 → (∀𝑗 ∈ (ℤ≥‘𝑀)(𝑘 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏) ↔ ∀𝑗 ∈ (ℤ≥‘𝑀)(𝑀 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏))) |
131 | 130 | rspcev 3282 |
. . . . . . . 8
⊢ ((𝑀 ∈ ℝ ∧
∀𝑗 ∈
(ℤ≥‘𝑀)(𝑀 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏)) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ (ℤ≥‘𝑀)(𝑘 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏)) |
132 | 112, 127,
131 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏) → ∃𝑘 ∈ ℝ ∀𝑗 ∈ (ℤ≥‘𝑀)(𝑘 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏)) |
133 | 132 | ex 449 |
. . . . . 6
⊢ (𝜑 → (∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏 → ∃𝑘 ∈ ℝ ∀𝑗 ∈ (ℤ≥‘𝑀)(𝑘 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏))) |
134 | 133 | reximdv 2999 |
. . . . 5
⊢ (𝜑 → (∃𝑏 ∈ ℝ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘(𝐹‘𝑥)) ≤ 𝑏 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ (ℤ≥‘𝑀)(𝑘 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏))) |
135 | 111, 134 | mpd 15 |
. . . 4
⊢ (𝜑 → ∃𝑏 ∈ ℝ ∃𝑘 ∈ ℝ ∀𝑗 ∈ (ℤ≥‘𝑀)(𝑘 ≤ 𝑗 → (abs‘(𝑆‘𝑗)) ≤ 𝑏)) |
136 | 4, 25, 108, 135 | limsupre 38708 |
. . 3
⊢ (𝜑 → (lim sup‘𝑆) ∈
ℝ) |
137 | 136 | recnd 9947 |
. 2
⊢ (𝜑 → (lim sup‘𝑆) ∈
ℂ) |
138 | | eluzelre 11574 |
. . . . . . . . 9
⊢ (𝑗 ∈
(ℤ≥‘𝑁) → 𝑗 ∈ ℝ) |
139 | 138 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 𝑗 ∈ ℝ) |
140 | | 0red 9920 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 0 ∈ ℝ) |
141 | 45 | peano2zd 11361 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((⌊‘(1 /
(𝐵 − 𝐴))) + 1) ∈
ℤ) |
142 | 5, 141 | syl5eqel 2692 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ ℤ) |
143 | 142 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑀 ∈
ℤ) |
144 | 143 | zred 11358 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑀 ∈
ℝ) |
145 | 144 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 𝑀 ∈ ℝ) |
146 | 67 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 0 < 𝑀) |
147 | | ioodvbdlimc1lem2.n |
. . . . . . . . . . . . . 14
⊢ 𝑁 = if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀) |
148 | | ioodvbdlimc1lem2.y |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑌 = sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) |
149 | | ioomidp 38587 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵) → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵)) |
150 | 7, 6, 9, 149 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵)) |
151 | | ne0i 3880 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅) |
152 | 150, 151 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝐴(,)𝐵) ≠ ∅) |
153 | | ioossre 12106 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝐴(,)𝐵) ⊆ ℝ |
154 | 153 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ) |
155 | | dvfre 23520 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
156 | 26, 154, 155 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ) |
157 | 109 | feq2d 5944 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ
D 𝐹):(𝐴(,)𝐵)⟶ℝ)) |
158 | 156, 157 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ) |
159 | 158 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ) |
160 | 159 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ) |
161 | 160 | abscld 14023 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → (abs‘((ℝ D 𝐹)‘𝑥)) ∈ ℝ) |
162 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))) = (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))) |
163 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ sup(ran
(𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) = sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) |
164 | 152, 161,
110, 162, 163 | suprnmpt 38350 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) ∈ ℝ ∧
∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ))) |
165 | 164 | simpld 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) ∈
ℝ) |
166 | 148, 165 | syl5eqel 2692 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑌 ∈ ℝ) |
167 | 166 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑌 ∈
ℝ) |
168 | | rpre 11715 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℝ) |
169 | 168 | rehalfcld 11156 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ℝ+
→ (𝑥 / 2) ∈
ℝ) |
170 | 169 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ∈
ℝ) |
171 | 168 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ∈
ℂ) |
172 | 171 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℂ) |
173 | | 2cnd 10970 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 2 ∈
ℂ) |
174 | | rpne0 11724 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ℝ+
→ 𝑥 ≠
0) |
175 | 174 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ≠ 0) |
176 | | 2ne0 10990 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 2 ≠
0 |
177 | 176 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 2 ≠
0) |
178 | 172, 173,
175, 177 | divne0d 10696 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ≠ 0) |
179 | 167, 170,
178 | redivcld 10732 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑌 / (𝑥 / 2)) ∈ ℝ) |
180 | 179 | flcld 12461 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
(⌊‘(𝑌 / (𝑥 / 2))) ∈
ℤ) |
181 | 180 | peano2zd 11361 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
((⌊‘(𝑌 / (𝑥 / 2))) + 1) ∈
ℤ) |
182 | 181, 143 | ifcld 4081 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀) ∈ ℤ) |
183 | 147, 182 | syl5eqel 2692 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑁 ∈
ℤ) |
184 | 183 | zred 11358 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑁 ∈
ℝ) |
185 | 184 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 𝑁 ∈ ℝ) |
186 | 181 | zred 11358 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
((⌊‘(𝑌 / (𝑥 / 2))) + 1) ∈
ℝ) |
187 | | max1 11890 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 ∈ ℝ ∧
((⌊‘(𝑌 / (𝑥 / 2))) + 1) ∈ ℝ)
→ 𝑀 ≤ if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀)) |
188 | 144, 186,
187 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑀 ≤ if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀)) |
189 | 188, 147 | syl6breqr 4625 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑀 ≤ 𝑁) |
190 | 189 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 𝑀 ≤ 𝑁) |
191 | | eluzle 11576 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈
(ℤ≥‘𝑁) → 𝑁 ≤ 𝑗) |
192 | 191 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 𝑁 ≤ 𝑗) |
193 | 145, 185,
139, 190, 192 | letrd 10073 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 𝑀 ≤ 𝑗) |
194 | 140, 145,
139, 146, 193 | ltletrd 10076 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 0 < 𝑗) |
