Step | Hyp | Ref
| Expression |
1 | | ovollb2.5 |
. . . 4
⊢ (𝜑 → 𝐴 ⊆ ∪ ran
([,] ∘ 𝐹)) |
2 | | ovollb2.4 |
. . . . 5
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
3 | | ovolficcss 23045 |
. . . . 5
⊢ (𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ∪ ran ([,] ∘
𝐹) ⊆
ℝ) |
4 | 2, 3 | syl 17 |
. . . 4
⊢ (𝜑 → ∪ ran ([,] ∘ 𝐹) ⊆ ℝ) |
5 | 1, 4 | sstrd 3578 |
. . 3
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
6 | | ovolcl 23053 |
. . 3
⊢ (𝐴 ⊆ ℝ →
(vol*‘𝐴) ∈
ℝ*) |
7 | 5, 6 | syl 17 |
. 2
⊢ (𝜑 → (vol*‘𝐴) ∈
ℝ*) |
8 | | ovolfcl 23042 |
. . . . . . . . . . . . 13
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛)))) |
9 | 2, 8 | sylan 487 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛)))) |
10 | 9 | simp1d 1066 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐹‘𝑛)) ∈
ℝ) |
11 | | ovollb2.6 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐵 ∈
ℝ+) |
12 | 11 | rphalfcld 11760 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐵 / 2) ∈
ℝ+) |
13 | 12 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐵 / 2) ∈
ℝ+) |
14 | | 2nn 11062 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℕ |
15 | | nnnn0 11176 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℕ0) |
16 | 15 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0) |
17 | | nnexpcl 12735 |
. . . . . . . . . . . . . . 15
⊢ ((2
∈ ℕ ∧ 𝑛
∈ ℕ0) → (2↑𝑛) ∈ ℕ) |
18 | 14, 16, 17 | sylancr 694 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈
ℕ) |
19 | 18 | nnrpd 11746 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈
ℝ+) |
20 | 13, 19 | rpdivcld 11765 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈
ℝ+) |
21 | 20 | rpred 11748 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑛)) ∈ ℝ) |
22 | 10, 21 | resubcld 10337 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ) |
23 | 9 | simp2d 1067 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐹‘𝑛)) ∈
ℝ) |
24 | 23, 21 | readdcld 9948 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd
‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ) |
25 | 10, 20 | ltsubrpd 11780 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < (1st ‘(𝐹‘𝑛))) |
26 | 9 | simp3d 1068 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛))) |
27 | 23, 20 | ltaddrpd 11781 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐹‘𝑛)) < ((2nd
‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))) |
28 | 10, 23, 24, 26, 27 | lelttrd 10074 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐹‘𝑛)) < ((2nd
‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))) |
29 | 22, 10, 24, 25, 28 | lttrd 10077 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) < ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))) |
30 | 22, 24, 29 | ltled 10064 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))) |
31 | | df-br 4584 |
. . . . . . . . 9
⊢
(((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ≤ ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ↔ 〈((1st
‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))〉 ∈ ≤ ) |
32 | 30, 31 | sylib 207 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) →
〈((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))〉 ∈ ≤ ) |
33 | | opelxpi 5072 |
. . . . . . . . 9
⊢
((((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ ∧ ((2nd
‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))) ∈ ℝ) →
〈((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))〉 ∈ (ℝ ×
ℝ)) |
34 | 22, 24, 33 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) →
〈((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))〉 ∈ (ℝ ×
ℝ)) |
