Step | Hyp | Ref
| Expression |
1 | | inss1 4162 |
. . . 4
⊢ (𝐸 ∩ 𝐵) ⊆ 𝐸 |
2 | | ioombl1.e |
. . . 4
⊢ (𝜑 → 𝐸 ⊆ ℝ) |
3 | | ioombl1.v |
. . . 4
⊢ (𝜑 → (vol*‘𝐸) ∈
ℝ) |
4 | | ovolsscl 24650 |
. . . 4
⊢ (((𝐸 ∩ 𝐵) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∩ 𝐵)) ∈
ℝ) |
5 | 1, 2, 3, 4 | mp3an2i 1465 |
. . 3
⊢ (𝜑 → (vol*‘(𝐸 ∩ 𝐵)) ∈ ℝ) |
6 | | difss 4066 |
. . . 4
⊢ (𝐸 ∖ 𝐵) ⊆ 𝐸 |
7 | | ovolsscl 24650 |
. . . 4
⊢ (((𝐸 ∖ 𝐵) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∖
𝐵)) ∈
ℝ) |
8 | 6, 2, 3, 7 | mp3an2i 1465 |
. . 3
⊢ (𝜑 → (vol*‘(𝐸 ∖ 𝐵)) ∈ ℝ) |
9 | 5, 8 | readdcld 11004 |
. 2
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ∈ ℝ) |
10 | | ioombl1.b |
. . 3
⊢ 𝐵 = (𝐴(,)+∞) |
11 | | ioombl1.a |
. . 3
⊢ (𝜑 → 𝐴 ∈ ℝ) |
12 | | ioombl1.c |
. . 3
⊢ (𝜑 → 𝐶 ∈
ℝ+) |
13 | | ioombl1.s |
. . 3
⊢ 𝑆 = seq1( + , ((abs ∘
− ) ∘ 𝐹)) |
14 | | ioombl1.t |
. . 3
⊢ 𝑇 = seq1( + , ((abs ∘
− ) ∘ 𝐺)) |
15 | | ioombl1.u |
. . 3
⊢ 𝑈 = seq1( + , ((abs ∘
− ) ∘ 𝐻)) |
16 | | ioombl1.f1 |
. . 3
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
17 | | ioombl1.f2 |
. . 3
⊢ (𝜑 → 𝐸 ⊆ ∪ ran
((,) ∘ 𝐹)) |
18 | | ioombl1.f3 |
. . 3
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤
((vol*‘𝐸) + 𝐶)) |
19 | | ioombl1.p |
. . 3
⊢ 𝑃 = (1st ‘(𝐹‘𝑛)) |
20 | | ioombl1.q |
. . 3
⊢ 𝑄 = (2nd ‘(𝐹‘𝑛)) |
21 | | ioombl1.g |
. . 3
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ 〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) |
22 | | ioombl1.h |
. . 3
⊢ 𝐻 = (𝑛 ∈ ℕ ↦ 〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) |
23 | 10, 11, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 | ioombl1lem2 24723 |
. 2
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ∈
ℝ) |
24 | 12 | rpred 12772 |
. . 3
⊢ (𝜑 → 𝐶 ∈ ℝ) |
25 | 3, 24 | readdcld 11004 |
. 2
⊢ (𝜑 → ((vol*‘𝐸) + 𝐶) ∈ ℝ) |
26 | 10, 11, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 | ioombl1lem1 24722 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ
× ℝ)))) |
27 | 26 | simpld 495 |
. . . . . . . 8
⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
28 | | eqid 2738 |
. . . . . . . . 9
⊢ ((abs
∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺) |
29 | 28, 14 | ovolsf 24636 |
. . . . . . . 8
⊢ (𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑇:ℕ⟶(0[,)+∞)) |
30 | 27, 29 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑇:ℕ⟶(0[,)+∞)) |
31 | 30 | frnd 6608 |
. . . . . 6
⊢ (𝜑 → ran 𝑇 ⊆ (0[,)+∞)) |
32 | | rge0ssre 13188 |
. . . . . 6
⊢
(0[,)+∞) ⊆ ℝ |
33 | 31, 32 | sstrdi 3933 |
. . . . 5
⊢ (𝜑 → ran 𝑇 ⊆ ℝ) |
34 | | 1nn 11984 |
. . . . . . . 8
⊢ 1 ∈
ℕ |
35 | 30 | fdmd 6611 |
. . . . . . . 8
⊢ (𝜑 → dom 𝑇 = ℕ) |
36 | 34, 35 | eleqtrrid 2846 |
. . . . . . 7
⊢ (𝜑 → 1 ∈ dom 𝑇) |
37 | 36 | ne0d 4269 |
. . . . . 6
⊢ (𝜑 → dom 𝑇 ≠ ∅) |
38 | | dm0rn0 5834 |
. . . . . . 7
⊢ (dom
𝑇 = ∅ ↔ ran
𝑇 =
∅) |
39 | 38 | necon3bii 2996 |
. . . . . 6
⊢ (dom
𝑇 ≠ ∅ ↔ ran
𝑇 ≠
∅) |
40 | 37, 39 | sylib 217 |
. . . . 5
⊢ (𝜑 → ran 𝑇 ≠ ∅) |
41 | 30 | ffvelrnda 6961 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ∈ (0[,)+∞)) |
42 | 32, 41 | sselid 3919 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ∈ ℝ) |
43 | | eqid 2738 |
. . . . . . . . . . . . 13
⊢ ((abs
∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) |
44 | 43, 13 | ovolsf 24636 |
. . . . . . . . . . . 12
⊢ (𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑆:ℕ⟶(0[,)+∞)) |
45 | 16, 44 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑆:ℕ⟶(0[,)+∞)) |
46 | 45 | ffvelrnda 6961 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) ∈ (0[,)+∞)) |
47 | 32, 46 | sselid 3919 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) ∈ ℝ) |
48 | 23 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → sup(ran 𝑆, ℝ*, < )
∈ ℝ) |
49 | | simpr 485 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) |
50 | | nnuz 12621 |
. . . . . . . . . . . 12
⊢ ℕ =
(ℤ≥‘1) |
51 | 49, 50 | eleqtrdi 2849 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈
(ℤ≥‘1)) |
52 | | simpl 483 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝜑) |
53 | | elfznn 13285 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ (1...𝑗) → 𝑛 ∈ ℕ) |
54 | 28 | ovolfsf 24635 |
. . . . . . . . . . . . . . 15
⊢ (𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐺):ℕ⟶(0[,)+∞)) |
55 | 27, 54 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((abs ∘ − )
∘ 𝐺):ℕ⟶(0[,)+∞)) |
56 | 55 | ffvelrnda 6961 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞)) |
57 | 32, 56 | sselid 3919 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ ℝ) |
58 | 52, 53, 57 | syl2an 596 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐺)‘𝑛) ∈ ℝ) |