195 | 194 | gt0ne0d 10471 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 𝑗 ≠ 0) |
196 | 139, 195 | rereccld 10731 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → (1 / 𝑗) ∈ ℝ) |
197 | 139, 194 | recgt0d 10837 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → 0 < (1 / 𝑗)) |
198 | 196, 197 | elrpd 11745 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) → (1 / 𝑗) ∈
ℝ+) |
199 | 198 | adantr 480 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) → (1 / 𝑗) ∈
ℝ+) |
200 | | ioodvbdlimc1lem2.ch |
. . . . . . . . 9
⊢ (𝜒 ↔ (((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗))) |
201 | 200 | biimpi 205 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜒 → (((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗))) |
202 | | simp-5l 804 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) → 𝜑) |
203 | 201, 202 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜒 → 𝜑) |
204 | 203, 26 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → 𝐹:(𝐴(,)𝐵)⟶ℝ) |
205 | 201 | simplrd 789 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → 𝑧 ∈ (𝐴(,)𝐵)) |
206 | 204, 205 | ffvelrnd 6268 |
. . . . . . . . . . . . . 14
⊢ (𝜒 → (𝐹‘𝑧) ∈ ℝ) |
207 | 206 | recnd 9947 |
. . . . . . . . . . . . 13
⊢ (𝜒 → (𝐹‘𝑧) ∈ ℂ) |
208 | 203, 108 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → 𝑆:(ℤ≥‘𝑀)⟶ℝ) |
209 | | simp-5r 805 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) → 𝑥 ∈ ℝ+) |
210 | 201, 209 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → 𝑥 ∈ ℝ+) |
211 | | eluz2 11569 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑁 ∈
(ℤ≥‘𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ≤ 𝑁)) |
212 | 143, 183,
189, 211 | syl3anbrc 1239 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑁 ∈
(ℤ≥‘𝑀)) |
213 | 203, 210,
212 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜒 → 𝑁 ∈ (ℤ≥‘𝑀)) |
214 | | uzss 11584 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (ℤ≥‘𝑁) ⊆
(ℤ≥‘𝑀)) |
215 | 213, 214 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜒 →
(ℤ≥‘𝑁) ⊆
(ℤ≥‘𝑀)) |
216 | | simp-4r 803 |
. . . . . . . . . . . . . . . . 17
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) → 𝑗 ∈ (ℤ≥‘𝑁)) |
217 | 201, 216 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜒 → 𝑗 ∈ (ℤ≥‘𝑁)) |
218 | 215, 217 | sseldd 3569 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → 𝑗 ∈ (ℤ≥‘𝑀)) |
219 | 208, 218 | ffvelrnd 6268 |
. . . . . . . . . . . . . 14
⊢ (𝜒 → (𝑆‘𝑗) ∈ ℝ) |
220 | 219 | recnd 9947 |
. . . . . . . . . . . . 13
⊢ (𝜒 → (𝑆‘𝑗) ∈ ℂ) |
221 | 203, 137 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜒 → (lim sup‘𝑆) ∈
ℂ) |
222 | 207, 220,
221 | npncand 10295 |
. . . . . . . . . . . 12
⊢ (𝜒 → (((𝐹‘𝑧) − (𝑆‘𝑗)) + ((𝑆‘𝑗) − (lim sup‘𝑆))) = ((𝐹‘𝑧) − (lim sup‘𝑆))) |
223 | 222 | eqcomd 2616 |
. . . . . . . . . . 11
⊢ (𝜒 → ((𝐹‘𝑧) − (lim sup‘𝑆)) = (((𝐹‘𝑧) − (𝑆‘𝑗)) + ((𝑆‘𝑗) − (lim sup‘𝑆)))) |
224 | 223 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝜒 → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) = (abs‘(((𝐹‘𝑧) − (𝑆‘𝑗)) + ((𝑆‘𝑗) − (lim sup‘𝑆))))) |
225 | 206, 219 | resubcld 10337 |
. . . . . . . . . . . . . 14
⊢ (𝜒 → ((𝐹‘𝑧) − (𝑆‘𝑗)) ∈ ℝ) |
226 | 203, 136 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → (lim sup‘𝑆) ∈
ℝ) |
227 | 219, 226 | resubcld 10337 |
. . . . . . . . . . . . . 14
⊢ (𝜒 → ((𝑆‘𝑗) − (lim sup‘𝑆)) ∈ ℝ) |
228 | 225, 227 | readdcld 9948 |
. . . . . . . . . . . . 13
⊢ (𝜒 → (((𝐹‘𝑧) − (𝑆‘𝑗)) + ((𝑆‘𝑗) − (lim sup‘𝑆))) ∈ ℝ) |
229 | 228 | recnd 9947 |
. . . . . . . . . . . 12
⊢ (𝜒 → (((𝐹‘𝑧) − (𝑆‘𝑗)) + ((𝑆‘𝑗) − (lim sup‘𝑆))) ∈ ℂ) |
230 | 229 | abscld 14023 |
. . . . . . . . . . 11
⊢ (𝜒 → (abs‘(((𝐹‘𝑧) − (𝑆‘𝑗)) + ((𝑆‘𝑗) − (lim sup‘𝑆)))) ∈ ℝ) |
231 | 225 | recnd 9947 |
. . . . . . . . . . . . 13
⊢ (𝜒 → ((𝐹‘𝑧) − (𝑆‘𝑗)) ∈ ℂ) |
232 | 231 | abscld 14023 |
. . . . . . . . . . . 12
⊢ (𝜒 → (abs‘((𝐹‘𝑧) − (𝑆‘𝑗))) ∈ ℝ) |
233 | 227 | recnd 9947 |
. . . . . . . . . . . . 13
⊢ (𝜒 → ((𝑆‘𝑗) − (lim sup‘𝑆)) ∈ ℂ) |
234 | 233 | abscld 14023 |
. . . . . . . . . . . 12
⊢ (𝜒 → (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) ∈ ℝ) |
235 | 232, 234 | readdcld 9948 |
. . . . . . . . . . 11
⊢ (𝜒 → ((abs‘((𝐹‘𝑧) − (𝑆‘𝑗))) + (abs‘((𝑆‘𝑗) − (lim sup‘𝑆)))) ∈ ℝ) |
236 | 210 | rpred 11748 |
. . . . . . . . . . 11
⊢ (𝜒 → 𝑥 ∈ ℝ) |
237 | 231, 233 | abstrid 14043 |
. . . . . . . . . . 11
⊢ (𝜒 → (abs‘(((𝐹‘𝑧) − (𝑆‘𝑗)) + ((𝑆‘𝑗) − (lim sup‘𝑆)))) ≤ ((abs‘((𝐹‘𝑧) − (𝑆‘𝑗))) + (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))))) |
238 | 236 | rehalfcld 11156 |
. . . . . . . . . . . . 13
⊢ (𝜒 → (𝑥 / 2) ∈ ℝ) |
239 | 207, 220 | abssubd 14040 |
. . . . . . . . . . . . . 14
⊢ (𝜒 → (abs‘((𝐹‘𝑧) − (𝑆‘𝑗))) = (abs‘((𝑆‘𝑗) − (𝐹‘𝑧)))) |
240 | 203, 218,
115 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜒 → (𝑆‘𝑗) = (𝐹‘(𝐴 + (1 / 𝑗)))) |
241 | 240 | oveq1d 6564 |
. . . . . . . . . . . . . . . 16
⊢ (𝜒 → ((𝑆‘𝑗) − (𝐹‘𝑧)) = ((𝐹‘(𝐴 + (1 / 𝑗))) − (𝐹‘𝑧))) |
242 | 241 | fveq2d 6107 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → (abs‘((𝑆‘𝑗) − (𝐹‘𝑧))) = (abs‘((𝐹‘(𝐴 + (1 / 𝑗))) − (𝐹‘𝑧)))) |
243 | 203, 218,
106 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜒 → (𝐹‘(𝐴 + (1 / 𝑗))) ∈ ℝ) |
244 | 243, 206 | resubcld 10337 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → ((𝐹‘(𝐴 + (1 / 𝑗))) − (𝐹‘𝑧)) ∈ ℝ) |
245 | 244 | recnd 9947 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜒 → ((𝐹‘(𝐴 + (1 / 𝑗))) − (𝐹‘𝑧)) ∈ ℂ) |
246 | 245 | abscld 14023 |
. . . . . . . . . . . . . . . 16
⊢ (𝜒 → (abs‘((𝐹‘(𝐴 + (1 / 𝑗))) − (𝐹‘𝑧))) ∈ ℝ) |
247 | 203, 166 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜒 → 𝑌 ∈ ℝ) |
248 | 203, 218,
58 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → (𝐴 + (1 / 𝑗)) ∈ ℝ) |
249 | 153, 205 | sseldi 3566 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → 𝑧 ∈ ℝ) |
250 | 248, 249 | resubcld 10337 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜒 → ((𝐴 + (1 / 𝑗)) − 𝑧) ∈ ℝ) |
251 | 247, 250 | remulcld 9949 |
. . . . . . . . . . . . . . . 16
⊢ (𝜒 → (𝑌 · ((𝐴 + (1 / 𝑗)) − 𝑧)) ∈ ℝ) |
252 | 203, 7 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → 𝐴 ∈ ℝ) |
253 | 203, 6 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → 𝐵 ∈ ℝ) |
254 | 203, 109 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → dom (ℝ D 𝐹) = (𝐴(,)𝐵)) |
255 | 164 | simprd 478 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )) |
256 | 148 | breq2i 4591 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑌 ↔ (abs‘((ℝ D 𝐹)‘𝑥)) ≤ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )) |
257 | 256 | ralbii 2963 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑥 ∈
(𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑌 ↔ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )) |
