Step | Hyp | Ref
| Expression |
1 | | nnuz 11599 |
. . 3
⊢ ℕ =
(ℤ≥‘1) |
2 | | eqid 2610 |
. . 3
⊢ seq1( + ,
((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = seq1( + , ((abs ∘ − )
∘ (𝐾 ∘ 𝐻))) |
3 | | 1zzd 11285 |
. . 3
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 1 ∈
ℤ) |
4 | | eqidd 2611 |
. . 3
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ (𝐾
∘ 𝐻))‘𝑛) = (((abs ∘ − )
∘ (𝐾 ∘ 𝐻))‘𝑛)) |
5 | | uniioombl.1 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
6 | | uniioombl.2 |
. . . . . . . . . . 11
⊢ (𝜑 → Disj 𝑥 ∈ ℕ
((,)‘(𝐹‘𝑥))) |
7 | | uniioombl.3 |
. . . . . . . . . . 11
⊢ 𝑆 = seq1( + , ((abs ∘
− ) ∘ 𝐹)) |
8 | | uniioombl.a |
. . . . . . . . . . 11
⊢ 𝐴 = ∪
ran ((,) ∘ 𝐹) |
9 | | uniioombl.e |
. . . . . . . . . . 11
⊢ (𝜑 → (vol*‘𝐸) ∈
ℝ) |
10 | | uniioombl.c |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐶 ∈
ℝ+) |
11 | | uniioombl.g |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐺:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
12 | | uniioombl.s |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐸 ⊆ ∪ ran
((,) ∘ 𝐺)) |
13 | | uniioombl.t |
. . . . . . . . . . 11
⊢ 𝑇 = seq1( + , ((abs ∘
− ) ∘ 𝐺)) |
14 | | uniioombl.v |
. . . . . . . . . . 11
⊢ (𝜑 → sup(ran 𝑇, ℝ*, < ) ≤
((vol*‘𝐸) + 𝐶)) |
15 | 5, 6, 7, 8, 9, 10,
11, 12, 13, 14 | uniioombllem2a 23156 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,)) |
16 | | inss2 3796 |
. . . . . . . . . . . . 13
⊢
(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ⊆ ((,)‘(𝐺‘𝐽)) |
17 | 16 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ⊆ ((,)‘(𝐺‘𝐽))) |
18 | | inss2 3796 |
. . . . . . . . . . . . . . . . 17
⊢ ( ≤
∩ (ℝ × ℝ)) ⊆ (ℝ ×
ℝ) |
19 | 11 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ ( ≤ ∩ (ℝ ×
ℝ))) |
20 | 18, 19 | sseldi 3566 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) ∈ (ℝ ×
ℝ)) |
21 | | 1st2nd2 7096 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺‘𝐽) ∈ (ℝ × ℝ) →
(𝐺‘𝐽) = 〈(1st ‘(𝐺‘𝐽)), (2nd ‘(𝐺‘𝐽))〉) |
22 | 20, 21 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐺‘𝐽) = 〈(1st ‘(𝐺‘𝐽)), (2nd ‘(𝐺‘𝐽))〉) |
23 | 22 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((,)‘〈(1st
‘(𝐺‘𝐽)), (2nd
‘(𝐺‘𝐽))〉)) |
24 | | df-ov 6552 |
. . . . . . . . . . . . . 14
⊢
((1st ‘(𝐺‘𝐽))(,)(2nd ‘(𝐺‘𝐽))) = ((,)‘〈(1st
‘(𝐺‘𝐽)), (2nd
‘(𝐺‘𝐽))〉) |
25 | 23, 24 | syl6eqr 2662 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) = ((1st ‘(𝐺‘𝐽))(,)(2nd ‘(𝐺‘𝐽)))) |
26 | | ioossre 12106 |
. . . . . . . . . . . . 13
⊢
((1st ‘(𝐺‘𝐽))(,)(2nd ‘(𝐺‘𝐽))) ⊆ ℝ |
27 | 25, 26 | syl6eqss 3618 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,)‘(𝐺‘𝐽)) ⊆ ℝ) |
28 | 25 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) →
(vol*‘((,)‘(𝐺‘𝐽))) = (vol*‘((1st
‘(𝐺‘𝐽))(,)(2nd
‘(𝐺‘𝐽))))) |
29 | | ovolfcl 23042 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝐽 ∈ ℕ) → ((1st
‘(𝐺‘𝐽)) ∈ ℝ ∧
(2nd ‘(𝐺‘𝐽)) ∈ ℝ ∧ (1st
‘(𝐺‘𝐽)) ≤ (2nd
‘(𝐺‘𝐽)))) |
30 | 11, 29 | sylan 487 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((1st
‘(𝐺‘𝐽)) ∈ ℝ ∧
(2nd ‘(𝐺‘𝐽)) ∈ ℝ ∧ (1st
‘(𝐺‘𝐽)) ≤ (2nd
‘(𝐺‘𝐽)))) |
31 | | ovolioo 23143 |