59 | 43 | ovolfsf 24635 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐹):ℕ⟶(0[,)+∞)) |
60 | 16, 59 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((abs ∘ − )
∘ 𝐹):ℕ⟶(0[,)+∞)) |
61 | 60 | ffvelrnda 6961 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞)) |
62 | | elrege0 13186 |
. . . . . . . . . . . . . 14
⊢ ((((abs
∘ − ) ∘ 𝐹)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs
∘ − ) ∘ 𝐹)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛))) |
63 | 61, 62 | sylib 217 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛))) |
64 | 63 | simpld 495 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ ℝ) |
65 | 52, 53, 64 | syl2an 596 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐹)‘𝑛) ∈ ℝ) |
66 | 26 | simprd 496 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
67 | | eqid 2738 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((abs
∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻) |
68 | 67 | ovolfsf 24635 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐻:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → ((abs ∘ − ) ∘ 𝐻):ℕ⟶(0[,)+∞)) |
69 | 66, 68 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((abs ∘ − )
∘ 𝐻):ℕ⟶(0[,)+∞)) |
70 | 69 | ffvelrnda 6961 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞)) |
71 | | elrege0 13186 |
. . . . . . . . . . . . . . . 16
⊢ ((((abs
∘ − ) ∘ 𝐻)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs
∘ − ) ∘ 𝐻)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐻)‘𝑛))) |
72 | 70, 71 | sylib 217 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐻)‘𝑛))) |
73 | 72 | simprd 496 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐻)‘𝑛)) |
74 | 72 | simpld 495 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ ℝ) |
75 | 57, 74 | addge01d 11563 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ (((abs
∘ − ) ∘ 𝐻)‘𝑛) ↔ (((abs ∘ − ) ∘
𝐺)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))) |
76 | 73, 75 | mpbid 231 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) |
77 | 10, 11, 2, 3, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 | ioombl1lem3 24724 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
78 | 76, 77 | breqtrd 5100 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
79 | 52, 53, 78 | syl2an 596 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐺)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
80 | 51, 58, 65, 79 | serle 13778 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐺))‘𝑗) ≤ (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑗)) |
81 | 14 | fveq1i 6775 |
. . . . . . . . . 10
⊢ (𝑇‘𝑗) = (seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑗) |
82 | 13 | fveq1i 6775 |
. . . . . . . . . 10
⊢ (𝑆‘𝑗) = (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑗) |
83 | 80, 81, 82 | 3brtr4g 5108 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ≤ (𝑆‘𝑗)) |
84 | | 1zzd 12351 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 1 ∈
ℤ) |
85 | | eqidd 2739 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) = (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
86 | 63 | simprd 496 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛)) |
87 | 45 | frnd 6608 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ran 𝑆 ⊆ (0[,)+∞)) |
88 | | icossxr 13164 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(0[,)+∞) ⊆ ℝ* |
89 | 87, 88 | sstrdi 3933 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ran 𝑆 ⊆
ℝ*) |
90 | 89 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ran 𝑆 ⊆
ℝ*) |
91 | 45 | ffnd 6601 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑆 Fn ℕ) |
92 | | fnfvelrn 6958 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑆 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ran 𝑆) |
93 | 91, 92 | sylan 580 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ∈ ran 𝑆) |
94 | | supxrub 13058 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ran
𝑆 ⊆
ℝ* ∧ (𝑆‘𝑘) ∈ ran 𝑆) → (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) |
95 | 90, 93, 94 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) |
96 | 95 | ralrimiva 3103 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, <
)) |
97 | | brralrspcev 5134 |
. . . . . . . . . . . . . . . 16
⊢ ((sup(ran
𝑆, ℝ*,
< ) ∈ ℝ ∧ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ sup(ran 𝑆, ℝ*, < )) →
∃𝑥 ∈ ℝ
∀𝑘 ∈ ℕ
(𝑆‘𝑘) ≤ 𝑥) |
98 | 23, 96, 97 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥) |
99 | 50, 13, 84, 85, 64, 86, 98 | isumsup2 15558 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝑆 ⇝ sup(ran 𝑆, ℝ, < )) |