258 | 255, 257 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑌) |
259 | 203, 258 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜒 → ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑌) |
260 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 = 𝑥 → ((ℝ D 𝐹)‘𝑤) = ((ℝ D 𝐹)‘𝑥)) |
261 | 260 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 = 𝑥 → (abs‘((ℝ D 𝐹)‘𝑤)) = (abs‘((ℝ D 𝐹)‘𝑥))) |
262 | 261 | breq1d 4593 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = 𝑥 → ((abs‘((ℝ D 𝐹)‘𝑤)) ≤ 𝑌 ↔ (abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑌)) |
263 | 262 | cbvralv 3147 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑤 ∈
(𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑤)) ≤ 𝑌 ↔ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ 𝑌) |
264 | 259, 263 | sylibr 223 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → ∀𝑤 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑤)) ≤ 𝑌) |
265 | 249 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜒 → 𝑧 ∈ ℝ*) |
266 | 203, 30 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜒 → 𝐵 ∈
ℝ*) |
267 | 249, 252 | resubcld 10337 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → (𝑧 − 𝐴) ∈ ℝ) |
268 | 267 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜒 → (𝑧 − 𝐴) ∈ ℂ) |
269 | 268 | abscld 14023 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → (abs‘(𝑧 − 𝐴)) ∈ ℝ) |
270 | 3, 218 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜒 → 𝑗 ∈ ℝ) |
271 | 203, 218,
56 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜒 → 𝑗 ≠ 0) |
272 | 270, 271 | rereccld 10731 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → (1 / 𝑗) ∈ ℝ) |
273 | 267 | leabsd 14001 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → (𝑧 − 𝐴) ≤ (abs‘(𝑧 − 𝐴))) |
274 | 201 | simprd 478 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) |
275 | 267, 269,
272, 273, 274 | lelttrd 10074 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒 → (𝑧 − 𝐴) < (1 / 𝑗)) |
276 | 249, 252,
272 | ltsubadd2d 10504 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒 → ((𝑧 − 𝐴) < (1 / 𝑗) ↔ 𝑧 < (𝐴 + (1 / 𝑗)))) |
277 | 275, 276 | mpbid 221 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜒 → 𝑧 < (𝐴 + (1 / 𝑗))) |
278 | 203, 218,
104 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜒 → (𝐴 + (1 / 𝑗)) < 𝐵) |
279 | 265, 266,
248, 277, 278 | eliood 38567 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → (𝐴 + (1 / 𝑗)) ∈ (𝑧(,)𝐵)) |
280 | 252, 253,
204, 254, 247, 264, 205, 279 | dvbdfbdioolem1 38818 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜒 → ((abs‘((𝐹‘(𝐴 + (1 / 𝑗))) − (𝐹‘𝑧))) ≤ (𝑌 · ((𝐴 + (1 / 𝑗)) − 𝑧)) ∧ (abs‘((𝐹‘(𝐴 + (1 / 𝑗))) − (𝐹‘𝑧))) ≤ (𝑌 · (𝐵 − 𝐴)))) |
281 | 280 | simpld 474 |
. . . . . . . . . . . . . . . 16
⊢ (𝜒 → (abs‘((𝐹‘(𝐴 + (1 / 𝑗))) − (𝐹‘𝑧))) ≤ (𝑌 · ((𝐴 + (1 / 𝑗)) − 𝑧))) |
282 | 203, 218,
57 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → (1 / 𝑗) ∈ ℝ) |
283 | 247, 282 | remulcld 9949 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜒 → (𝑌 · (1 / 𝑗)) ∈ ℝ) |
284 | 158, 150 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℝ) |
285 | 284 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)) ∈ ℂ) |
286 | 285 | abscld 14023 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (abs‘((ℝ D
𝐹)‘((𝐴 + 𝐵) / 2))) ∈ ℝ) |
287 | 285 | absge0d 14031 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 0 ≤
(abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)))) |
288 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) |
289 | 288 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → (abs‘((ℝ D 𝐹)‘𝑥)) = (abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2)))) |
290 | 148 | eqcomi 2619 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ sup(ran
(𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) = 𝑌 |
291 | 290 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) = 𝑌) |
292 | 289, 291 | breq12d 4596 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = ((𝐴 + 𝐵) / 2) → ((abs‘((ℝ D 𝐹)‘𝑥)) ≤ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < ) ↔
(abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝑌)) |
293 | 292 | rspcva 3280 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐴 + 𝐵) / 2) ∈ (𝐴(,)𝐵) ∧ ∀𝑥 ∈ (𝐴(,)𝐵)(abs‘((ℝ D 𝐹)‘𝑥)) ≤ sup(ran (𝑥 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑥))), ℝ, < )) →
(abs‘((ℝ D 𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝑌) |
294 | 150, 255,
293 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (abs‘((ℝ D
𝐹)‘((𝐴 + 𝐵) / 2))) ≤ 𝑌) |
295 | 14, 286, 166, 287, 294 | letrd 10073 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 0 ≤ 𝑌) |
296 | 203, 295 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → 0 ≤ 𝑌) |
297 | 203, 28 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜒 → 𝐴 ∈
ℝ*) |
298 | | ioogtlb 38564 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝑧
∈ (𝐴(,)𝐵)) → 𝐴 < 𝑧) |
299 | 297, 266,
205, 298 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → 𝐴 < 𝑧) |
300 | 252, 249,
248, 299 | ltsub2dd 10519 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒 → ((𝐴 + (1 / 𝑗)) − 𝑧) < ((𝐴 + (1 / 𝑗)) − 𝐴)) |
301 | 203, 98 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → 𝐴 ∈ ℂ) |
302 | 282 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → (1 / 𝑗) ∈ ℂ) |
303 | 301, 302 | pncan2d 10273 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒 → ((𝐴 + (1 / 𝑗)) − 𝐴) = (1 / 𝑗)) |
304 | 300, 303 | breqtrd 4609 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜒 → ((𝐴 + (1 / 𝑗)) − 𝑧) < (1 / 𝑗)) |
305 | 250, 272,
304 | ltled 10064 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜒 → ((𝐴 + (1 / 𝑗)) − 𝑧) ≤ (1 / 𝑗)) |
306 | 250, 272,
247, 296, 305 | lemul2ad 10843 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜒 → (𝑌 · ((𝐴 + (1 / 𝑗)) − 𝑧)) ≤ (𝑌 · (1 / 𝑗))) |
307 | 283 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜒 ∧ 𝑌 = 0) → (𝑌 · (1 / 𝑗)) ∈ ℝ) |
308 | 238 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜒 ∧ 𝑌 = 0) → (𝑥 / 2) ∈ ℝ) |
309 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑌 = 0 → (𝑌 · (1 / 𝑗)) = (0 · (1 / 𝑗))) |
310 | 302 | mul02d 10113 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → (0 · (1 / 𝑗)) = 0) |
311 | 309, 310 | sylan9eqr 2666 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜒 ∧ 𝑌 = 0) → (𝑌 · (1 / 𝑗)) = 0) |
312 | 210 | rphalfcld 11760 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜒 → (𝑥 / 2) ∈
ℝ+) |
313 | 312 | rpgt0d 11751 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → 0 < (𝑥 / 2)) |