. . . . . . . . . . . . . . 15
⊢
(((1st ‘(𝐺‘𝐽)) ∈ ℝ ∧ (2nd
‘(𝐺‘𝐽)) ∈ ℝ ∧
(1st ‘(𝐺‘𝐽)) ≤ (2nd ‘(𝐺‘𝐽))) → (vol*‘((1st
‘(𝐺‘𝐽))(,)(2nd
‘(𝐺‘𝐽)))) = ((2nd
‘(𝐺‘𝐽)) − (1st
‘(𝐺‘𝐽)))) |
32 | 30, 31 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) →
(vol*‘((1st ‘(𝐺‘𝐽))(,)(2nd ‘(𝐺‘𝐽)))) = ((2nd ‘(𝐺‘𝐽)) − (1st ‘(𝐺‘𝐽)))) |
33 | 28, 32 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) →
(vol*‘((,)‘(𝐺‘𝐽))) = ((2nd ‘(𝐺‘𝐽)) − (1st ‘(𝐺‘𝐽)))) |
34 | 30 | simp2d 1067 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (2nd
‘(𝐺‘𝐽)) ∈
ℝ) |
35 | 30 | simp1d 1066 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (1st
‘(𝐺‘𝐽)) ∈
ℝ) |
36 | 34, 35 | resubcld 10337 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((2nd
‘(𝐺‘𝐽)) − (1st
‘(𝐺‘𝐽))) ∈
ℝ) |
37 | 33, 36 | eqeltrd 2688 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) →
(vol*‘((,)‘(𝐺‘𝐽))) ∈ ℝ) |
38 | | ovolsscl 23061 |
. . . . . . . . . . . 12
⊢
(((((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ⊆ ((,)‘(𝐺‘𝐽)) ∧ ((,)‘(𝐺‘𝐽)) ⊆ ℝ ∧
(vol*‘((,)‘(𝐺‘𝐽))) ∈ ℝ) →
(vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ℝ) |
39 | 17, 27, 37, 38 | syl3anc 1318 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) →
(vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ℝ) |
40 | 39 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) →
(vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ℝ) |
41 | | uniioombllem2.k |
. . . . . . . . . . 11
⊢ 𝐾 = (𝑥 ∈ ran (,) ↦ if(𝑥 = ∅, 〈0, 0〉, 〈inf(𝑥, ℝ*, < ),
sup(𝑥, ℝ*,
< )〉)) |
42 | 41 | ioorcl 23151 |
. . . . . . . . . 10
⊢
(((((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,) ∧
(vol*‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ℝ) → (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ( ≤ ∩ (ℝ ×
ℝ))) |
43 | 15, 40, 42 | syl2anc 691 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ ( ≤ ∩ (ℝ ×
ℝ))) |
44 | | eqid 2610 |
. . . . . . . . 9
⊢ (𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) = (𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) |
45 | 43, 44 | fmptd 6292 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))):ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
46 | | uniioombllem2.h |
. . . . . . . . . . 11
⊢ 𝐻 = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) |
47 | 46 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐻 = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) |
48 | 41 | ioorf 23147 |
. . . . . . . . . . . 12
⊢ 𝐾:ran (,)⟶( ≤ ∩
(ℝ* × ℝ*)) |
49 | 48 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐾:ran (,)⟶( ≤ ∩
(ℝ* × ℝ*))) |
50 | 49 | feqmptd 6159 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐾 = (𝑦 ∈ ran (,) ↦ (𝐾‘𝑦))) |
51 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑦 = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) → (𝐾‘𝑦) = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) |
52 | 15, 47, 50, 51 | fmptco 6303 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐾 ∘ 𝐻) = (𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))) |
53 | 52 | feq1d 5943 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((𝐾 ∘ 𝐻):ℕ⟶( ≤ ∩ (ℝ