100 | 87, 32 | sstrdi 3933 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ran 𝑆 ⊆ ℝ) |
101 | 45 | fdmd 6611 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → dom 𝑆 = ℕ) |
102 | 34, 101 | eleqtrrid 2846 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 1 ∈ dom 𝑆) |
103 | 102 | ne0d 4269 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → dom 𝑆 ≠ ∅) |
104 | | dm0rn0 5834 |
. . . . . . . . . . . . . . . . 17
⊢ (dom
𝑆 = ∅ ↔ ran
𝑆 =
∅) |
105 | 104 | necon3bii 2996 |
. . . . . . . . . . . . . . . 16
⊢ (dom
𝑆 ≠ ∅ ↔ ran
𝑆 ≠
∅) |
106 | 103, 105 | sylib 217 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ran 𝑆 ≠ ∅) |
107 | | breq1 5077 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = (𝑆‘𝑘) → (𝑧 ≤ 𝑥 ↔ (𝑆‘𝑘) ≤ 𝑥)) |
108 | 107 | ralrn 6964 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑆 Fn ℕ →
(∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)) |
109 | 91, 108 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)) |
110 | 109 | rexbidv 3226 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑘 ∈ ℕ (𝑆‘𝑘) ≤ 𝑥)) |
111 | 98, 110 | mpbird 256 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥) |
112 | | supxrre 13061 |
. . . . . . . . . . . . . . 15
⊢ ((ran
𝑆 ⊆ ℝ ∧ ran
𝑆 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑆 𝑧 ≤ 𝑥) → sup(ran 𝑆, ℝ*, < ) = sup(ran
𝑆, ℝ, <
)) |
113 | 100, 106,
111, 112 | syl3anc 1370 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) = sup(ran
𝑆, ℝ, <
)) |
114 | 99, 113 | breqtrrd 5102 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑆 ⇝ sup(ran 𝑆, ℝ*, <
)) |
115 | 114 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑆 ⇝ sup(ran 𝑆, ℝ*, <
)) |
116 | 13, 115 | eqbrtrrid 5110 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → seq1( + , ((abs
∘ − ) ∘ 𝐹)) ⇝ sup(ran 𝑆, ℝ*, <
)) |
117 | 64 | adantlr 712 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) ∈ ℝ) |
118 | 86 | adantlr 712 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐹)‘𝑛)) |
119 | 50, 49, 116, 117, 118 | climserle 15374 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐹))‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
120 | 82, 119 | eqbrtrid 5109 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
121 | 42, 47, 48, 83, 120 | letrd 11132 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
122 | 121 | ralrimiva 3103 |
. . . . . . 7
⊢ (𝜑 → ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
123 | | brralrspcev 5134 |
. . . . . . 7
⊢ ((sup(ran
𝑆, ℝ*,
< ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ sup(ran 𝑆, ℝ*, < )) →
∃𝑥 ∈ ℝ
∀𝑗 ∈ ℕ
(𝑇‘𝑗) ≤ 𝑥) |
124 | 23, 122, 123 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥) |
125 | 30 | ffnd 6601 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 Fn ℕ) |
126 | | breq1 5077 |
. . . . . . . . 9
⊢ (𝑧 = (𝑇‘𝑗) → (𝑧 ≤ 𝑥 ↔ (𝑇‘𝑗) ≤ 𝑥)) |
127 | 126 | ralrn 6964 |
. . . . . . . 8
⊢ (𝑇 Fn ℕ →
(∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥)) |
128 | 125, 127 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥)) |
129 | 128 | rexbidv 3226 |
. . . . . 6
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑇‘𝑗) ≤ 𝑥)) |
130 | 124, 129 | mpbird 256 |
. . . . 5
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥) |
131 | 33, 40, 130 | suprcld 11938 |
. . . 4
⊢ (𝜑 → sup(ran 𝑇, ℝ, < ) ∈
ℝ) |
132 | 67, 15 | ovolsf 24636 |
. . . . . . . 8
⊢ (𝐻:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → 𝑈:ℕ⟶(0[,)+∞)) |
133 | 66, 132 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑈:ℕ⟶(0[,)+∞)) |
134 | 133 | frnd 6608 |
. . . . . 6
⊢ (𝜑 → ran 𝑈 ⊆ (0[,)+∞)) |
135 | 134, 32 | sstrdi 3933 |
. . . . 5
⊢ (𝜑 → ran 𝑈 ⊆ ℝ) |
136 | 133 | fdmd 6611 |
. . . . . . . 8
⊢ (𝜑 → dom 𝑈 = ℕ) |
137 | 34, 136 | eleqtrrid 2846 |
. . . . . . 7
⊢ (𝜑 → 1 ∈ dom 𝑈) |
138 | 137 | ne0d 4269 |
. . . . . 6
⊢ (𝜑 → dom 𝑈 ≠ ∅) |
139 | | dm0rn0 5834 |
. . . . . . 7
⊢ (dom
𝑈 = ∅ ↔ ran
𝑈 =
∅) |
140 | 139 | necon3bii 2996 |
. . . . . 6
⊢ (dom
𝑈 ≠ ∅ ↔ ran
𝑈 ≠
∅) |
141 | 138, 140 | sylib 217 |
. . . . 5
⊢ (𝜑 → ran 𝑈 ≠ ∅) |
142 | 133 | ffvelrnda 6961 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ∈ (0[,)+∞)) |
143 | 32, 142 | sselid 3919 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ∈ ℝ) |
144 | 52, 53, 74 | syl2an 596 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐻)‘𝑛) ∈ ℝ) |
145 | | elrege0 13186 |
. . . . . . . . . . . . . . . 16
⊢ ((((abs
∘ − ) ∘ 𝐺)‘𝑛) ∈ (0[,)+∞) ↔ ((((abs
∘ − ) ∘ 𝐺)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐺)‘𝑛))) |
146 | 56, 145 | sylib 217 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ 𝐺)‘𝑛))) |