314 | 313 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜒 ∧ 𝑌 = 0) → 0 < (𝑥 / 2)) |
315 | 311, 314 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜒 ∧ 𝑌 = 0) → (𝑌 · (1 / 𝑗)) < (𝑥 / 2)) |
316 | 307, 308,
315 | ltled 10064 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜒 ∧ 𝑌 = 0) → (𝑌 · (1 / 𝑗)) ≤ (𝑥 / 2)) |
317 | 247 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜒 ∧ ¬ 𝑌 = 0) → 𝑌 ∈ ℝ) |
318 | 296 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜒 ∧ ¬ 𝑌 = 0) → 0 ≤ 𝑌) |
319 | | neqne 2790 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (¬
𝑌 = 0 → 𝑌 ≠ 0) |
320 | 319 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜒 ∧ ¬ 𝑌 = 0) → 𝑌 ≠ 0) |
321 | 317, 318,
320 | ne0gt0d 10053 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜒 ∧ ¬ 𝑌 = 0) → 0 < 𝑌) |
322 | 283 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 · (1 / 𝑗)) ∈ ℝ) |
323 | 3, 213 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜒 → 𝑁 ∈ ℝ) |
324 | | 0red 9920 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜒 → 0 ∈
ℝ) |
325 | 203, 210,
144 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜒 → 𝑀 ∈ ℝ) |
326 | 203, 67 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜒 → 0 < 𝑀) |
327 | 203, 210,
189 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜒 → 𝑀 ≤ 𝑁) |
328 | 324, 325,
323, 326, 327 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜒 → 0 < 𝑁) |
329 | 328 | gt0ne0d 10471 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜒 → 𝑁 ≠ 0) |
330 | 323, 329 | rereccld 10731 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜒 → (1 / 𝑁) ∈ ℝ) |
331 | 247, 330 | remulcld 9949 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒 → (𝑌 · (1 / 𝑁)) ∈ ℝ) |
332 | 331 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 · (1 / 𝑁)) ∈ ℝ) |
333 | 238 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑥 / 2) ∈ ℝ) |
334 | 282 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜒 ∧ 0 < 𝑌) → (1 / 𝑗) ∈ ℝ) |
335 | 330 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜒 ∧ 0 < 𝑌) → (1 / 𝑁) ∈ ℝ) |
336 | 247 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜒 ∧ 0 < 𝑌) → 𝑌 ∈ ℝ) |
337 | 296 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜒 ∧ 0 < 𝑌) → 0 ≤ 𝑌) |
338 | 323, 328 | elrpd 11745 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜒 → 𝑁 ∈
ℝ+) |
339 | 203, 218,
59 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜒 → 𝑗 ∈ ℝ+) |
340 | | 1red 9934 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜒 → 1 ∈
ℝ) |
341 | 76 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜒 → 0 ≤ 1) |
342 | 217, 191 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜒 → 𝑁 ≤ 𝑗) |
343 | 338, 339,
340, 341, 342 | lediv2ad 11770 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜒 → (1 / 𝑗) ≤ (1 / 𝑁)) |
344 | 343 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜒 ∧ 0 < 𝑌) → (1 / 𝑗) ≤ (1 / 𝑁)) |
345 | 334, 335,
336, 337, 344 | lemul2ad 10843 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 · (1 / 𝑗)) ≤ (𝑌 · (1 / 𝑁))) |
346 | 236 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜒 → 𝑥 ∈ ℂ) |
347 | | 2cnd 10970 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜒 → 2 ∈
ℂ) |
348 | 210 | rpne0d 11753 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜒 → 𝑥 ≠ 0) |
349 | 176 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜒 → 2 ≠ 0) |
350 | 346, 347,
348, 349 | divne0d 10696 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜒 → (𝑥 / 2) ≠ 0) |
351 | 247, 238,
350 | redivcld 10732 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜒 → (𝑌 / (𝑥 / 2)) ∈ ℝ) |
352 | 351 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 / (𝑥 / 2)) ∈ ℝ) |
353 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜒 ∧ 0 < 𝑌) → 0 < 𝑌) |
354 | 313 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜒 ∧ 0 < 𝑌) → 0 < (𝑥 / 2)) |
355 | 336, 333,
353, 354 | divgt0d 10838 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜒 ∧ 0 < 𝑌) → 0 < (𝑌 / (𝑥 / 2))) |
356 | 352, 355 | elrpd 11745 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 / (𝑥 / 2)) ∈
ℝ+) |
357 | 356 | rprecred 11759 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜒 ∧ 0 < 𝑌) → (1 / (𝑌 / (𝑥 / 2))) ∈ ℝ) |
358 | 338 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → 𝑁 ∈
ℝ+) |
359 | | 1red 9934 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → 1 ∈ ℝ) |
360 | 76 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → 0 ≤ 1) |
361 | 351 | flcld 12461 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜒 → (⌊‘(𝑌 / (𝑥 / 2))) ∈ ℤ) |
362 | 361 | peano2zd 11361 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜒 → ((⌊‘(𝑌 / (𝑥 / 2))) + 1) ∈ ℤ) |
363 | 362 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜒 → ((⌊‘(𝑌 / (𝑥 / 2))) + 1) ∈ ℝ) |
364 | 203, 142 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜒 → 𝑀 ∈ ℤ) |
365 | 362, 364 | ifcld 4081 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜒 → if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀) ∈ ℤ) |
366 | 147, 365 | syl5eqel 2692 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜒 → 𝑁 ∈ ℤ) |
367 | 366 | zred 11358 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜒 → 𝑁 ∈ ℝ) |
368 | | flltp1 12463 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑌 / (𝑥 / 2)) ∈ ℝ → (𝑌 / (𝑥 / 2)) < ((⌊‘(𝑌 / (𝑥 / 2))) + 1)) |
369 | 351, 368 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜒 → (𝑌 / (𝑥 / 2)) < ((⌊‘(𝑌 / (𝑥 / 2))) + 1)) |
370 | 203, 62 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜒 → 𝑀 ∈ ℝ) |
371 | | max2 11892 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑀 ∈ ℝ ∧
((⌊‘(𝑌 / (𝑥 / 2))) + 1) ∈ ℝ)
→ ((⌊‘(𝑌 /
(𝑥 / 2))) + 1) ≤
if(𝑀 ≤
((⌊‘(𝑌 / (𝑥 / 2))) + 1),
((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀)) |
372 | 370, 363,
371 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜒 → ((⌊‘(𝑌 / (𝑥 / 2))) + 1) ≤ if(𝑀 ≤ ((⌊‘(𝑌 / (𝑥 / 2))) + 1), ((⌊‘(𝑌 / (𝑥 / 2))) + 1), 𝑀)) |
373 | 372, 147 | syl6breqr 4625 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜒 → ((⌊‘(𝑌 / (𝑥 / 2))) + 1) ≤ 𝑁) |
374 | 351, 363,
367, 369, 373 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜒 → (𝑌 / (𝑥 / 2)) < 𝑁) |
375 | 351, 323,
374 | ltled 10064 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜒 → (𝑌 / (𝑥 / 2)) ≤ 𝑁) |
376 | 375 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 / (𝑥 / 2)) ≤ 𝑁) |
377 | 356, 358,
359, 360, 376 | lediv2ad 11770 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜒 ∧ 0 < 𝑌) → (1 / 𝑁) ≤ (1 / (𝑌 / (𝑥 / 2)))) |
378 | 335, 357,
336, 337, 377 | lemul2ad 10843 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 · (1 / 𝑁)) ≤ (𝑌 · (1 / (𝑌 / (𝑥 / 2))))) |