× ℝ)) ↔ (𝑧
∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))):ℕ⟶( ≤ ∩ (ℝ
× ℝ)))) |
54 | 45, 53 | mpbird 246 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝐾 ∘ 𝐻):ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
55 | | eqid 2610 |
. . . . . . . 8
⊢ ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)) = ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)) |
56 | 55 | ovolfsf 23047 |
. . . . . . 7
⊢ ((𝐾 ∘ 𝐻):ℕ⟶( ≤ ∩ (ℝ
× ℝ)) → ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)):ℕ⟶(0[,)+∞)) |
57 | 54, 56 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((abs ∘
− ) ∘ (𝐾
∘ 𝐻)):ℕ⟶(0[,)+∞)) |
58 | 57 | ffvelrnda 6267 |
. . . . 5
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ (𝐾
∘ 𝐻))‘𝑛) ∈
(0[,)+∞)) |
59 | | elrege0 12149 |
. . . . 5
⊢ ((((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛) ∈ (0[,)+∞) ↔ ((((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛) ∈ ℝ ∧ 0 ≤ (((abs ∘
− ) ∘ (𝐾
∘ 𝐻))‘𝑛))) |
60 | 58, 59 | sylib 207 |
. . . 4
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → ((((abs ∘
− ) ∘ (𝐾
∘ 𝐻))‘𝑛) ∈ ℝ ∧ 0 ≤
(((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))‘𝑛))) |
61 | 60 | simpld 474 |
. . 3
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → (((abs ∘
− ) ∘ (𝐾
∘ 𝐻))‘𝑛) ∈
ℝ) |
62 | 60 | simprd 478 |
. . 3
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑛 ∈ ℕ) → 0 ≤ (((abs ∘
− ) ∘ (𝐾
∘ 𝐻))‘𝑛)) |
63 | 52 | fveq1d 6105 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((𝐾 ∘ 𝐻)‘𝑧) = ((𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))‘𝑧)) |
64 | | fvex 6113 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ V |
65 | 44 | fvmpt2 6200 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑧 ∈ ℕ ∧ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ V) → ((𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))‘𝑧) = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) |
66 | 64, 65 | mpan2 703 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ ℕ → ((𝑧 ∈ ℕ ↦ (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))‘𝑧) = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) |
67 | 63, 66 | sylan9eq 2664 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((𝐾 ∘ 𝐻)‘𝑧) = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) |
68 | 67 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))) |
69 | 41 | ioorinv 23150 |
. . . . . . . . . . . . . 14
⊢
((((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ ran (,) → ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) |
70 | 15, 69 | syl 17 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) |
71 | 68, 70 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) |
72 | 71 | ralrimiva 2949 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∀𝑧 ∈ ℕ
((,)‘((𝐾 ∘
𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) |
73 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑥 → ((𝐾 ∘ 𝐻)‘𝑧) = ((𝐾 ∘ 𝐻)‘𝑥)) |
74 | 73 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑥 → ((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = ((,)‘((𝐾 ∘ 𝐻)‘𝑥))) |