147 | 146 | simprd 496 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ 𝐺)‘𝑛)) |
148 | 74, 57 | addge02d 11564 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (0 ≤ (((abs
∘ − ) ∘ 𝐺)‘𝑛) ↔ (((abs ∘ − ) ∘
𝐻)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)))) |
149 | 147, 148 | mpbid 231 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ≤ ((((abs ∘ − ) ∘
𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) |
150 | 149, 77 | breqtrd 5100 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
151 | 52, 53, 150 | syl2an 596 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐻)‘𝑛) ≤ (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |
152 | 51, 144, 65, 151 | serle 13778 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐻))‘𝑗) ≤ (seq1( + , ((abs ∘ − )
∘ 𝐹))‘𝑗)) |
153 | 15 | fveq1i 6775 |
. . . . . . . . . 10
⊢ (𝑈‘𝑗) = (seq1( + , ((abs ∘ − )
∘ 𝐻))‘𝑗) |
154 | 152, 153,
82 | 3brtr4g 5108 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ≤ (𝑆‘𝑗)) |
155 | 143, 47, 48, 154, 120 | letrd 11132 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
156 | 155 | ralrimiva 3103 |
. . . . . . 7
⊢ (𝜑 → ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, <
)) |
157 | | brralrspcev 5134 |
. . . . . . 7
⊢ ((sup(ran
𝑆, ℝ*,
< ) ∈ ℝ ∧ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ sup(ran 𝑆, ℝ*, < )) →
∃𝑥 ∈ ℝ
∀𝑗 ∈ ℕ
(𝑈‘𝑗) ≤ 𝑥) |
158 | 23, 156, 157 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥) |
159 | 133 | ffnd 6601 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 Fn ℕ) |
160 | | breq1 5077 |
. . . . . . . . 9
⊢ (𝑧 = (𝑈‘𝑗) → (𝑧 ≤ 𝑥 ↔ (𝑈‘𝑗) ≤ 𝑥)) |
161 | 160 | ralrn 6964 |
. . . . . . . 8
⊢ (𝑈 Fn ℕ →
(∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥)) |
162 | 159, 161 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥 ↔ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥)) |
163 | 162 | rexbidv 3226 |
. . . . . 6
⊢ (𝜑 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗 ∈ ℕ (𝑈‘𝑗) ≤ 𝑥)) |
164 | 158, 163 | mpbird 256 |
. . . . 5
⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥) |
165 | 135, 141,
164 | suprcld 11938 |
. . . 4
⊢ (𝜑 → sup(ran 𝑈, ℝ, < ) ∈
ℝ) |
166 | | ssralv 3987 |
. . . . . . . . . 10
⊢ ((𝐸 ∩ 𝐵) ⊆ 𝐸 → (∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))) |
167 | 1, 166 | ax-mp 5 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))) |
168 | 19 | breq1i 5081 |
. . . . . . . . . . . . 13
⊢ (𝑃 < 𝑥 ↔ (1st ‘(𝐹‘𝑛)) < 𝑥) |
169 | | ovolfcl 24630 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛)))) |
170 | 16, 169 | sylan 580 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛)))) |
171 | 170 | simp1d 1141 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐹‘𝑛)) ∈
ℝ) |
172 | 19, 171 | eqeltrid 2843 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ) |
173 | 172 | adantlr 712 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ) |
174 | 1, 2 | sstrid 3932 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐸 ∩ 𝐵) ⊆ ℝ) |
175 | 174 | sselda 3921 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) → 𝑥 ∈ ℝ) |
176 | 175 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ) |
177 | | ltle 11063 |
. . . . . . . . . . . . . . 15
⊢ ((𝑃 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑃 < 𝑥 → 𝑃 ≤ 𝑥)) |
178 | 173, 176,
177 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → 𝑃 ≤ 𝑥)) |
179 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
180 | | opex 5379 |
. . . . . . . . . . . . . . . . . . . 20
⊢
〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉 ∈ V |
181 | 21 | fvmpt2 6886 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑛 ∈ ℕ ∧
〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉 ∈ V) → (𝐺‘𝑛) = 〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) |
182 | 179, 180,
181 | sylancl 586 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = 〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) |
183 | 182 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐺‘𝑛)) = (1st
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉)) |
184 | 11 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
185 | 184, 172 | ifcld 4505 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ) |
186 | 170 | simp2d 1142 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐹‘𝑛)) ∈
ℝ) |
187 | 20, 186 | eqeltrid 2843 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ) |
188 | 185, 187 | ifcld 4505 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) |
189 | | op1stg 7843 |
. . . . . . . . . . . . . . . . . . 19
⊢