379 | 336 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → 𝑌 ∈ ℂ) |
380 | 352 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 / (𝑥 / 2)) ∈ ℂ) |
381 | 355 | gt0ne0d 10471 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 / (𝑥 / 2)) ≠ 0) |
382 | 379, 380,
381 | divrecd 10683 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 / (𝑌 / (𝑥 / 2))) = (𝑌 · (1 / (𝑌 / (𝑥 / 2))))) |
383 | 333 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑥 / 2) ∈ ℂ) |
384 | 353 | gt0ne0d 10471 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → 𝑌 ≠ 0) |
385 | 350 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑥 / 2) ≠ 0) |
386 | 379, 383,
384, 385 | ddcand 10700 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 / (𝑌 / (𝑥 / 2))) = (𝑥 / 2)) |
387 | 382, 386 | eqtr3d 2646 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 · (1 / (𝑌 / (𝑥 / 2)))) = (𝑥 / 2)) |
388 | 378, 387 | breqtrd 4609 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 · (1 / 𝑁)) ≤ (𝑥 / 2)) |
389 | 322, 332,
333, 345, 388 | letrd 10073 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜒 ∧ 0 < 𝑌) → (𝑌 · (1 / 𝑗)) ≤ (𝑥 / 2)) |
390 | 321, 389 | syldan 486 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜒 ∧ ¬ 𝑌 = 0) → (𝑌 · (1 / 𝑗)) ≤ (𝑥 / 2)) |
391 | 316, 390 | pm2.61dan 828 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜒 → (𝑌 · (1 / 𝑗)) ≤ (𝑥 / 2)) |
392 | 251, 283,
238, 306, 391 | letrd 10073 |
. . . . . . . . . . . . . . . 16
⊢ (𝜒 → (𝑌 · ((𝐴 + (1 / 𝑗)) − 𝑧)) ≤ (𝑥 / 2)) |
393 | 246, 251,
238, 281, 392 | letrd 10073 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → (abs‘((𝐹‘(𝐴 + (1 / 𝑗))) − (𝐹‘𝑧))) ≤ (𝑥 / 2)) |
394 | 242, 393 | eqbrtrd 4605 |
. . . . . . . . . . . . . 14
⊢ (𝜒 → (abs‘((𝑆‘𝑗) − (𝐹‘𝑧))) ≤ (𝑥 / 2)) |
395 | 239, 394 | eqbrtrd 4605 |
. . . . . . . . . . . . 13
⊢ (𝜒 → (abs‘((𝐹‘𝑧) − (𝑆‘𝑗))) ≤ (𝑥 / 2)) |
396 | | simpllr 795 |
. . . . . . . . . . . . . 14
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) → (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) |
397 | 201, 396 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜒 → (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) |
398 | 232, 234,
238, 238, 395, 397 | leltaddd 10528 |
. . . . . . . . . . . 12
⊢ (𝜒 → ((abs‘((𝐹‘𝑧) − (𝑆‘𝑗))) + (abs‘((𝑆‘𝑗) − (lim sup‘𝑆)))) < ((𝑥 / 2) + (𝑥 / 2))) |
399 | 346 | 2halvesd 11155 |
. . . . . . . . . . . 12
⊢ (𝜒 → ((𝑥 / 2) + (𝑥 / 2)) = 𝑥) |
400 | 398, 399 | breqtrd 4609 |
. . . . . . . . . . 11
⊢ (𝜒 → ((abs‘((𝐹‘𝑧) − (𝑆‘𝑗))) + (abs‘((𝑆‘𝑗) − (lim sup‘𝑆)))) < 𝑥) |
401 | 230, 235,
236, 237, 400 | lelttrd 10074 |
. . . . . . . . . 10
⊢ (𝜒 → (abs‘(((𝐹‘𝑧) − (𝑆‘𝑗)) + ((𝑆‘𝑗) − (lim sup‘𝑆)))) < 𝑥) |
402 | 224, 401 | eqbrtrd 4605 |
. . . . . . . . 9
⊢ (𝜒 → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥) |
403 | 200, 402 | sylbir 224 |
. . . . . . . 8
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥) |
404 | 403 | adantrl 748 |
. . . . . . 7
⊢
((((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) ∧ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗))) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥) |
405 | 404 | ex 449 |
. . . . . 6
⊢
(((((𝜑 ∧ 𝑥 ∈ ℝ+)
∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ 𝑧 ∈ (𝐴(,)𝐵)) → ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥)) |
406 | 405 | ralrimiva 2949 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) → ∀𝑧 ∈ (𝐴(,)𝐵)((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥)) |
407 | | breq2 4587 |
. . . . . . . . 9
⊢ (𝑦 = (1 / 𝑗) → ((abs‘(𝑧 − 𝐴)) < 𝑦 ↔ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗))) |
408 | 407 | anbi2d 736 |
. . . . . . . 8
⊢ (𝑦 = (1 / 𝑗) → ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑦) ↔ (𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)))) |
409 | 408 | imbi1d 330 |
. . . . . . 7
⊢ (𝑦 = (1 / 𝑗) → (((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑦) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥) ↔ ((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥))) |
410 | 409 | ralbidv 2969 |
. . . . . 6
⊢ (𝑦 = (1 / 𝑗) → (∀𝑧 ∈ (𝐴(,)𝐵)((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑦) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥) ↔ ∀𝑧 ∈ (𝐴(,)𝐵)((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥))) |
411 | 410 | rspcev 3282 |
. . . . 5
⊢ (((1 /
𝑗) ∈
ℝ+ ∧ ∀𝑧 ∈ (𝐴(,)𝐵)((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < (1 / 𝑗)) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥)) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ (𝐴(,)𝐵)((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑦) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥)) |
412 | 199, 406,
411 | syl2anc 691 |
. . . 4
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑗 ∈
(ℤ≥‘𝑁)) ∧ (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) → ∃𝑦 ∈ ℝ+ ∀𝑧 ∈ (𝐴(,)𝐵)((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑦) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥)) |
413 | | simpr 476 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ≤ 𝑁) → 𝑏 ≤ 𝑁) |
414 | 413 | iftrued 4044 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ≤ 𝑁) → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) = 𝑁) |
415 | | uzid 11578 |
. . . . . . . . . . . 12
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
(ℤ≥‘𝑁)) |
416 | 183, 415 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑁 ∈
(ℤ≥‘𝑁)) |
417 | 416 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ≤ 𝑁) → 𝑁 ∈ (ℤ≥‘𝑁)) |
418 | 414, 417 | eqeltrd 2688 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ≤ 𝑁) → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ (ℤ≥‘𝑁)) |
419 | 418 | adantlr 747 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ 𝑏 ≤ 𝑁) → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ (ℤ≥‘𝑁)) |
420 | | iffalse 4045 |
. . . . . . . . . 10
⊢ (¬
𝑏 ≤ 𝑁 → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) = 𝑏) |
421 | 420 | adantl 481 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) = 𝑏) |
422 | 183 | ad2antrr 758 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → 𝑁 ∈ ℤ) |
423 | | simplr 788 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → 𝑏 ∈ ℤ) |
424 | 422 | zred 11358 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → 𝑁 ∈ ℝ) |
425 | 423 | zred 11358 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → 𝑏 ∈ ℝ) |
426 | | simpr 476 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → ¬ 𝑏 ≤ 𝑁) |
427 | 424, 425 | ltnled 10063 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → (𝑁 < 𝑏 ↔ ¬ 𝑏 ≤ 𝑁)) |