75 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥)) |
76 | 75 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑥 → ((,)‘(𝐹‘𝑧)) = ((,)‘(𝐹‘𝑥))) |
77 | 76 | ineq1d 3775 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑥 → (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) = (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽)))) |
78 | 74, 77 | eqeq12d 2625 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑥 → (((,)‘((𝐾 ∘ 𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ↔ ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) = (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽))))) |
79 | 78 | rspccva 3281 |
. . . . . . . . . . 11
⊢
((∀𝑧 ∈
ℕ ((,)‘((𝐾
∘ 𝐻)‘𝑧)) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∧ 𝑥 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) = (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽)))) |
80 | 72, 79 | sylan 487 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) = (((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽)))) |
81 | | inss1 3795 |
. . . . . . . . . 10
⊢
(((,)‘(𝐹‘𝑥)) ∩ ((,)‘(𝐺‘𝐽))) ⊆ ((,)‘(𝐹‘𝑥)) |
82 | 80, 81 | syl6eqss 3618 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑥 ∈ ℕ) → ((,)‘((𝐾 ∘ 𝐻)‘𝑥)) ⊆ ((,)‘(𝐹‘𝑥))) |
83 | 82 | ralrimiva 2949 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∀𝑥 ∈ ℕ
((,)‘((𝐾 ∘
𝐻)‘𝑥)) ⊆ ((,)‘(𝐹‘𝑥))) |
84 | 6 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → Disj 𝑥 ∈ ℕ
((,)‘(𝐹‘𝑥))) |
85 | | disjss2 4556 |
. . . . . . . 8
⊢
(∀𝑥 ∈
ℕ ((,)‘((𝐾
∘ 𝐻)‘𝑥)) ⊆ ((,)‘(𝐹‘𝑥)) → (Disj 𝑥 ∈ ℕ ((,)‘(𝐹‘𝑥)) → Disj 𝑥 ∈ ℕ ((,)‘((𝐾 ∘ 𝐻)‘𝑥)))) |
86 | 83, 84, 85 | sylc 63 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → Disj 𝑥 ∈ ℕ
((,)‘((𝐾 ∘
𝐻)‘𝑥))) |
87 | 54, 86, 2 | uniioovol 23153 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol*‘∪ ran ((,) ∘ (𝐾 ∘ 𝐻))) = sup(ran seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻))),
ℝ*, < )) |
88 | 70 | mpteq2dva 4672 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (𝑧 ∈ ℕ ↦ ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))) = (𝑧 ∈ ℕ ↦ (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))) |
89 | | rexpssxrxp 9963 |
. . . . . . . . . . . . . 14
⊢ (ℝ
× ℝ) ⊆ (ℝ* ×
ℝ*) |
90 | 18, 89 | sstri 3577 |
. . . . . . . . . . . . 13
⊢ ( ≤
∩ (ℝ × ℝ)) ⊆ (ℝ* ×
ℝ*) |
91 | 90, 43 | sseldi 3566 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) ∈ (ℝ* ×
ℝ*)) |
92 | | ioof 12142 |
. . . . . . . . . . . . . 14
⊢
(,):(ℝ* × ℝ*)⟶𝒫
ℝ |
93 | 92 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) →
(,):(ℝ* × ℝ*)⟶𝒫
ℝ) |
94 | 93 | feqmptd 6159 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (,) = (𝑦 ∈ (ℝ*
× ℝ*) ↦ ((,)‘𝑦))) |
95 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑦 = (𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) → ((,)‘𝑦) = ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))))) |
96 | 91, 52, 94, 95 | fmptco 6303 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,) ∘ (𝐾 ∘ 𝐻)) = (𝑧 ∈ ℕ ↦ ((,)‘(𝐾‘(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))))))) |
