Proof of Theorem ioombl1lem3
Step | Hyp | Ref
| Expression |
1 | | ioombl1.q |
. . . . 5
⊢ 𝑄 = (2nd ‘(𝐹‘𝑛)) |
2 | | ioombl1.f1 |
. . . . . . 7
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
3 | | ovolfcl 24387 |
. . . . . . 7
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛)))) |
4 | 2, 3 | sylan 583 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1st
‘(𝐹‘𝑛)) ∈ ℝ ∧
(2nd ‘(𝐹‘𝑛)) ∈ ℝ ∧ (1st
‘(𝐹‘𝑛)) ≤ (2nd
‘(𝐹‘𝑛)))) |
5 | 4 | simp2d 1145 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐹‘𝑛)) ∈
ℝ) |
6 | 1, 5 | eqeltrid 2843 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℝ) |
7 | 6 | recnd 10885 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑄 ∈ ℂ) |
8 | | ioombl1.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℝ) |
9 | 8 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
10 | | ioombl1.p |
. . . . . . 7
⊢ 𝑃 = (1st ‘(𝐹‘𝑛)) |
11 | 4 | simp1d 1144 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐹‘𝑛)) ∈
ℝ) |
12 | 10, 11 | eqeltrid 2843 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℝ) |
13 | 9, 12 | ifcld 4499 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ∈ ℝ) |
14 | 13, 6 | ifcld 4499 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℝ) |
15 | 14 | recnd 10885 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) ∈ ℂ) |
16 | 12 | recnd 10885 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑃 ∈ ℂ) |
17 | 7, 15, 16 | npncand 11237 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑄 − 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) − 𝑃)) = (𝑄 − 𝑃)) |
18 | | ioombl1.b |
. . . . . . 7
⊢ 𝐵 = (𝐴(,)+∞) |
19 | | ioombl1.e |
. . . . . . 7
⊢ (𝜑 → 𝐸 ⊆ ℝ) |
20 | | ioombl1.v |
. . . . . . 7
⊢ (𝜑 → (vol*‘𝐸) ∈
ℝ) |
21 | | ioombl1.c |
. . . . . . 7
⊢ (𝜑 → 𝐶 ∈
ℝ+) |
22 | | ioombl1.s |
. . . . . . 7
⊢ 𝑆 = seq1( + , ((abs ∘
− ) ∘ 𝐹)) |
23 | | ioombl1.t |
. . . . . . 7
⊢ 𝑇 = seq1( + , ((abs ∘
− ) ∘ 𝐺)) |
24 | | ioombl1.u |
. . . . . . 7
⊢ 𝑈 = seq1( + , ((abs ∘
− ) ∘ 𝐻)) |
25 | | ioombl1.f2 |
. . . . . . 7
⊢ (𝜑 → 𝐸 ⊆ ∪ ran
((,) ∘ 𝐹)) |
26 | | ioombl1.f3 |
. . . . . . 7
⊢ (𝜑 → sup(ran 𝑆, ℝ*, < ) ≤
((vol*‘𝐸) + 𝐶)) |
27 | | ioombl1.g |
. . . . . . 7
⊢ 𝐺 = (𝑛 ∈ ℕ ↦ 〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) |
28 | | ioombl1.h |
. . . . . . 7
⊢ 𝐻 = (𝑛 ∈ ℕ ↦ 〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) |
29 | 18, 8, 19, 20, 21, 22, 23, 24, 2, 25, 26, 10, 1, 27, 28 | ioombl1lem1 24479 |
. . . . . 6
⊢ (𝜑 → (𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ)) ∧ 𝐻:ℕ⟶( ≤ ∩ (ℝ
× ℝ)))) |
30 | 29 | simpld 498 |
. . . . 5
⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
31 | | eqid 2738 |
. . . . . 6
⊢ ((abs
∘ − ) ∘ 𝐺) = ((abs ∘ − ) ∘ 𝐺) |
32 | 31 | ovolfsval 24391 |
. . . . 5
⊢ ((𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛)))) |
33 | 30, 32 | sylan 583 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) = ((2nd ‘(𝐺‘𝑛)) − (1st ‘(𝐺‘𝑛)))) |
34 | | simpr 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
35 | | opex 5362 |
. . . . . . . 8
⊢
〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉 ∈ V |
36 | 27 | fvmpt2 6847 |
. . . . . . . 8
⊢ ((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〉) |
37 | 34, 35, 36 | sylancl 589 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = 〈if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) |
38 | 37 | fveq2d 6739 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = (2nd
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉)) |
39 | | op2ndg 7792 |
. . . . . . 7
⊢
((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) |
40 | 14, 6, 39 | syl2anc 587 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉) = 𝑄) |
41 | 38, 40 | eqtrd 2778 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐺‘𝑛)) = 𝑄) |