428 | 426, 427 | mpbird 246 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → 𝑁 < 𝑏) |
429 | 424, 425,
428 | ltled 10064 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → 𝑁 ≤ 𝑏) |
430 | | eluz2 11569 |
. . . . . . . . . 10
⊢ (𝑏 ∈
(ℤ≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑁 ≤ 𝑏)) |
431 | 422, 423,
429, 430 | syl3anbrc 1239 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → 𝑏 ∈ (ℤ≥‘𝑁)) |
432 | 421, 431 | eqeltrd 2688 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧ ¬
𝑏 ≤ 𝑁) → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ (ℤ≥‘𝑁)) |
433 | 419, 432 | pm2.61dan 828 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ (ℤ≥‘𝑁)) |
434 | 433 | adantr 480 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧
∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2))) → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ (ℤ≥‘𝑁)) |
435 | | simpr 476 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧
∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2))) → ∀𝑐 ∈ (ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2))) |
436 | | simpr 476 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈
ℤ) |
437 | 183 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) → 𝑁 ∈
ℤ) |
438 | 437, 436 | ifcld 4081 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ ℤ) |
439 | 436 | zred 11358 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) → 𝑏 ∈
ℝ) |
440 | 437 | zred 11358 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) → 𝑁 ∈
ℝ) |
441 | | max1 11890 |
. . . . . . . . . . 11
⊢ ((𝑏 ∈ ℝ ∧ 𝑁 ∈ ℝ) → 𝑏 ≤ if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) |
442 | 439, 440,
441 | syl2anc 691 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) → 𝑏 ≤ if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) |
443 | | eluz2 11569 |
. . . . . . . . . 10
⊢ (if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ (ℤ≥‘𝑏) ↔ (𝑏 ∈ ℤ ∧ if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ ℤ ∧ 𝑏 ≤ if(𝑏 ≤ 𝑁, 𝑁, 𝑏))) |
444 | 436, 438,
442, 443 | syl3anbrc 1239 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ (ℤ≥‘𝑏)) |
445 | 444 | adantr 480 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧
∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2))) → if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ (ℤ≥‘𝑏)) |
446 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑐 = if(𝑏 ≤ 𝑁, 𝑁, 𝑏) → (𝑆‘𝑐) = (𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏))) |
447 | 446 | eleq1d 2672 |
. . . . . . . . . 10
⊢ (𝑐 = if(𝑏 ≤ 𝑁, 𝑁, 𝑏) → ((𝑆‘𝑐) ∈ ℂ ↔ (𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) ∈ ℂ)) |
448 | 446 | oveq1d 6564 |
. . . . . . . . . . . 12
⊢ (𝑐 = if(𝑏 ≤ 𝑁, 𝑁, 𝑏) → ((𝑆‘𝑐) − (lim sup‘𝑆)) = ((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆))) |
449 | 448 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ (𝑐 = if(𝑏 ≤ 𝑁, 𝑁, 𝑏) → (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) = (abs‘((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆)))) |
450 | 449 | breq1d 4593 |
. . . . . . . . . 10
⊢ (𝑐 = if(𝑏 ≤ 𝑁, 𝑁, 𝑏) → ((abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2) ↔ (abs‘((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆))) < (𝑥 / 2))) |
451 | 447, 450 | anbi12d 743 |
. . . . . . . . 9
⊢ (𝑐 = if(𝑏 ≤ 𝑁, 𝑁, 𝑏) → (((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2)) ↔ ((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) ∈ ℂ ∧ (abs‘((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆))) < (𝑥 / 2)))) |
452 | 451 | rspccva 3281 |
. . . . . . . 8
⊢
((∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2)) ∧ if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ (ℤ≥‘𝑏)) → ((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) ∈ ℂ ∧ (abs‘((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆))) < (𝑥 / 2))) |
453 | 435, 445,
452 | syl2anc 691 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧
∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2))) → ((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) ∈ ℂ ∧ (abs‘((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆))) < (𝑥 / 2))) |
454 | 453 | simprd 478 |
. . . . . 6
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧
∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2))) → (abs‘((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆))) < (𝑥 / 2)) |
455 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑗 = if(𝑏 ≤ 𝑁, 𝑁, 𝑏) → (𝑆‘𝑗) = (𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏))) |
456 | 455 | oveq1d 6564 |
. . . . . . . . 9
⊢ (𝑗 = if(𝑏 ≤ 𝑁, 𝑁, 𝑏) → ((𝑆‘𝑗) − (lim sup‘𝑆)) = ((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆))) |
457 | 456 | fveq2d 6107 |
. . . . . . . 8
⊢ (𝑗 = if(𝑏 ≤ 𝑁, 𝑁, 𝑏) → (abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) = (abs‘((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆)))) |
458 | 457 | breq1d 4593 |
. . . . . . 7
⊢ (𝑗 = if(𝑏 ≤ 𝑁, 𝑁, 𝑏) → ((abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2) ↔ (abs‘((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆))) < (𝑥 / 2))) |
459 | 458 | rspcev 3282 |
. . . . . 6
⊢
((if(𝑏 ≤ 𝑁, 𝑁, 𝑏) ∈ (ℤ≥‘𝑁) ∧ (abs‘((𝑆‘if(𝑏 ≤ 𝑁, 𝑁, 𝑏)) − (lim sup‘𝑆))) < (𝑥 / 2)) → ∃𝑗 ∈ (ℤ≥‘𝑁)(abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) |
460 | 434, 454,
459 | syl2anc 691 |
. . . . 5
⊢ ((((𝜑 ∧ 𝑥 ∈ ℝ+) ∧ 𝑏 ∈ ℤ) ∧
∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2))) → ∃𝑗 ∈ (ℤ≥‘𝑁)(abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) |
461 | | ax-resscn 9872 |
. . . . . . . . . . . . . 14
⊢ ℝ
⊆ ℂ |
462 | 461 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ℝ ⊆
ℂ) |
463 | 26, 462 | fssd 5970 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ) |
464 | | dvcn 23490 |
. . . . . . . . . . . . . 14
⊢
(((ℝ ⊆ ℂ ∧ 𝐹:(𝐴(,)𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ ℝ) ∧ dom (ℝ D
𝐹) = (𝐴(,)𝐵)) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
465 | 462, 463,
154, 109, 464 | syl31anc 1321 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
466 | | cncffvrn 22509 |
. . . . . . . . . . . . 13
⊢ ((ℝ
⊆ ℂ ∧ 𝐹
∈ ((𝐴(,)𝐵)–cn→ℂ)) → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ)) |
467 | 462, 465,
466 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ 𝐹:(𝐴(,)𝐵)⟶ℝ)) |
468 | 26, 467 | mpbird 246 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ)) |
469 | | ioodvbdlimc1lem2.r |
. . . . . . . . . . . 12
⊢ 𝑅 = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐴 + (1 / 𝑗))) |
470 | 105, 469 | fmptd 6292 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅:(ℤ≥‘𝑀)⟶(𝐴(,)𝐵)) |
471 | | eqid 2610 |
. . . . . . . . . . 11
⊢ (𝑗 ∈