35 | 32, 34 | elind 3760 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) →
〈((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))〉 ∈ ( ≤ ∩ (ℝ
× ℝ))) |
36 | | ovollb2.2 |
. . . . . . 7
⊢ 𝐺 = (𝑛 ∈ ℕ ↦
〈((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))〉) |
37 | 35, 36 | fmptd 6292 |
. . . . . 6
⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
38 | | eqid 2610 |
. . . . . . 7
⊢ ((abs
∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺) |
39 | | ovollb2.3 |
. . . . . . 7
⊢ 𝑇 = seq1( + , ((abs ∘
− ) ∘ 𝐺)) |
40 | 38, 39 | ovolsf 23048 |
. . . . . 6
⊢ (𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞)) |
41 | 37, 40 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞)) |
42 | | frn 5966 |
. . . . 5
⊢ (𝑇:ℕ⟶(0[,)+∞)
→ ran 𝑇 ⊆
(0[,)+∞)) |
43 | 41, 42 | syl 17 |
. . . 4
⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞)) |
44 | | icossxr 12129 |
. . . 4
⊢
(0[,)+∞) ⊆ ℝ* |
45 | 43, 44 | syl6ss 3580 |
. . 3
⊢ (𝜑 → ran 𝑇 ⊆
ℝ*) |
46 | | supxrcl 12017 |
. . 3
⊢ (ran
𝑇 ⊆
ℝ* → sup(ran 𝑇, ℝ*, < ) ∈
ℝ*) |
47 | 45, 46 | syl 17 |
. 2
⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ∈
ℝ*) |
48 | | ovollb2.7 |
. . . 4
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈
ℝ) |
49 | 11 | rpred 11748 |
. . . 4
⊢ (𝜑 → 𝐵 ∈ ℝ) |
50 | 48, 49 | readdcld 9948 |
. . 3
⊢ (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈
ℝ) |
51 | 50 | rexrd 9968 |
. 2
⊢ (𝜑 → (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈
ℝ*) |
52 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚)) |
53 | 52 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑚 → (1st ‘(𝐹‘𝑛)) = (1st ‘(𝐹‘𝑚))) |
54 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑚 → (2↑𝑛) = (2↑𝑚)) |
55 | 54 | oveq2d 6565 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑚 → ((𝐵 / 2) / (2↑𝑛)) = ((𝐵 / 2) / (2↑𝑚))) |
56 | 53, 55 | oveq12d 6567 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑚 → ((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))) = ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) |
57 | 52 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑚 → (2nd ‘(𝐹‘𝑛)) = (2nd ‘(𝐹‘𝑚))) |
58 | 57, 55 | oveq12d 6567 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑚 → ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛))) = ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))) |
59 | 56, 58 | opeq12d 4348 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → 〈((1st ‘(𝐹‘𝑛)) − ((𝐵 / 2) / (2↑𝑛))), ((2nd ‘(𝐹‘𝑛)) + ((𝐵 / 2) / (2↑𝑛)))〉 = 〈((1st
‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))〉) |
60 | | opex 4859 |
. . . . . . . . . . . . . . 15
⊢
〈((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))〉 ∈ V |
61 | 59, 36, 60 | fvmpt 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑚 ∈ ℕ → (𝐺‘𝑚) = 〈((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))〉) |
62 | 61 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) = 〈((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))〉) |
63 | 62 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st
‘(𝐺‘𝑚)) = (1st
‘〈((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))〉)) |
64 | | ovex 6577 |
. . . . . . . . . . . . 13
⊢
((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ V |
65 | | ovex 6577 |
. . . . . . . . . . . . 13
⊢
((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚))) ∈ V |
66 | 64, 65 | op1st 7067 |
. . . . . . . . . . . 12
⊢
(1st ‘〈((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))〉) = ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))) |
67 | 63, 66 | syl6eq 2660 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st
‘(𝐺‘𝑚)) = ((1st
‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) |
68 | | ovolfcl 23042 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st
‘(𝐹‘𝑚)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑚)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑚)) ≤ (2nd
‘(𝐹‘𝑚)))) |
69 | 2, 68 | sylan 487 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st
‘(𝐹‘𝑚)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑚)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑚)) ≤ (2nd
‘(𝐹‘𝑚)))) |
70 | 69 | simp1d 1066 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st
‘(𝐹‘𝑚)) ∈
ℝ) |
71 | 12 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 / 2) ∈
ℝ+) |
72 | | nnnn0 11176 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℕ0) |
73 | 72 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ0) |
74 | | nnexpcl 12735 |
. . . . . . . . . . . . . . 15
⊢ ((2
∈ ℕ ∧ 𝑚
∈ ℕ0) → (2↑𝑚) ∈ ℕ) |
75 | 14, 73, 74 | sylancr 694 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑𝑚) ∈
ℕ) |
76 | 75 | nnrpd 11746 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑𝑚) ∈
ℝ+) |
77 | 71, 76 | rpdivcld 11765 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈
ℝ+) |
78 | 70, 77 | ltsubrpd 11780 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st
‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))) < (1st ‘(𝐹‘𝑚))) |
79 | 67, 78 | eqbrtrd 4605 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st
‘(𝐺‘𝑚)) < (1st
‘(𝐹‘𝑚))) |
80 | 79 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (1st
‘(𝐺‘𝑚)) < (1st
‘(𝐹‘𝑚))) |
81 | | ovolfcl 23042 |
. . . . . . . . . . . . 13
⊢ ((𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → ((1st
‘(𝐺‘𝑚)) ∈ ℝ ∧
(2nd ‘(𝐺‘𝑚)) ∈ ℝ ∧ (1st
‘(𝐺‘𝑚)) ≤ (2nd
‘(𝐺‘𝑚)))) |
82 | 37, 81 | sylan 487 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st
‘(𝐺‘𝑚)) ∈ ℝ ∧
(2nd ‘(𝐺‘𝑚)) ∈ ℝ ∧ (1st
‘(𝐺‘𝑚)) ≤ (2nd
‘(𝐺‘𝑚)))) |
83 | 82 | simp1d 1066 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st
‘(𝐺‘𝑚)) ∈
ℝ) |
84 | 83 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (1st
‘(𝐺‘𝑚)) ∈
ℝ) |
85 | 70 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (1st
‘(𝐹‘𝑚)) ∈
ℝ) |
86 | 5 | sselda 3568 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ ℝ) |
87 | 86 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → 𝑧 ∈ ℝ) |
88 | | ltletr 10008 |
. . . . . . . . . 10
⊢
(((1st ‘(𝐺‘𝑚)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑚)) ∈ ℝ ∧ 𝑧 ∈ ℝ) →
(((1st ‘(𝐺‘𝑚)) < (1st ‘(𝐹‘𝑚)) ∧ (1st ‘(𝐹‘𝑚)) ≤ 𝑧) → (1st ‘(𝐺‘𝑚)) < 𝑧)) |
89 | 84, 85, 87, 88 | syl3anc 1318 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (((1st
‘(𝐺‘𝑚)) < (1st
‘(𝐹‘𝑚)) ∧ (1st
‘(𝐹‘𝑚)) ≤ 𝑧) → (1st ‘(𝐺‘𝑚)) < 𝑧)) |
90 | 80, 89 | mpand 707 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → ((1st
‘(𝐹‘𝑚)) ≤ 𝑧 → (1st ‘(𝐺‘𝑚)) < 𝑧)) |
91 | 69 | simp2d 1067 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd
‘(𝐹‘𝑚)) ∈
ℝ) |
92 | 91, 77 | ltaddrpd 11781 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd
‘(𝐹‘𝑚)) < ((2nd
‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))) |
93 | 62 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd
‘(𝐺‘𝑚)) = (2nd
‘〈((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))〉)) |
94 | 64, 65 | op2nd 7068 |
. . . . . . . . . . . 12
⊢
(2nd ‘〈((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))), ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))〉) = ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚))) |
95 | 93, 94 | syl6eq 2660 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd
‘(𝐺‘𝑚)) = ((2nd
‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚)))) |
96 | 92, 95 | breqtrrd 4611 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd
‘(𝐹‘𝑚)) < (2nd
‘(𝐺‘𝑚))) |
97 | 96 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (2nd
‘(𝐹‘𝑚)) < (2nd
‘(𝐺‘𝑚))) |
98 | 91 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (2nd
‘(𝐹‘𝑚)) ∈
ℝ) |
99 | 82 | simp2d 1067 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd
‘(𝐺‘𝑚)) ∈
ℝ) |
100 | 99 | adantlr 747 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (2nd
‘(𝐺‘𝑚)) ∈
ℝ) |
101 | | lelttr 10007 |
. . . . . . . . . 10
⊢ ((𝑧 ∈ ℝ ∧
(2nd ‘(𝐹‘𝑚)) ∈ ℝ ∧ (2nd
‘(𝐺‘𝑚)) ∈ ℝ) →
((𝑧 ≤ (2nd
‘(𝐹‘𝑚)) ∧ (2nd
‘(𝐹‘𝑚)) < (2nd
‘(𝐺‘𝑚))) → 𝑧 < (2nd ‘(𝐺‘𝑚)))) |
102 | 87, 98, 100, 101 | syl3anc 1318 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → ((𝑧 ≤ (2nd ‘(𝐹‘𝑚)) ∧ (2nd ‘(𝐹‘𝑚)) < (2nd ‘(𝐺‘𝑚))) → 𝑧 < (2nd ‘(𝐺‘𝑚)))) |
103 | 97, 102 | mpan2d 706 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (𝑧 ≤ (2nd ‘(𝐹‘𝑚)) → 𝑧 < (2nd ‘(𝐺‘𝑚)))) |
104 | 90, 103 | anim12d 584 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑚 ∈ ℕ) → (((1st
‘(𝐹‘𝑚)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑚))) → ((1st ‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))))) |
105 | 104 | reximdva 3000 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (∃𝑚 ∈ ℕ ((1st
‘(𝐹‘𝑚)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑚))) → ∃𝑚 ∈ ℕ ((1st
‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))))) |
106 | 105 | ralimdva 2945 |
. . . . 5
⊢ (𝜑 → (∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st
‘(𝐹‘𝑚)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑚))) → ∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st
‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))))) |
107 | | ovolficc 23044 |
. . . . . 6
⊢ ((𝐴 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ))) → (𝐴 ⊆ ∪ ran
([,] ∘ 𝐹) ↔
∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st
‘(𝐹‘𝑚)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑚))))) |
108 | 5, 2, 107 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → (𝐴 ⊆ ∪ ran
([,] ∘ 𝐹) ↔
∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st
‘(𝐹‘𝑚)) ≤ 𝑧 ∧ 𝑧 ≤ (2nd ‘(𝐹‘𝑚))))) |
109 | | ovolfioo 23043 |
. . . . . 6
⊢ ((𝐴 ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ))) → (𝐴 ⊆ ∪ ran
((,) ∘ 𝐺) ↔
∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st
‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))))) |
110 | 5, 37, 109 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → (𝐴 ⊆ ∪ ran
((,) ∘ 𝐺) ↔
∀𝑧 ∈ 𝐴 ∃𝑚 ∈ ℕ ((1st
‘(𝐺‘𝑚)) < 𝑧 ∧ 𝑧 < (2nd ‘(𝐺‘𝑚))))) |
111 | 106, 108,
110 | 3imtr4d 282 |
. . . 4
⊢ (𝜑 → (𝐴 ⊆ ∪ ran
([,] ∘ 𝐹) →
𝐴 ⊆ ∪ ran ((,) ∘ 𝐺))) |
112 | 1, 111 | mpd 15 |
. . 3
⊢ (𝜑 → 𝐴 ⊆ ∪ ran
((,) ∘ 𝐺)) |
113 | 39 | ovollb 23054 |
. . 3
⊢ ((𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝐴 ⊆ ∪ ran
((,) ∘ 𝐺)) →
(vol*‘𝐴) ≤ sup(ran
𝑇, ℝ*,
< )) |
114 | 37, 112, 113 | syl2anc 691 |
. 2
⊢ (𝜑 → (vol*‘𝐴) ≤ sup(ran 𝑇, ℝ*, <
)) |
115 | 39 | fveq1i 6104 |
. . . . . . 7
⊢ (𝑇‘𝑘) = (seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑘) |
116 | | fzfid 12634 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1...𝑘) ∈ Fin) |
117 | | rge0ssre 12151 |
. . . . . . . . . . 11
⊢
(0[,)+∞) ⊆ ℝ |
118 | | eqid 2610 |
. . . . . . . . . . . . . . 15
⊢ ((abs
∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) |
119 | 118 | ovolfsf 23047 |
. . . . . . . . . . . . . 14
⊢ (𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞)) |
120 | 2, 119 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((abs ∘ − )
∘ 𝐹):ℕ⟶(0[,)+∞)) |
121 | 120 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((abs ∘ −
) ∘ 𝐹):ℕ⟶(0[,)+∞)) |