97 | 88, 96, 47 | 3eqtr4d 2654 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,) ∘ (𝐾 ∘ 𝐻)) = 𝐻) |
98 | 97 | rneqd 5274 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran ((,) ∘
(𝐾 ∘ 𝐻)) = ran 𝐻) |
99 | 98 | unieqd 4382 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ ran ((,) ∘ (𝐾 ∘ 𝐻)) = ∪ ran 𝐻) |
100 | | fvex 6113 |
. . . . . . . . . . . . . 14
⊢
((,)‘(𝐹‘𝑧)) ∈ V |
101 | 100 | inex1 4727 |
. . . . . . . . . . . . 13
⊢
(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ V |
102 | 46 | fvmpt2 6200 |
. . . . . . . . . . . . 13
⊢ ((𝑧 ∈ ℕ ∧
(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) ∈ V) → (𝐻‘𝑧) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) |
103 | 101, 102 | mpan2 703 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈ ℕ → (𝐻‘𝑧) = (((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽)))) |
104 | | incom 3767 |
. . . . . . . . . . . 12
⊢
(((,)‘(𝐹‘𝑧)) ∩ ((,)‘(𝐺‘𝐽))) = (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧))) |
105 | 103, 104 | syl6eq 2660 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ ℕ → (𝐻‘𝑧) = (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧)))) |
106 | 105 | iuneq2i 4475 |
. . . . . . . . . 10
⊢ ∪ 𝑧 ∈ ℕ (𝐻‘𝑧) = ∪ 𝑧 ∈ ℕ
(((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧))) |
107 | | iunin2 4520 |
. . . . . . . . . 10
⊢ ∪ 𝑧 ∈ ℕ (((,)‘(𝐺‘𝐽)) ∩ ((,)‘(𝐹‘𝑧))) = (((,)‘(𝐺‘𝐽)) ∩ ∪
𝑧 ∈ ℕ
((,)‘(𝐹‘𝑧))) |
108 | 106, 107 | eqtri 2632 |
. . . . . . . . 9
⊢ ∪ 𝑧 ∈ ℕ (𝐻‘𝑧) = (((,)‘(𝐺‘𝐽)) ∩ ∪
𝑧 ∈ ℕ
((,)‘(𝐹‘𝑧))) |
109 | 15, 46 | fmptd 6292 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐻:ℕ⟶ran (,)) |
110 | | ffn 5958 |
. . . . . . . . . . 11
⊢ (𝐻:ℕ⟶ran (,) →
𝐻 Fn
ℕ) |
111 | 109, 110 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐻 Fn ℕ) |
112 | | fniunfv 6409 |
. . . . . . . . . 10
⊢ (𝐻 Fn ℕ → ∪ 𝑧 ∈ ℕ (𝐻‘𝑧) = ∪ ran 𝐻) |
113 | 111, 112 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ (𝐻‘𝑧) = ∪ ran 𝐻) |
114 | 108, 113 | syl5eqr 2658 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (((,)‘(𝐺‘𝐽)) ∩ ∪
𝑧 ∈ ℕ
((,)‘(𝐹‘𝑧))) = ∪ ran 𝐻) |
115 | 5 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐹:ℕ⟶( ≤ ∩ (ℝ
× ℝ))) |
116 | | fvco3 6185 |
. . . . . . . . . . . 12
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ 𝑧 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑧) = ((,)‘(𝐹‘𝑧))) |
117 | 115, 116 | sylan 487 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑧 ∈ ℕ) → (((,) ∘ 𝐹)‘𝑧) = ((,)‘(𝐹‘𝑧))) |
118 | 117 | iuneq2dv 4478 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ (((,) ∘ 𝐹)‘𝑧) = ∪ 𝑧 ∈ ℕ
((,)‘(𝐹‘𝑧))) |
119 | | ffn 5958 |
. . . . . . . . . . . . . 14
⊢
((,):(ℝ* × ℝ*)⟶𝒫
ℝ → (,) Fn (ℝ* ×
ℝ*)) |
120 | 92, 119 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ (,) Fn
(ℝ* × ℝ*) |
121 | | fss 5969 |
. . . . . . . . . . . . . 14
⊢ ((𝐹:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ ( ≤ ∩ (ℝ × ℝ))
⊆ (ℝ* × ℝ*)) → 𝐹:ℕ⟶(ℝ* ×
ℝ*)) |
122 | 115, 90, 121 | sylancl 693 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 𝐹:ℕ⟶(ℝ* ×
ℝ*)) |
123 | | fnfco 5982 |
. . . . . . . . . . . . 13