((if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (1st
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
190 | 188, 187,
189 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
191 | 183, 190 | eqtrd 2778 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐺‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
192 | 191 | ad2ant2r 744 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (1st ‘(𝐺‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
193 | 188 | ad2ant2r 744 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) |
194 | 185 | ad2ant2r 744 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ) |
195 | 174 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (𝐸 ∩ 𝐵) ⊆ ℝ) |
196 | | simplr 766 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑥 ∈ (𝐸 ∩ 𝐵)) |
197 | 195, 196 | sseldd 3922 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑥 ∈ ℝ) |
198 | 187 | ad2ant2r 744 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑄 ∈ ℝ) |
199 | | min1 12923 |
. . . . . . . . . . . . . . . . . 18
⊢
((if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ ∧ 𝑄 ∈ ℝ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) |
200 | 194, 198,
199 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) |
201 | 11 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝐴 ∈ ℝ) |
202 | | elinel2 4130 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (𝐸 ∩ 𝐵) → 𝑥 ∈ 𝐵) |
203 | 202 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑥 ∈ 𝐵) |
204 | 11 | rexrd 11025 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐴 ∈
ℝ*) |
205 | | pnfxr 11029 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ +∞
∈ ℝ* |
206 | | elioo2 13120 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐴 ∈ ℝ*
∧ +∞ ∈ ℝ*) → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞))) |
207 | 204, 205,
206 | sylancl 586 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑥 ∈ (𝐴(,)+∞) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞))) |
208 | 10 | eleq2i 2830 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝐴(,)+∞)) |
209 | | ltpnf 12856 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 ∈ ℝ → 𝑥 < +∞) |
210 | 209 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝑥 < +∞) |
211 | 210 | pm4.71i 560 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞)) |
212 | | df-3an 1088 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞) ↔ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ∧ 𝑥 < +∞)) |
213 | 211, 212 | bitr4i 277 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥 ∧ 𝑥 < +∞)) |
214 | 207, 208,
213 | 3bitr4g 314 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥))) |
215 | | simpr 485 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ ℝ ∧ 𝐴 < 𝑥) → 𝐴 < 𝑥) |
216 | 214, 215 | syl6bi 252 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝐴 < 𝑥)) |
217 | 216 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (𝑥 ∈ 𝐵 → 𝐴 < 𝑥)) |
218 | 203, 217 | mpd 15 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝐴 < 𝑥) |
219 | 201, 197,
218 | ltled 11123 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝐴 ≤ 𝑥) |
220 | | simprr 770 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → 𝑃 ≤ 𝑥) |
221 | | breq1 5077 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 = if(𝑃 ≤ 𝐴, 𝐴, 𝑃) → (𝐴 ≤ 𝑥 ↔ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥)) |
222 | | breq1 5077 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑃 = if(𝑃 ≤ 𝐴, 𝐴, 𝑃) → (𝑃 ≤ 𝑥 ↔ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥)) |
223 | 221, 222 | ifboth 4498 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ≤ 𝑥 ∧ 𝑃 ≤ 𝑥) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥) |
224 | 219, 220,
223 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑥) |
225 | 193, 194,
197, 200, 224 | letrd 11132 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ≤ 𝑥) |
226 | 192, 225 | eqbrtrd 5096 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑃 ≤ 𝑥)) → (1st ‘(𝐺‘𝑛)) ≤ 𝑥) |
227 | 226 | expr 457 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 ≤ 𝑥 → (1st ‘(𝐺‘𝑛)) ≤ 𝑥)) |
228 | 178, 227 | syld 47 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → (1st ‘(𝐺‘𝑛)) ≤ 𝑥)) |
229 | 168, 228 | syl5bir 242 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) < 𝑥 → (1st ‘(𝐺‘𝑛)) ≤ 𝑥)) |
230 | 20 | breq2i 5082 |
. . . . . . . . . . . . . 14
⊢ (𝑥 < 𝑄 ↔ 𝑥 < (2nd ‘(𝐹‘𝑛))) |
231 | 187 | adantlr 712 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ) |
232 | | ltle 11063 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ ∧ 𝑄 ∈ ℝ) → (𝑥 < 𝑄 → 𝑥 ≤ 𝑄)) |
233 | 176, 231,