122 | | elfznn 12241 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ (1...𝑘) → 𝑚 ∈ ℕ) |
123 | | ffvelrn 6265 |
. . . . . . . . . . . 12
⊢ ((((abs
∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞) ∧ 𝑚 ∈ ℕ) → (((abs
∘ − ) ∘ 𝐹)‘𝑚) ∈ (0[,)+∞)) |
124 | 121, 122,
123 | syl2an 493 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘
𝐹)‘𝑚) ∈ (0[,)+∞)) |
125 | 117, 124 | sseldi 3566 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘
𝐹)‘𝑚) ∈ ℝ) |
126 | 125 | recnd 9947 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘
𝐹)‘𝑚) ∈ ℂ) |
127 | 11 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈
ℝ+) |
128 | 127, 76 | rpdivcld 11765 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈
ℝ+) |
129 | 128 | rpcnd 11750 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 / (2↑𝑚)) ∈ ℂ) |
130 | 122, 129 | sylan2 490 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ) |
131 | 130 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (𝐵 / (2↑𝑚)) ∈ ℂ) |
132 | 116, 126,
131 | fsumadd 14317 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)))) |
133 | 38 | ovolfsval 23046 |
. . . . . . . . . . . . 13
⊢ ((𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺‘𝑚)) − (1st ‘(𝐺‘𝑚)))) |
134 | 37, 133 | sylan 487 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑚) = ((2nd ‘(𝐺‘𝑚)) − (1st ‘(𝐺‘𝑚)))) |
135 | 91 | recnd 9947 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2nd
‘(𝐹‘𝑚)) ∈
ℂ) |
136 | 77 | rpcnd 11750 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐵 / 2) / (2↑𝑚)) ∈ ℂ) |
137 | 70 | recnd 9947 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1st
‘(𝐹‘𝑚)) ∈
ℂ) |
138 | 137, 136 | subcld 10271 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((1st
‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))) ∈ ℂ) |
139 | 135, 136,
138 | addsubassd 10291 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((2nd
‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((2nd ‘(𝐹‘𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))))) |
140 | 95, 67 | oveq12d 6567 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2nd
‘(𝐺‘𝑚)) − (1st
‘(𝐺‘𝑚))) = (((2nd
‘(𝐹‘𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))))) |
141 | 135, 137,
129 | subadd23d 10293 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((2nd
‘(𝐹‘𝑚)) − (1st
‘(𝐹‘𝑚))) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹‘𝑚))))) |
142 | 118 | ovolfsval 23046 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹‘𝑚)) − (1st ‘(𝐹‘𝑚)))) |
143 | 2, 142 | sylan 487 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑚) = ((2nd ‘(𝐹‘𝑚)) − (1st ‘(𝐹‘𝑚)))) |
144 | 143 | oveq1d 6564 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (((2nd ‘(𝐹‘𝑚)) − (1st ‘(𝐹‘𝑚))) + (𝐵 / (2↑𝑚)))) |
145 | 136, 137,
136 | subsub3d 10301 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹‘𝑚)))) |
146 | 71 | rpcnd 11750 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐵 / 2) ∈ ℂ) |
147 | 75 | nncnd 10913 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑𝑚) ∈
ℂ) |
148 | 75 | nnne0d 10942 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (2↑𝑚) ≠ 0) |
149 | 146, 146,
147, 148 | divdird 10718 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚)))) |
150 | 127 | rpcnd 11750 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐵 ∈ ℂ) |
151 | 150 | 2halvesd 11155 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐵 / 2) + (𝐵 / 2)) = 𝐵) |
152 | 151 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐵 / 2) + (𝐵 / 2)) / (2↑𝑚)) = (𝐵 / (2↑𝑚))) |
153 | 149, 152 | eqtr3d 2646 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) = (𝐵 / (2↑𝑚))) |