⊢ (((,) Fn
(ℝ* × ℝ*) ∧ 𝐹:ℕ⟶(ℝ* ×
ℝ*)) → ((,) ∘ 𝐹) Fn ℕ) |
124 | 120, 122,
123 | sylancr 694 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((,) ∘ 𝐹) Fn ℕ) |
125 | | fniunfv 6409 |
. . . . . . . . . . . 12
⊢ (((,)
∘ 𝐹) Fn ℕ
→ ∪ 𝑧 ∈ ℕ (((,) ∘ 𝐹)‘𝑧) = ∪ ran ((,)
∘ 𝐹)) |
126 | 124, 125 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ (((,) ∘ 𝐹)‘𝑧) = ∪ ran ((,)
∘ 𝐹)) |
127 | 126, 8 | syl6eqr 2662 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ (((,) ∘ 𝐹)‘𝑧) = 𝐴) |
128 | 118, 127 | eqtr3d 2646 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ 𝑧 ∈ ℕ ((,)‘(𝐹‘𝑧)) = 𝐴) |
129 | 128 | ineq2d 3776 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (((,)‘(𝐺‘𝐽)) ∩ ∪
𝑧 ∈ ℕ
((,)‘(𝐹‘𝑧))) = (((,)‘(𝐺‘𝐽)) ∩ 𝐴)) |
130 | 99, 114, 129 | 3eqtr2d 2650 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∪ ran ((,) ∘ (𝐾 ∘ 𝐻)) = (((,)‘(𝐺‘𝐽)) ∩ 𝐴)) |
131 | 130 | fveq2d 6107 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol*‘∪ ran ((,) ∘ (𝐾 ∘ 𝐻))) = (vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴))) |
132 | 87, 131 | eqtr3d 2646 |
. . . . 5
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ*, < ) =
(vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴))) |
133 | | inss1 3795 |
. . . . . . 7
⊢
(((,)‘(𝐺‘𝐽)) ∩ 𝐴) ⊆ ((,)‘(𝐺‘𝐽)) |
134 | 133 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (((,)‘(𝐺‘𝐽)) ∩ 𝐴) ⊆ ((,)‘(𝐺‘𝐽))) |
135 | | ovolsscl 23061 |
. . . . . 6
⊢
(((((,)‘(𝐺‘𝐽)) ∩ 𝐴) ⊆ ((,)‘(𝐺‘𝐽)) ∧ ((,)‘(𝐺‘𝐽)) ⊆ ℝ ∧
(vol*‘((,)‘(𝐺‘𝐽))) ∈ ℝ) →
(vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)) ∈ ℝ) |
136 | 134, 27, 37, 135 | syl3anc 1318 |
. . . . 5
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) →
(vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴)) ∈ ℝ) |
137 | 132, 136 | eqeltrd 2688 |
. . . 4
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ*, < ) ∈
ℝ) |
138 | 55, 2 | ovolsf 23048 |
. . . . . . . . 9
⊢ ((𝐾 ∘ 𝐻):ℕ⟶( ≤ ∩ (ℝ
× ℝ)) → seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))):ℕ⟶(0[,)+∞)) |
139 | 54, 138 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))):ℕ⟶(0[,)+∞)) |
140 | | ffn 5958 |
. . . . . . . 8
⊢ (seq1( +
, ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))):ℕ⟶(0[,)+∞) →
seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) Fn ℕ) |
141 | 139, 140 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) Fn ℕ) |
142 | | fnfvelrn 6264 |
. . . . . . 7
⊢ ((seq1( +
, ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) Fn ℕ ∧ 𝑦 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ∈ ran seq1( + , ((abs ∘ −
) ∘ (𝐾 ∘ 𝐻)))) |
143 | 141, 142 | sylan 487 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ∈ ran seq1( + , ((abs ∘ −
) ∘ (𝐾 ∘ 𝐻)))) |
144 | | frn 5966 |
. . . . . . . . 9
⊢ (seq1( +
, ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))):ℕ⟶(0[,)+∞) → ran
seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ (0[,)+∞)) |
145 | 139, 144 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ (0[,)+∞)) |
146 | | icossxr 12129 |
. . . . . . . 8
⊢
(0[,)+∞) ⊆ ℝ* |