232 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄 → 𝑥 ≤ 𝑄)) |
234 | 230, 233 | syl5bir 242 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹‘𝑛)) → 𝑥 ≤ 𝑄)) |
235 | 182 | fveq2d 6778 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = (2nd
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉)) |
236 | | op2ndg 7844 |
. . . . . . . . . . . . . . . . 17
⊢
((if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ ∧ 𝑄 ∈ ℝ) → (2nd
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) = 𝑄) |
237 | 188, 187,
236 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) = 𝑄) |
238 | 235, 237 | eqtrd 2778 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = 𝑄) |
239 | 238 | adantlr 712 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = 𝑄) |
240 | 239 | breq2d 5086 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 ≤ (2nd ‘(𝐺‘𝑛)) ↔ 𝑥 ≤ 𝑄)) |
241 | 234, 240 | sylibrd 258 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹‘𝑛)) → 𝑥 ≤ (2nd ‘(𝐺‘𝑛)))) |
242 | 229, 241 | anim12d 609 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ((1st ‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
243 | 242 | reximdva 3203 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 ∩ 𝐵)) → (∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
244 | 243 | ralimdva 3108 |
. . . . . . . . 9
⊢ (𝜑 → (∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
245 | 167, 244 | syl5 34 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
246 | | ovolfioo 24631 |
. . . . . . . . 9
⊢ ((𝐸 ⊆ ℝ ∧ 𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ))) → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) ↔
∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))) |
247 | 2, 16, 246 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) ↔
∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))) |
248 | | ovolficc 24632 |
. . . . . . . . 9
⊢ (((𝐸 ∩ 𝐵) ⊆ ℝ ∧ 𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) → ((𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺) ↔
∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
249 | 174, 27, 248 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺) ↔
∀𝑥 ∈ (𝐸 ∩ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐺‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐺‘𝑛))))) |
250 | 245, 247,
249 | 3imtr4d 294 |
. . . . . . 7
⊢ (𝜑 → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) →
(𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺))) |
251 | 17, 250 | mpd 15 |
. . . . . 6
⊢ (𝜑 → (𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺)) |
252 | 14 | ovollb2 24653 |
. . . . . 6
⊢ ((𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ (𝐸 ∩ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐺)) →
(vol*‘(𝐸 ∩ 𝐵)) ≤ sup(ran 𝑇, ℝ*, <
)) |
253 | 27, 251, 252 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → (vol*‘(𝐸 ∩ 𝐵)) ≤ sup(ran 𝑇, ℝ*, <
)) |
254 | | supxrre 13061 |
. . . . . 6
⊢ ((ran
𝑇 ⊆ ℝ ∧ ran
𝑇 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑇 𝑧 ≤ 𝑥) → sup(ran 𝑇, ℝ*, < ) = sup(ran
𝑇, ℝ, <
)) |
255 | 33, 40, 130, 254 | syl3anc 1370 |
. . . . 5
⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) = sup(ran
𝑇, ℝ, <
)) |
256 | 253, 255 | breqtrd 5100 |
. . . 4
⊢ (𝜑 → (vol*‘(𝐸 ∩ 𝐵)) ≤ sup(ran 𝑇, ℝ, < )) |
257 | | ssralv 3987 |
. . . . . . . . . 10
⊢ ((𝐸 ∖ 𝐵) ⊆ 𝐸 → (∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))))) |
258 | 6, 257 | ax-mp 5 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛)))) |
259 | 172 | adantlr 712 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ) |
260 | 6, 2 | sstrid 3932 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐸 ∖ 𝐵) ⊆ ℝ) |
261 | 260 | sselda 3921 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) → 𝑥 ∈ ℝ) |
262 | 261 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑥 ∈ ℝ) |
263 | 259, 262,
177 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑃 < 𝑥 → 𝑃 ≤ 𝑥)) |
264 | 168, 263 | syl5bir 242 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) < 𝑥 → 𝑃 ≤ 𝑥)) |
265 | | opex 5379 |
. . . . . . . . . . . . . . . . . 18
⊢
〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉 ∈ V |
266 | 22 | fvmpt2 6886 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ ℕ ∧
〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉 ∈ V) → (𝐻‘𝑛) = 〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) |
267 | 179, 265,
266 | sylancl 586 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) = 〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) |
268 | 267 | fveq2d 6778 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = (1st
