Step | Hyp | Ref
| Expression |
1 | | difss 4065 |
. . . 4
⊢ (𝐸 ∖ ∪ ran 𝐹) ⊆ 𝐸 |
2 | | voliunlem1.7 |
. . . 4
⊢ (𝜑 → 𝐸 ⊆ ℝ) |
3 | | voliunlem1.8 |
. . . . 5
⊢ (𝜑 → (vol*‘𝐸) ∈
ℝ) |
4 | 3 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘𝐸) ∈
ℝ) |
5 | | ovolsscl 24660 |
. . . 4
⊢ (((𝐸 ∖ ∪ ran 𝐹) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∖
∪ ran 𝐹)) ∈ ℝ) |
6 | 1, 2, 4, 5 | mp3an2ani 1467 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∖ ∪ ran 𝐹)) ∈ ℝ) |
7 | | difss 4065 |
. . . 4
⊢ (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸 |
8 | | ovolsscl 24660 |
. . . 4
⊢ (((𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∖
∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ) |
9 | 7, 2, 4, 8 | mp3an2ani 1467 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ) |
10 | | inss1 4162 |
. . . 4
⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸 |
11 | | ovolsscl 24660 |
. . . 4
⊢ (((𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ) |
12 | 10, 2, 4, 11 | mp3an2ani 1467 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℝ) |
13 | | elfznn 13295 |
. . . . . . . . 9
⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ∈ ℕ) |
14 | | voliunlem.3 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐹:ℕ⟶dom vol) |
15 | 14 | ffnd 6593 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 Fn ℕ) |
16 | | fnfvelrn 6950 |
. . . . . . . . . . 11
⊢ ((𝐹 Fn ℕ ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹) |
17 | 15, 16 | sylan 580 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ran 𝐹) |
18 | | elssuni 4871 |
. . . . . . . . . 10
⊢ ((𝐹‘𝑛) ∈ ran 𝐹 → (𝐹‘𝑛) ⊆ ∪ ran
𝐹) |
19 | 17, 18 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ⊆ ∪ ran
𝐹) |
20 | 13, 19 | sylan2 593 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑘)) → (𝐹‘𝑛) ⊆ ∪ ran
𝐹) |
21 | 20 | ralrimiva 3108 |
. . . . . . 7
⊢ (𝜑 → ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran
𝐹) |
22 | 21 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran
𝐹) |
23 | | iunss 4974 |
. . . . . 6
⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran
𝐹 ↔ ∀𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran
𝐹) |
24 | 22, 23 | sylibr 233 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ⊆ ∪ ran
𝐹) |
25 | 24 | sscond 4075 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∖ ∪ ran
𝐹) ⊆ (𝐸 ∖ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) |
26 | 2 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐸 ⊆ ℝ) |
27 | 7, 26 | sstrid 3931 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ ℝ) |
28 | | ovolss 24659 |
. . . 4
⊢ (((𝐸 ∖ ∪ ran 𝐹) ⊆ (𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ∧ (𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) ⊆ ℝ) → (vol*‘(𝐸 ∖ ∪ ran 𝐹)) ≤ (vol*‘(𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) |
29 | 25, 27, 28 | syl2anc 584 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∖ ∪ ran 𝐹)) ≤ (vol*‘(𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) |
30 | 6, 9, 12, 29 | leadd2dd 11600 |
. 2
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪ ran
𝐹))) ≤
((vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))) |
31 | | oveq2 7275 |
. . . . . . . . . . 11
⊢ (𝑧 = 1 → (1...𝑧) = (1...1)) |
32 | 31 | iuneq1d 4951 |
. . . . . . . . . 10
⊢ (𝑧 = 1 → ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛)) |
33 | 32 | eleq1d 2823 |
. . . . . . . . 9
⊢ (𝑧 = 1 → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol)) |
34 | 32 | ineq2d 4146 |
. . . . . . . . . . 11
⊢ (𝑧 = 1 → (𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪
𝑛 ∈ (1...1)(𝐹‘𝑛))) |
35 | 34 | fveq2d 6770 |
. . . . . . . . . 10
⊢ (𝑧 = 1 → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...1)(𝐹‘𝑛)))) |
36 | | fveq2 6766 |
. . . . . . . . . 10
⊢ (𝑧 = 1 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘1)) |
37 | 35, 36 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑧 = 1 → ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1))) |
38 | 33, 37 | anbi12d 631 |
. . . . . . . 8
⊢ (𝑧 = 1 → ((∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ (∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1)))) |
39 | 38 | imbi2d 341 |
. . . . . . 7
⊢ (𝑧 = 1 → ((𝜑 → (∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → (∪
𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1))))) |
40 | | oveq2 7275 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑘 → (1...𝑧) = (1...𝑘)) |
41 | 40 | iuneq1d 4951 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑘 → ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) |
42 | 41 | eleq1d 2823 |
. . . . . . . . 9
⊢ (𝑧 = 𝑘 → (∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol)) |
43 | 41 | ineq2d 4146 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑘 → (𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) |
44 | 43 | fveq2d 6770 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑘 → (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) |
45 | | fveq2 6766 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑘 → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘𝑘)) |
46 | 44, 45 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑧 = 𝑘 → ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))) |
47 | 42, 46 | anbi12d 631 |
. . . . . . . 8
⊢ (𝑧 = 𝑘 → ((∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)))) |
48 | 47 | imbi2d 341 |
. . . . . . 7
⊢ (𝑧 = 𝑘 → ((𝜑 → (∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → (∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))))) |
49 | | oveq2 7275 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝑘 + 1) → (1...𝑧) = (1...(𝑘 + 1))) |
50 | 49 | iuneq1d 4951 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑘 + 1) → ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) = ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) |
51 | 50 | eleq1d 2823 |
. . . . . . . . 9
⊢ (𝑧 = (𝑘 + 1) → (∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ↔ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol)) |
52 | 50 | ineq2d 4146 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝑘 + 1) → (𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛)) = (𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) |
53 | 52 | fveq2d 6770 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑘 + 1) → (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)))) |
54 | | fveq2 6766 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑘 + 1) → (seq1( + , 𝐻)‘𝑧) = (seq1( + , 𝐻)‘(𝑘 + 1))) |
55 | 53, 54 | eqeq12d 2754 |
. . . . . . . . 9
⊢ (𝑧 = (𝑘 + 1) → ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧) ↔ (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))) |
56 | 51, 55 | anbi12d 631 |
. . . . . . . 8
⊢ (𝑧 = (𝑘 + 1) → ((∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧)) ↔ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))) |
57 | 56 | imbi2d 341 |
. . . . . . 7
⊢ (𝑧 = (𝑘 + 1) → ((𝜑 → (∪
𝑛 ∈ (1...𝑧)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑧)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑧))) ↔ (𝜑 → (∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))))) |