154 | 153 | oveq1d 6564 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((𝐵 / 2) / (2↑𝑚)) + ((𝐵 / 2) / (2↑𝑚))) − (1st ‘(𝐹‘𝑚))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹‘𝑚)))) |
155 | 145, 154 | eqtrd 2644 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))) = ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹‘𝑚)))) |
156 | 155 | oveq2d 6565 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2nd
‘(𝐹‘𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚))))) = ((2nd ‘(𝐹‘𝑚)) + ((𝐵 / (2↑𝑚)) − (1st ‘(𝐹‘𝑚))))) |
157 | 141, 144,
156 | 3eqtr4d 2654 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = ((2nd ‘(𝐹‘𝑚)) + (((𝐵 / 2) / (2↑𝑚)) − ((1st ‘(𝐹‘𝑚)) − ((𝐵 / 2) / (2↑𝑚)))))) |
158 | 139, 140,
157 | 3eqtr4d 2654 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((2nd
‘(𝐺‘𝑚)) − (1st
‘(𝐺‘𝑚))) = ((((abs ∘ − )
∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚)))) |
159 | 134, 158 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚)))) |
160 | 122, 159 | sylan2 490 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘
𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚)))) |
161 | 160 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘
𝐺)‘𝑚) = ((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚)))) |
162 | | simpr 476 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) |
163 | | nnuz 11599 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
164 | 162, 163 | syl6eleq 2698 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈
(ℤ≥‘1)) |
165 | 126, 131 | addcld 9938 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → ((((abs ∘ − ) ∘
𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) ∈ ℂ) |
166 | 161, 164,
165 | fsumser 14308 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)((((abs ∘ − ) ∘ 𝐹)‘𝑚) + (𝐵 / (2↑𝑚))) = (seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑘)) |
167 | | eqidd 2611 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑘)) → (((abs ∘ − ) ∘
𝐹)‘𝑚) = (((abs ∘ − ) ∘ 𝐹)‘𝑚)) |
168 | 167, 164,
126 | fsumser 14308 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑘)) |
169 | | ovollb2.1 |
. . . . . . . . . . 11
⊢ 𝑆 = seq1( + , ((abs ∘
− ) ∘ 𝐹)) |
170 | 169 | fveq1i 6104 |
. . . . . . . . . 10
⊢ (𝑆‘𝑘) = (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑘) |
171 | 168, 170 | syl6eqr 2662 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) = (𝑆‘𝑘)) |
172 | 11 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈
ℝ+) |
173 | 172 | rpcnd 11750 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℂ) |
174 | | geo2sum 14443 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℕ ∧ 𝐵 ∈ ℂ) →
Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘)))) |
175 | 162, 173,
174 | syl2anc 691 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚)) = (𝐵 − (𝐵 / (2↑𝑘)))) |
176 | 171, 175 | oveq12d 6567 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (Σ𝑚 ∈ (1...𝑘)(((abs ∘ − ) ∘ 𝐹)‘𝑚) + Σ𝑚 ∈ (1...𝑘)(𝐵 / (2↑𝑚))) = ((𝑆‘𝑘) + (𝐵 − (𝐵 / (2↑𝑘))))) |
177 | 132, 166,
176 | 3eqtr3d 2652 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐺))‘𝑘) = ((𝑆‘𝑘) + (𝐵 − (𝐵 / (2↑𝑘))))) |
178 | 115, 177 | syl5eq 2656 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑇‘𝑘) = ((𝑆‘𝑘) + (𝐵 − (𝐵 / (2↑𝑘))))) |
179 | 118, 169 | ovolsf 23048 |
. . . . . . . . . 10
⊢ (𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
180 | 2, 179 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞)) |
181 | 180 | ffvelrnda 6267 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ (0[,)+∞)) |
182 | 117, 181 | sseldi 3566 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ℝ) |