147 | 145, 146 | syl6ss 3580 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆
ℝ*) |
148 | | supxrub 12026 |
. . . . . . 7
⊢ ((ran
seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ ℝ* ∧ (seq1(
+ , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ∈ ran seq1( + , ((abs ∘ −
) ∘ (𝐾 ∘ 𝐻)))) → (seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻))),
ℝ*, < )) |
149 | 147, 148 | sylan 487 |
. . . . . 6
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ (seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ∈ ran seq1( + , ((abs ∘ −
) ∘ (𝐾 ∘ 𝐻)))) → (seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻))),
ℝ*, < )) |
150 | 143, 149 | syldan 486 |
. . . . 5
⊢ (((𝜑 ∧ 𝐽 ∈ ℕ) ∧ 𝑦 ∈ ℕ) → (seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻))),
ℝ*, < )) |
151 | 150 | ralrimiva 2949 |
. . . 4
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∀𝑦 ∈ ℕ (seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻))),
ℝ*, < )) |
152 | | breq2 4587 |
. . . . . 6
⊢ (𝑥 = sup(ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ*, < ) →
((seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥 ↔ (seq1( + , ((abs ∘ − )
∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻))),
ℝ*, < ))) |
153 | 152 | ralbidv 2969 |
. . . . 5
⊢ (𝑥 = sup(ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ*, < ) →
(∀𝑦 ∈ ℕ
(seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥 ↔ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ*, <
))) |
154 | 153 | rspcev 3282 |
. . . 4
⊢ ((sup(ran
seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ*, < ) ∈
ℝ ∧ ∀𝑦
∈ ℕ (seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻))),
ℝ*, < )) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻)))‘𝑦) ≤ 𝑥) |
155 | 137, 151,
154 | syl2anc 691 |
. . 3
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℕ (seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥) |
156 | 1, 2, 3, 4, 61, 62, 155 | isumsup2 14417 |
. 2
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) ⇝ sup(ran seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻))), ℝ, <
)) |
157 | 55 | ovolfs2 23145 |
. . . . 5
⊢ ((𝐾 ∘ 𝐻):ℕ⟶( ≤ ∩ (ℝ
× ℝ)) → ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)) = ((vol* ∘ (,)) ∘ (𝐾 ∘ 𝐻))) |
158 | 54, 157 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((abs ∘
− ) ∘ (𝐾
∘ 𝐻)) = ((vol*
∘ (,)) ∘ (𝐾
∘ 𝐻))) |
159 | | coass 5571 |
. . . . 5
⊢ ((vol*
∘ (,)) ∘ (𝐾
∘ 𝐻)) = (vol* ∘
((,) ∘ (𝐾 ∘
𝐻))) |
160 | 97 | coeq2d 5206 |
. . . . 5
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (vol* ∘ ((,)
∘ (𝐾 ∘ 𝐻))) = (vol* ∘ 𝐻)) |
161 | 159, 160 | syl5eq 2656 |
. . . 4
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((vol* ∘ (,))
∘ (𝐾 ∘ 𝐻)) = (vol* ∘ 𝐻)) |
162 | 158, 161 | eqtrd 2644 |
. . 3
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ((abs ∘
− ) ∘ (𝐾
∘ 𝐻)) = (vol* ∘
𝐻)) |
163 | 162 | seqeq3d 12671 |
. 2
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) = seq1( + , (vol* ∘ 𝐻))) |
164 | | rge0ssre 12151 |
. . . . 5
⊢
(0[,)+∞) ⊆ ℝ |
165 | 145, 164 | syl6ss 3580 |
. . . 4
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ ℝ) |