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉)) |
269 | | op1stg 7843 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑃 ∈ ℝ ∧
if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) → (1st
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) = 𝑃) |
270 | 172, 188,
269 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) = 𝑃) |
271 | 268, 270 | eqtrd 2778 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = 𝑃) |
272 | 271 | adantlr 712 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = 𝑃) |
273 | 272 | breq1d 5084 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ↔ 𝑃 ≤ 𝑥)) |
274 | 264, 273 | sylibrd 258 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) < 𝑥 → (1st ‘(𝐻‘𝑛)) ≤ 𝑥)) |
275 | 187 | adantlr 712 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ) |
276 | 262, 275,
232 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄 → 𝑥 ≤ 𝑄)) |
277 | 260 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝐸 ∖ 𝐵) ⊆ ℝ) |
278 | | simplr 766 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ∈ (𝐸 ∖ 𝐵)) |
279 | 277, 278 | sseldd 3922 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ∈ ℝ) |
280 | 11 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝐴 ∈ ℝ) |
281 | 172 | ad2ant2r 744 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑃 ∈ ℝ) |
282 | 280, 281 | ifcld 4505 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ) |
283 | | eldifn 4062 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (𝐸 ∖ 𝐵) → ¬ 𝑥 ∈ 𝐵) |
284 | 283 | ad2antlr 724 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → ¬ 𝑥 ∈ 𝐵) |
285 | 279 | biantrurd 533 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝐴 < 𝑥 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥))) |
286 | 214 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ ℝ ∧ 𝐴 < 𝑥))) |
287 | 285, 286 | bitr4d 281 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (𝐴 < 𝑥 ↔ 𝑥 ∈ 𝐵)) |
288 | 284, 287 | mtbird 325 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → ¬ 𝐴 < 𝑥) |
289 | 279, 280,
288 | nltled 11125 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ 𝐴) |
290 | | max2 12921 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑃 ∈ ℝ ∧ 𝐴 ∈ ℝ) → 𝐴 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) |
291 | 281, 280,
290 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝐴 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) |
292 | 279, 280,
282, 289, 291 | letrd 11132 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃)) |
293 | | simprr 770 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ 𝑄) |
294 | | breq2 5078 |
. . . . . . . . . . . . . . . . . 18
⊢ (if(𝑃 ≤ 𝐴, 𝐴, 𝑃) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) → (𝑥 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ↔ 𝑥 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))) |
295 | | breq2 5078 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑄 = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) → (𝑥 ≤ 𝑄 ↔ 𝑥 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))) |
296 | 294, 295 | ifboth 4498 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ≤ if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∧ 𝑥 ≤ 𝑄) → 𝑥 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
297 | 292, 293,
296 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
298 | 267 | fveq2d 6778 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐻‘𝑛)) = (2nd
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉)) |
299 | | op2ndg 7844 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑃 ∈ ℝ ∧
if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) → (2nd
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
300 | 172, 188,
299 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
301 | 298, 300 | eqtrd 2778 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐻‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
302 | 301 | ad2ant2r 744 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → (2nd ‘(𝐻‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
303 | 297, 302 | breqtrrd 5102 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ (𝑛 ∈ ℕ ∧ 𝑥 ≤ 𝑄)) → 𝑥 ≤ (2nd ‘(𝐻‘𝑛))) |
304 | 303 | expr 457 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 ≤ 𝑄 → 𝑥 ≤ (2nd ‘(𝐻‘𝑛)))) |
305 | 276, 304 | syld 47 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < 𝑄 → 𝑥 ≤ (2nd ‘(𝐻‘𝑛)))) |
306 | 230, 305 | syl5bir 242 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (𝑥 < (2nd ‘(𝐹‘𝑛)) → 𝑥 ≤ (2nd ‘(𝐻‘𝑛)))) |