(ℤ≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗))) = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗))) |
472 | | climrel 14071 |
. . . . . . . . . . . . 13
⊢ Rel
⇝ |
473 | 472 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → Rel ⇝
) |
474 | | fvex 6113 |
. . . . . . . . . . . . . . . . 17
⊢
(ℤ≥‘𝑀) ∈ V |
475 | 474 | mptex 6390 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈
(ℤ≥‘𝑀) ↦ 𝐴) ∈ V |
476 | 475 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴) ∈ V) |
477 | | eqidd 2611 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → (𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴) = (𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴)) |
478 | | eqidd 2611 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) ∧ 𝑗 = 𝑚) → 𝐴 = 𝐴) |
479 | | simpr 476 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → 𝑚 ∈ (ℤ≥‘𝑀)) |
480 | 7 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℝ) |
481 | 477, 478,
479, 480 | fvmptd 6197 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴)‘𝑚) = 𝐴) |
482 | 23, 142, 476, 98, 481 | climconst 14122 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴) ⇝ 𝐴) |
483 | 474 | mptex 6390 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈
(ℤ≥‘𝑀) ↦ (𝐴 + (1 / 𝑗))) ∈ V |
484 | 469, 483 | eqeltri 2684 |
. . . . . . . . . . . . . . 15
⊢ 𝑅 ∈ V |
485 | 484 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑅 ∈ V) |
486 | | 1cnd 9935 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ∈
ℂ) |
487 | | elnnnn0b 11214 |
. . . . . . . . . . . . . . . 16
⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℕ0
∧ 0 < 𝑀)) |
488 | 21, 67, 487 | sylanbrc 695 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑀 ∈ ℕ) |
489 | | divcnvg 38694 |
. . . . . . . . . . . . . . 15
⊢ ((1
∈ ℂ ∧ 𝑀
∈ ℕ) → (𝑗
∈ (ℤ≥‘𝑀) ↦ (1 / 𝑗)) ⇝ 0) |
490 | 486, 488,
489 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝑀) ↦ (1 / 𝑗)) ⇝ 0) |
491 | | eqidd 2611 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → (𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴) = (𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴)) |
492 | | eqidd 2611 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) ∧ 𝑗 = 𝑖) → 𝐴 = 𝐴) |
493 | | simpr 476 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 𝑖 ∈ (ℤ≥‘𝑀)) |
494 | 7 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℝ) |
495 | 491, 492,
493, 494 | fvmptd 6197 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → ((𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴)‘𝑖) = 𝐴) |
496 | 98 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 𝐴 ∈ ℂ) |
497 | 495, 496 | eqeltrd 2688 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → ((𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴)‘𝑖) ∈ ℂ) |
498 | | eqidd 2611 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → (𝑗 ∈ (ℤ≥‘𝑀) ↦ (1 / 𝑗)) = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (1 / 𝑗))) |
499 | | oveq2 6557 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑖 → (1 / 𝑗) = (1 / 𝑖)) |
500 | 499 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) ∧ 𝑗 = 𝑖) → (1 / 𝑗) = (1 / 𝑖)) |
501 | 3, 493 | sseldi 3566 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 𝑖 ∈ ℝ) |
502 | | 0red 9920 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 0 ∈
ℝ) |
503 | 62 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℝ) |
504 | 67 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 0 < 𝑀) |
505 | | eluzle 11576 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈
(ℤ≥‘𝑀) → 𝑀 ≤ 𝑖) |
506 | 505 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 𝑀 ≤ 𝑖) |
507 | 502, 503,
501, 504, 506 | ltletrd 10076 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 0 < 𝑖) |
508 | 507 | gt0ne0d 10471 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 𝑖 ≠ 0) |
509 | 501, 508 | rereccld 10731 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → (1 / 𝑖) ∈
ℝ) |
510 | 498, 500,
493, 509 | fvmptd 6197 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → ((𝑗 ∈ (ℤ≥‘𝑀) ↦ (1 / 𝑗))‘𝑖) = (1 / 𝑖)) |
511 | 501 | recnd 9947 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 𝑖 ∈ ℂ) |
512 | 511, 508 | reccld 10673 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → (1 / 𝑖) ∈
ℂ) |
513 | 510, 512 | eqeltrd 2688 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → ((𝑗 ∈ (ℤ≥‘𝑀) ↦ (1 / 𝑗))‘𝑖) ∈ ℂ) |
514 | 469 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → 𝑅 = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐴 + (1 / 𝑗)))) |
515 | 499 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑖 → (𝐴 + (1 / 𝑗)) = (𝐴 + (1 / 𝑖))) |
516 | 515 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) ∧ 𝑗 = 𝑖) → (𝐴 + (1 / 𝑗)) = (𝐴 + (1 / 𝑖))) |
517 | 494, 509 | readdcld 9948 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → (𝐴 + (1 / 𝑖)) ∈ ℝ) |
518 | 514, 516,
493, 517 | fvmptd 6197 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → (𝑅‘𝑖) = (𝐴 + (1 / 𝑖))) |
519 | 495, 510 | oveq12d 6567 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → (((𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴)‘𝑖) + ((𝑗 ∈ (ℤ≥‘𝑀) ↦ (1 / 𝑗))‘𝑖)) = (𝐴 + (1 / 𝑖))) |
520 | 518, 519 | eqtr4d 2647 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ (ℤ≥‘𝑀)) → (𝑅‘𝑖) = (((𝑗 ∈ (ℤ≥‘𝑀) ↦ 𝐴)‘𝑖) + ((𝑗 ∈ (ℤ≥‘𝑀) ↦ (1 / 𝑗))‘𝑖))) |
521 | 23, 142, 482, 485, 490, 497, 513, 520 | climadd 14210 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑅 ⇝ (𝐴 + 0)) |
522 | 98 | addid1d 10115 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 + 0) = 𝐴) |
523 | 521, 522 | breqtrd 4609 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑅 ⇝ 𝐴) |
524 | | releldm 5279 |
. . . . . . . . . . . 12
⊢ ((Rel
⇝ ∧ 𝑅 ⇝
𝐴) → 𝑅 ∈ dom ⇝ ) |
525 | 473, 523,
524 | syl2anc 691 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ dom ⇝ ) |
526 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑙 = 𝑘 → (ℤ≥‘𝑙) =
(ℤ≥‘𝑘)) |
527 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑙 = 𝑘 → (𝑅‘𝑙) = (𝑅‘𝑘)) |
528 | 527 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑙 = 𝑘 → ((𝑅‘ℎ) − (𝑅‘𝑙)) = ((𝑅‘ℎ) − (𝑅‘𝑘))) |
529 | 528 | fveq2d 6107 |
. . . . . . . . . . . . . . . 16
⊢ (𝑙 = 𝑘 → (abs‘((𝑅‘ℎ) − (𝑅‘𝑙))) = (abs‘((𝑅‘ℎ) − (𝑅‘𝑘)))) |
530 | 529 | breq1d 4593 |
. . . . . . . . . . . . . . 15
⊢ (𝑙 = 𝑘 → ((abs‘((𝑅‘ℎ) − (𝑅‘𝑙))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ↔
(abs‘((𝑅‘ℎ) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))) |
531 | 526, 530 | raleqbidv 3129 |
. . . . . . . . . . . . . 14
⊢ (𝑙 = 𝑘 → (∀ℎ ∈ (ℤ≥‘𝑙)(abs‘((𝑅‘ℎ) − (𝑅‘𝑙))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ↔
∀ℎ ∈
(ℤ≥‘𝑘)(abs‘((𝑅‘ℎ) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))) |