166 | | 1nn 10908 |
. . . . . . 7
⊢ 1 ∈
ℕ |
167 | | fdm 5964 |
. . . . . . . 8
⊢ (seq1( +
, ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))):ℕ⟶(0[,)+∞) → dom
seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = ℕ) |
168 | 139, 167 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → dom seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) = ℕ) |
169 | 166, 168 | syl5eleqr 2695 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → 1 ∈ dom seq1( +
, ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻)))) |
170 | | ne0i 3880 |
. . . . . 6
⊢ (1 ∈
dom seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) → dom seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻))) ≠
∅) |
171 | 169, 170 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → dom seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅) |
172 | | dm0rn0 5263 |
. . . . . 6
⊢ (dom
seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) = ∅ ↔ ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) = ∅) |
173 | 172 | necon3bii 2834 |
. . . . 5
⊢ (dom
seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅ ↔ ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅) |
174 | 171, 173 | sylib 207 |
. . . 4
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅) |
175 | | breq1 4586 |
. . . . . . . 8
⊢ (𝑧 = (seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻)))‘𝑦) → (𝑧 ≤ 𝑥 ↔ (seq1( + , ((abs ∘ − )
∘ (𝐾 ∘ 𝐻)))‘𝑦) ≤ 𝑥)) |
176 | 175 | ralrn 6270 |
. . . . . . 7
⊢ (seq1( +
, ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) Fn ℕ → (∀𝑧 ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻)))‘𝑦) ≤ 𝑥)) |
177 | 141, 176 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (∀𝑧 ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))𝑧 ≤ 𝑥 ↔ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻)))‘𝑦) ≤ 𝑥)) |
178 | 177 | rexbidv 3034 |
. . . . 5
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → (∃𝑥 ∈ ℝ ∀𝑧 ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ℕ (seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻)))‘𝑦) ≤ 𝑥)) |
179 | 155, 178 | mpbird 246 |
. . . 4
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))𝑧 ≤ 𝑥) |
180 | | supxrre 12029 |
. . . 4
⊢ ((ran
seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ⊆ ℝ ∧ ran seq1( + ,
((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))) ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ ran seq1( + , ((abs
∘ − ) ∘ (𝐾 ∘ 𝐻)))𝑧 ≤ 𝑥) → sup(ran seq1( + , ((abs ∘
− ) ∘ (𝐾
∘ 𝐻))),
ℝ*, < ) = sup(ran seq1( + , ((abs ∘ − ) ∘
(𝐾 ∘ 𝐻))), ℝ, <
)) |
181 | 165, 174,
179, 180 | syl3anc 1318 |
. . 3
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ*, < ) = sup(ran
seq1( + , ((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ, < )) |
182 | 181, 132 | eqtr3d 2646 |
. 2
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → sup(ran seq1( + ,
((abs ∘ − ) ∘ (𝐾 ∘ 𝐻))), ℝ, < ) =
(vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴))) |
183 | 156, 163,
182 | 3brtr3d 4614 |
1
⊢ ((𝜑 ∧ 𝐽 ∈ ℕ) → seq1( + , (vol*
∘ 𝐻)) ⇝
(vol*‘(((,)‘(𝐺‘𝐽)) ∩ 𝐴))) |