307 | 274, 306 | anim12d 609 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) ∧ 𝑛 ∈ ℕ) → (((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ((1st ‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
308 | 307 | reximdva 3203 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (𝐸 ∖ 𝐵)) → (∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
309 | 308 | ralimdva 3108 |
. . . . . . . . 9
⊢ (𝜑 → (∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
310 | 258, 309 | syl5 34 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑥 ∈ 𝐸 ∃𝑛 ∈ ℕ ((1st
‘(𝐹‘𝑛)) < 𝑥 ∧ 𝑥 < (2nd ‘(𝐹‘𝑛))) → ∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
311 | | ovolficc 24632 |
. . . . . . . . 9
⊢ (((𝐸 ∖ 𝐵) ⊆ ℝ ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) → ((𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻) ↔
∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
312 | 260, 66, 311 | syl2anc 584 |
. . . . . . . 8
⊢ (𝜑 → ((𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻) ↔
∀𝑥 ∈ (𝐸 ∖ 𝐵)∃𝑛 ∈ ℕ ((1st
‘(𝐻‘𝑛)) ≤ 𝑥 ∧ 𝑥 ≤ (2nd ‘(𝐻‘𝑛))))) |
313 | 310, 247,
312 | 3imtr4d 294 |
. . . . . . 7
⊢ (𝜑 → (𝐸 ⊆ ∪ ran
((,) ∘ 𝐹) →
(𝐸 ∖ 𝐵) ⊆ ∪ ran ([,] ∘ 𝐻))) |
314 | 17, 313 | mpd 15 |
. . . . . 6
⊢ (𝜑 → (𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻)) |
315 | 15 | ovollb2 24653 |
. . . . . 6
⊢ ((𝐻:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ (𝐸 ∖ 𝐵) ⊆ ∪ ran
([,] ∘ 𝐻)) →
(vol*‘(𝐸 ∖
𝐵)) ≤ sup(ran 𝑈, ℝ*, <
)) |
316 | 66, 314, 315 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → (vol*‘(𝐸 ∖ 𝐵)) ≤ sup(ran 𝑈, ℝ*, <
)) |
317 | | supxrre 13061 |
. . . . . 6
⊢ ((ran
𝑈 ⊆ ℝ ∧ ran
𝑈 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑧 ∈ ran 𝑈 𝑧 ≤ 𝑥) → sup(ran 𝑈, ℝ*, < ) = sup(ran
𝑈, ℝ, <
)) |
318 | 135, 141,
164, 317 | syl3anc 1370 |
. . . . 5
⊢ (𝜑 → sup(ran 𝑈, ℝ*, < ) = sup(ran
𝑈, ℝ, <
)) |
319 | 316, 318 | breqtrd 5100 |
. . . 4
⊢ (𝜑 → (vol*‘(𝐸 ∖ 𝐵)) ≤ sup(ran 𝑈, ℝ, < )) |
320 | 5, 8, 131, 165, 256, 319 | le2addd 11594 |
. . 3
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ≤ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, <
))) |
321 | | eqidd 2739 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) = (((abs ∘ − ) ∘ 𝐺)‘𝑛)) |
322 | 50, 14, 84, 321, 57, 147, 124 | isumsup2 15558 |
. . . . 5
⊢ (𝜑 → 𝑇 ⇝ sup(ran 𝑇, ℝ, < )) |
323 | | seqex 13723 |
. . . . . . 7
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝐹)) ∈ V |
324 | 13, 323 | eqeltri 2835 |
. . . . . 6
⊢ 𝑆 ∈ V |
325 | 324 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝑆 ∈ V) |
326 | | eqidd 2739 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) = (((abs ∘ − ) ∘ 𝐻)‘𝑛)) |
327 | 50, 15, 84, 326, 74, 73, 158 | isumsup2 15558 |
. . . . 5
⊢ (𝜑 → 𝑈 ⇝ sup(ran 𝑈, ℝ, < )) |
328 | 42 | recnd 11003 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑗) ∈ ℂ) |
329 | 143 | recnd 11003 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑈‘𝑗) ∈ ℂ) |
330 | 57 | recnd 11003 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) ∈ ℂ) |
331 | 52, 53, 330 | syl2an 596 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐺)‘𝑛) ∈ ℂ) |
332 | 74 | recnd 11003 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) ∈ ℂ) |
333 | 52, 53, 332 | syl2an 596 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐻)‘𝑛) ∈ ℂ) |
334 | 77 | eqcomd 2744 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) |
335 | 52, 53, 334 | syl2an 596 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑗)) → (((abs ∘ − ) ∘
𝐹)‘𝑛) = ((((abs ∘ − ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛))) |
336 | 51, 331, 333, 335 | seradd 13765 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ 𝐹))‘𝑗) = ((seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘
− ) ∘ 𝐻))‘𝑗))) |
337 | 81, 153 | oveq12i 7287 |
. . . . . 6
⊢ ((𝑇‘𝑗) + (𝑈‘𝑗)) = ((seq1( + , ((abs ∘ − )
∘ 𝐺))‘𝑗) + (seq1( + , ((abs ∘
− ) ∘ 𝐻))‘𝑗)) |
338 | 336, 82, 337 | 3eqtr4g 2803 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑆‘𝑗) = ((𝑇‘𝑗) + (𝑈‘𝑗))) |
339 | 50, 84, 322, 325, 327, 328, 329, 338 | climadd 15341 |
. . . 4
⊢ (𝜑 → 𝑆 ⇝ (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, <
))) |
340 | | climuni 15261 |
. . . 4
⊢ ((𝑆 ⇝ (sup(ran 𝑇, ℝ, < ) + sup(ran
𝑈, ℝ, < )) ∧
𝑆 ⇝ sup(ran 𝑆, ℝ*, < ))
→ (sup(ran 𝑇, ℝ,
< ) + sup(ran 𝑈,
ℝ, < )) = sup(ran 𝑆, ℝ*, <
)) |
341 | 339, 114,
340 | syl2anc 584 |
. . 3
⊢ (𝜑 → (sup(ran 𝑇, ℝ, < ) + sup(ran 𝑈, ℝ, < )) = sup(ran
𝑆, ℝ*,
< )) |
342 | 320, 341 | breqtrd 5100 |
. 2
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ≤ sup(ran 𝑆, ℝ*, <
)) |
343 | 9, 23, 25, 342, 18 | letrd 11132 |
1
⊢ (𝜑 → ((vol*‘(𝐸 ∩ 𝐵)) + (vol*‘(𝐸 ∖ 𝐵))) ≤ ((vol*‘𝐸) + 𝐶)) |