183 | 172 | rpred 11748 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐵 ∈ ℝ) |
184 | | nnnn0 11176 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℕ0) |
185 | 184 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ0) |
186 | | nnexpcl 12735 |
. . . . . . . . . . . 12
⊢ ((2
∈ ℕ ∧ 𝑘
∈ ℕ0) → (2↑𝑘) ∈ ℕ) |
187 | 14, 185, 186 | sylancr 694 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2↑𝑘) ∈
ℕ) |
188 | 187 | nnrpd 11746 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (2↑𝑘) ∈
ℝ+) |
189 | 172, 188 | rpdivcld 11765 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈
ℝ+) |
190 | 189 | rpred 11748 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵 / (2↑𝑘)) ∈ ℝ) |
191 | 183, 190 | resubcld 10337 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ∈ ℝ) |
192 | 48 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → sup(ran 𝑆, ℝ*, < )
∈ ℝ) |
193 | | frn 5966 |
. . . . . . . . . . 11
⊢ (𝑆:ℕ⟶(0[,)+∞)
→ ran 𝑆 ⊆
(0[,)+∞)) |
194 | 180, 193 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞)) |
195 | 194, 44 | syl6ss 3580 |
. . . . . . . . 9
⊢ (𝜑 → ran 𝑆 ⊆
ℝ*) |
196 | 195 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran 𝑆 ⊆
ℝ*) |
197 | | ffn 5958 |
. . . . . . . . . 10
⊢ (𝑆:ℕ⟶(0[,)+∞)
→ 𝑆 Fn
ℕ) |
198 | 180, 197 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑆 Fn ℕ) |
199 | | fnfvelrn 6264 |
. . . . . . . . 9
⊢ ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ran 𝑆) |
200 | 198, 199 | sylan 487 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ran 𝑆) |
201 | | supxrub 12026 |
. . . . . . . 8
⊢ ((ran
𝑆 ⊆
ℝ* ∧ (𝑆‘𝑘) ∈ ran 𝑆) → (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) |
202 | 196, 200,
201 | syl2anc 691 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) |
203 | 183, 189 | ltsubrpd 11780 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) < 𝐵) |
204 | 191, 183,
203 | ltled 10064 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐵 − (𝐵 / (2↑𝑘))) ≤ 𝐵) |
205 | 182, 191,
192, 183, 202, 204 | le2addd 10525 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑆‘𝑘) + (𝐵 − (𝐵 / (2↑𝑘)))) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)) |
206 | 178, 205 | eqbrtrd 4605 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑇‘𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)) |
207 | 206 | ralrimiva 2949 |
. . . 4
⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)) |
208 | | ffn 5958 |
. . . . 5
⊢ (𝑇:ℕ⟶(0[,)+∞)
→ 𝑇 Fn
ℕ) |
209 | | breq1 4586 |
. . . . . 6
⊢ (𝑦 = (𝑇‘𝑘) → (𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ (𝑇‘𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))) |
210 | 209 | ralrn 6270 |
. . . . 5
⊢ (𝑇 Fn ℕ →
(∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))) |
211 | 41, 208, 210 | 3syl 18 |
. . . 4
⊢ (𝜑 → (∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑘 ∈ ℕ (𝑇‘𝑘) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))) |
212 | 207, 211 | mpbird 246 |
. . 3
⊢ (𝜑 → ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵)) |
213 | | supxrleub 12028 |
. . . 4
⊢ ((ran
𝑇 ⊆
ℝ* ∧ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ∈ ℝ*)
→ (sup(ran 𝑇,
ℝ*, < ) ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵) ↔ ∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))) |
214 | 45, 51, 213 | syl2anc 691 |
. . 3
⊢ (𝜑 → (sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran
𝑆, ℝ*,
< ) + 𝐵) ↔
∀𝑦 ∈ ran 𝑇 𝑦 ≤ (sup(ran 𝑆, ℝ*, < ) + 𝐵))) |
215 | 212, 214 | mpbird 246 |
. 2
⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤ (sup(ran
𝑆, ℝ*,
< ) + 𝐵)) |
216 | 7, 47, 51, 114, 215 | xrletrd 11869 |
1
⊢ (𝜑 → (vol*‘𝐴) ≤ (sup(ran 𝑆, ℝ*, < ) +
𝐵)) |