532 | 531 | cbvrabv 3172 |
. . . . . . . . . . . . 13
⊢ {𝑙 ∈
(ℤ≥‘𝑀) ∣ ∀ℎ ∈ (ℤ≥‘𝑙)(abs‘((𝑅‘ℎ) − (𝑅‘𝑙))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} = {𝑘 ∈
(ℤ≥‘𝑀) ∣ ∀ℎ ∈ (ℤ≥‘𝑘)(abs‘((𝑅‘ℎ) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} |
533 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = 𝑖 → (𝑅‘ℎ) = (𝑅‘𝑖)) |
534 | 533 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = 𝑖 → ((𝑅‘ℎ) − (𝑅‘𝑘)) = ((𝑅‘𝑖) − (𝑅‘𝑘))) |
535 | 534 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (ℎ = 𝑖 → (abs‘((𝑅‘ℎ) − (𝑅‘𝑘))) = (abs‘((𝑅‘𝑖) − (𝑅‘𝑘)))) |
536 | 535 | breq1d 4593 |
. . . . . . . . . . . . . . . 16
⊢ (ℎ = 𝑖 → ((abs‘((𝑅‘ℎ) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ↔
(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)))) |
537 | 536 | cbvralv 3147 |
. . . . . . . . . . . . . . 15
⊢
(∀ℎ ∈
(ℤ≥‘𝑘)(abs‘((𝑅‘ℎ) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ↔
∀𝑖 ∈
(ℤ≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))) |
538 | 537 | rgenw 2908 |
. . . . . . . . . . . . . 14
⊢
∀𝑘 ∈
(ℤ≥‘𝑀)(∀ℎ ∈ (ℤ≥‘𝑘)(abs‘((𝑅‘ℎ) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ↔
∀𝑖 ∈
(ℤ≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))) |
539 | | rabbi 3097 |
. . . . . . . . . . . . . 14
⊢
(∀𝑘 ∈
(ℤ≥‘𝑀)(∀ℎ ∈ (ℤ≥‘𝑘)(abs‘((𝑅‘ℎ) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1)) ↔
∀𝑖 ∈
(ℤ≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))) ↔ {𝑘 ∈
(ℤ≥‘𝑀) ∣ ∀ℎ ∈ (ℤ≥‘𝑘)(abs‘((𝑅‘ℎ) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} = {𝑘 ∈
(ℤ≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}) |
540 | 538, 539 | mpbi 219 |
. . . . . . . . . . . . 13
⊢ {𝑘 ∈
(ℤ≥‘𝑀) ∣ ∀ℎ ∈ (ℤ≥‘𝑘)(abs‘((𝑅‘ℎ) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} = {𝑘 ∈
(ℤ≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} |
541 | 532, 540 | eqtri 2632 |
. . . . . . . . . . . 12
⊢ {𝑙 ∈
(ℤ≥‘𝑀) ∣ ∀ℎ ∈ (ℤ≥‘𝑙)(abs‘((𝑅‘ℎ) − (𝑅‘𝑙))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} = {𝑘 ∈
(ℤ≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))} |
542 | 541 | infeq1i 8267 |
. . . . . . . . . . 11
⊢
inf({𝑙 ∈
(ℤ≥‘𝑀) ∣ ∀ℎ ∈ (ℤ≥‘𝑙)(abs‘((𝑅‘ℎ) − (𝑅‘𝑙))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}, ℝ, < )
= inf({𝑘 ∈
(ℤ≥‘𝑀) ∣ ∀𝑖 ∈ (ℤ≥‘𝑘)(abs‘((𝑅‘𝑖) − (𝑅‘𝑘))) < (𝑥 / (sup(ran (𝑧 ∈ (𝐴(,)𝐵) ↦ (abs‘((ℝ D 𝐹)‘𝑧))), ℝ, < ) + 1))}, ℝ, <
) |
543 | 7, 6, 9, 468, 109, 110, 22, 470, 471, 525, 542 | ioodvbdlimc1lem1 38821 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗))) ⇝ (lim sup‘(𝑗 ∈
(ℤ≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗))))) |
544 | 469 | fvmpt2 6200 |
. . . . . . . . . . . . . . 15
⊢ ((𝑗 ∈
(ℤ≥‘𝑀) ∧ (𝐴 + (1 / 𝑗)) ∈ ℝ) → (𝑅‘𝑗) = (𝐴 + (1 / 𝑗))) |
545 | 113, 58, 544 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝑅‘𝑗) = (𝐴 + (1 / 𝑗))) |
546 | 545 | eqcomd 2616 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐴 + (1 / 𝑗)) = (𝑅‘𝑗)) |
547 | 546 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ≥‘𝑀)) → (𝐹‘(𝐴 + (1 / 𝑗))) = (𝐹‘(𝑅‘𝑗))) |
548 | 547 | mpteq2dva 4672 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐹‘(𝐴 + (1 / 𝑗)))) = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗)))) |
549 | 107, 548 | syl5eq 2656 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 = (𝑗 ∈ (ℤ≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗)))) |
550 | 549 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝜑 → (lim sup‘𝑆) = (lim sup‘(𝑗 ∈
(ℤ≥‘𝑀) ↦ (𝐹‘(𝑅‘𝑗))))) |
551 | 543, 549,
550 | 3brtr4d 4615 |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 ⇝ (lim sup‘𝑆)) |
552 | 474 | mptex 6390 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈
(ℤ≥‘𝑀) ↦ (𝐹‘(𝐴 + (1 / 𝑗)))) ∈ V |
553 | 107, 552 | eqeltri 2684 |
. . . . . . . . . . 11
⊢ 𝑆 ∈ V |
554 | 553 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ∈ V) |
555 | | eqidd 2611 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ℤ) → (𝑆‘𝑐) = (𝑆‘𝑐)) |
556 | 554, 555 | clim 14073 |
. . . . . . . . 9
⊢ (𝜑 → (𝑆 ⇝ (lim sup‘𝑆) ↔ ((lim sup‘𝑆) ∈ ℂ ∧ ∀𝑎 ∈ ℝ+
∃𝑏 ∈ ℤ
∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < 𝑎)))) |
557 | 551, 556 | mpbid 221 |
. . . . . . . 8
⊢ (𝜑 → ((lim sup‘𝑆) ∈ ℂ ∧
∀𝑎 ∈
ℝ+ ∃𝑏 ∈ ℤ ∀𝑐 ∈ (ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < 𝑎))) |
558 | 557 | simprd 478 |
. . . . . . 7
⊢ (𝜑 → ∀𝑎 ∈ ℝ+ ∃𝑏 ∈ ℤ ∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < 𝑎)) |
559 | 558 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∀𝑎 ∈
ℝ+ ∃𝑏 ∈ ℤ ∀𝑐 ∈ (ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < 𝑎)) |
560 | | simpr 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → 𝑥 ∈
ℝ+) |
561 | 560 | rphalfcld 11760 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ∈
ℝ+) |
562 | | breq2 4587 |
. . . . . . . . 9
⊢ (𝑎 = (𝑥 / 2) → ((abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < 𝑎 ↔ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2))) |
563 | 562 | anbi2d 736 |
. . . . . . . 8
⊢ (𝑎 = (𝑥 / 2) → (((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < 𝑎) ↔ ((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2)))) |
564 | 563 | rexralbidv 3040 |
. . . . . . 7
⊢ (𝑎 = (𝑥 / 2) → (∃𝑏 ∈ ℤ ∀𝑐 ∈ (ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < 𝑎) ↔ ∃𝑏 ∈ ℤ ∀𝑐 ∈ (ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2)))) |
565 | 564 | rspccva 3281 |
. . . . . 6
⊢
((∀𝑎 ∈
ℝ+ ∃𝑏 ∈ ℤ ∀𝑐 ∈ (ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < 𝑎) ∧ (𝑥 / 2) ∈ ℝ+) →
∃𝑏 ∈ ℤ
∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2))) |
566 | 559, 561,
565 | syl2anc 691 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑏 ∈ ℤ
∀𝑐 ∈
(ℤ≥‘𝑏)((𝑆‘𝑐) ∈ ℂ ∧ (abs‘((𝑆‘𝑐) − (lim sup‘𝑆))) < (𝑥 / 2))) |
567 | 460, 566 | r19.29a 3060 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑗 ∈
(ℤ≥‘𝑁)(abs‘((𝑆‘𝑗) − (lim sup‘𝑆))) < (𝑥 / 2)) |
568 | 412, 567 | r19.29a 3060 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ+) →
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ (𝐴(,)𝐵)((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑦) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥)) |
569 | 568 | ralrimiva 2949 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ℝ+
∀𝑧 ∈ (𝐴(,)𝐵)((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑦) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥)) |
570 | | ioosscn 38563 |
. . . 4
⊢ (𝐴(,)𝐵) ⊆ ℂ |
571 | 570 | a1i 11 |
. . 3
⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ) |
572 | 463, 571,
98 | ellimc3 23449 |
. 2
⊢ (𝜑 → ((lim sup‘𝑆) ∈ (𝐹 limℂ 𝐴) ↔ ((lim sup‘𝑆) ∈ ℂ ∧ ∀𝑥 ∈ ℝ+
∃𝑦 ∈
ℝ+ ∀𝑧 ∈ (𝐴(,)𝐵)((𝑧 ≠ 𝐴 ∧ (abs‘(𝑧 − 𝐴)) < 𝑦) → (abs‘((𝐹‘𝑧) − (lim sup‘𝑆))) < 𝑥)))) |
573 | 137, 569,
572 | mpbir2and 959 |
1
⊢ (𝜑 → (lim sup‘𝑆) ∈ (𝐹 limℂ 𝐴)) |