42 | 37 | fveq2d 6739 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐺‘𝑛)) = (1st
‘〈if(if(𝑃 ≤
𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄), 𝑄〉)) |
43 | | op1stg 7791 |
. . . . . . 7
⊢
((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)) |
44 | 14, 6, 43 | syl2anc 587 |
. . . . . 6
⊢ ((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)) |
45 | 42, 44 | eqtrd 2778 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐺‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
46 | 41, 45 | oveq12d 7249 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd
‘(𝐺‘𝑛)) − (1st
‘(𝐺‘𝑛))) = (𝑄 − if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))) |
47 | 33, 46 | eqtrd 2778 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐺)‘𝑛) = (𝑄 − if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄))) |
48 | 29 | simprd 499 |
. . . . 5
⊢ (𝜑 → 𝐻:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
49 | | eqid 2738 |
. . . . . 6
⊢ ((abs
∘ − ) ∘ 𝐻) = ((abs ∘ − ) ∘ 𝐻) |
50 | 49 | ovolfsval 24391 |
. . . . 5
⊢ ((𝐻:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) = ((2nd ‘(𝐻‘𝑛)) − (1st ‘(𝐻‘𝑛)))) |
51 | 48, 50 | sylan 583 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) = ((2nd ‘(𝐻‘𝑛)) − (1st ‘(𝐻‘𝑛)))) |
52 | | opex 5362 |
. . . . . . . 8
⊢
〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉 ∈ V |
53 | 28 | fvmpt2 6847 |
. . . . . . . 8
⊢ ((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)〉) |
54 | 34, 52, 53 | sylancl 589 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘𝑛) = 〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) |
55 | 54 | fveq2d 6739 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐻‘𝑛)) = (2nd
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉)) |
56 | | op2ndg 7792 |
. . . . . . 7
⊢ ((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)) |
57 | 12, 14, 56 | syl2anc 587 |
. . . . . 6
⊢ ((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)) |
58 | 55, 57 | eqtrd 2778 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2nd
‘(𝐻‘𝑛)) = if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)) |
59 | 54 | fveq2d 6739 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = (1st
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉)) |
60 | | op1stg 7791 |
. . . . . . 7
⊢ ((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) |
61 | 12, 14, 60 | syl2anc 587 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘〈𝑃, if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄)〉) = 𝑃) |
62 | 59, 61 | eqtrd 2778 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (1st
‘(𝐻‘𝑛)) = 𝑃) |
63 | 58, 62 | oveq12d 7249 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((2nd
‘(𝐻‘𝑛)) − (1st
‘(𝐻‘𝑛))) = (if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) − 𝑃)) |
64 | 51, 63 | eqtrd 2778 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐻)‘𝑛) = (if(if(𝑃 ≤ 𝐴, 𝐴, 𝑃) ≤ 𝑄, if(𝑃 ≤ 𝐴, 𝐴, 𝑃), 𝑄) − 𝑃)) |
65 | 47, 64 | oveq12d 7249 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = ((𝑄 − 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) − 𝑃))) |
66 | | eqid 2738 |
. . . . 5
⊢ ((abs
∘ − ) ∘ 𝐹) = ((abs ∘ − ) ∘ 𝐹) |
67 | 66 | ovolfsval 24391 |
. . . 4
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛)))) |
68 | 2, 67 | sylan 583 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛)))) |
69 | 1, 10 | oveq12i 7243 |
. . 3
⊢ (𝑄 − 𝑃) = ((2nd ‘(𝐹‘𝑛)) − (1st ‘(𝐹‘𝑛))) |
70 | 68, 69 | eqtr4di 2797 |
. 2
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ 𝐹)‘𝑛) = (𝑄 − 𝑃)) |
71 | 17, 65, 70 | 3eqtr4d 2788 |
1
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ 𝐺)‘𝑛) + (((abs ∘ − ) ∘ 𝐻)‘𝑛)) = (((abs ∘ − ) ∘ 𝐹)‘𝑛)) |