58 | | 1z 12360 |
. . . . . . . . . . 11
⊢ 1 ∈
ℤ |
59 | | fzsn 13308 |
. . . . . . . . . . 11
⊢ (1 ∈
ℤ → (1...1) = {1}) |
60 | | iuneq1 4940 |
. . . . . . . . . . 11
⊢ ((1...1)
= {1} → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = ∪ 𝑛 ∈ {1} (𝐹‘𝑛)) |
61 | 58, 59, 60 | mp2b 10 |
. . . . . . . . . 10
⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = ∪ 𝑛 ∈ {1} (𝐹‘𝑛) |
62 | | 1ex 10981 |
. . . . . . . . . . 11
⊢ 1 ∈
V |
63 | | fveq2 6766 |
. . . . . . . . . . 11
⊢ (𝑛 = 1 → (𝐹‘𝑛) = (𝐹‘1)) |
64 | 62, 63 | iunxsn 5019 |
. . . . . . . . . 10
⊢ ∪ 𝑛 ∈ {1} (𝐹‘𝑛) = (𝐹‘1) |
65 | 61, 64 | eqtri 2766 |
. . . . . . . . 9
⊢ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) = (𝐹‘1) |
66 | | 1nn 11994 |
. . . . . . . . . 10
⊢ 1 ∈
ℕ |
67 | | ffvelrn 6951 |
. . . . . . . . . 10
⊢ ((𝐹:ℕ⟶dom vol ∧ 1
∈ ℕ) → (𝐹‘1) ∈ dom vol) |
68 | 14, 66, 67 | sylancl 586 |
. . . . . . . . 9
⊢ (𝜑 → (𝐹‘1) ∈ dom vol) |
69 | 65, 68 | eqeltrid 2843 |
. . . . . . . 8
⊢ (𝜑 → ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol) |
70 | 63 | ineq2d 4146 |
. . . . . . . . . . . 12
⊢ (𝑛 = 1 → (𝐸 ∩ (𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘1))) |
71 | 70 | fveq2d 6770 |
. . . . . . . . . . 11
⊢ (𝑛 = 1 → (vol*‘(𝐸 ∩ (𝐹‘𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1)))) |
72 | | voliunlem1.6 |
. . . . . . . . . . 11
⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝐸 ∩ (𝐹‘𝑛)))) |
73 | | fvex 6779 |
. . . . . . . . . . 11
⊢
(vol*‘(𝐸 ∩
(𝐹‘1))) ∈
V |
74 | 71, 72, 73 | fvmpt 6867 |
. . . . . . . . . 10
⊢ (1 ∈
ℕ → (𝐻‘1)
= (vol*‘(𝐸 ∩
(𝐹‘1)))) |
75 | 66, 74 | ax-mp 5 |
. . . . . . . . 9
⊢ (𝐻‘1) = (vol*‘(𝐸 ∩ (𝐹‘1))) |
76 | | seq1 13744 |
. . . . . . . . . 10
⊢ (1 ∈
ℤ → (seq1( + , 𝐻)‘1) = (𝐻‘1)) |
77 | 58, 76 | ax-mp 5 |
. . . . . . . . 9
⊢ (seq1( +
, 𝐻)‘1) = (𝐻‘1) |
78 | 65 | ineq2i 4143 |
. . . . . . . . . 10
⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘1)) |
79 | 78 | fveq2i 6769 |
. . . . . . . . 9
⊢
(vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘1))) |
80 | 75, 77, 79 | 3eqtr4ri 2777 |
. . . . . . . 8
⊢
(vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1) |
81 | 69, 80 | jctir 521 |
. . . . . . 7
⊢ (𝜑 → (∪ 𝑛 ∈ (1...1)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...1)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘1))) |
82 | | peano2nn 11995 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
83 | | ffvelrn 6951 |
. . . . . . . . . . . . 13
⊢ ((𝐹:ℕ⟶dom vol ∧
(𝑘 + 1) ∈ ℕ)
→ (𝐹‘(𝑘 + 1)) ∈ dom
vol) |
84 | 14, 82, 83 | syl2an 596 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ∈ dom vol) |
85 | | unmbl 24711 |
. . . . . . . . . . . . 13
⊢
((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (𝐹‘(𝑘 + 1)) ∈ dom vol) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol) |
86 | 85 | ex 413 |
. . . . . . . . . . . 12
⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol → ((𝐹‘(𝑘 + 1)) ∈ dom vol → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)) |
87 | 84, 86 | syl5com 31 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)) |
88 | | simpr 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ) |
89 | | nnuz 12631 |
. . . . . . . . . . . . . . 15
⊢ ℕ =
(ℤ≥‘1) |
90 | 88, 89 | eleqtrdi 2849 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈
(ℤ≥‘1)) |
91 | | fzsuc 13313 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈
(ℤ≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)})) |
92 | | iuneq1 4940 |
. . . . . . . . . . . . . 14
⊢
((1...(𝑘 + 1)) =
((1...𝑘) ∪ {(𝑘 + 1)}) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛)) |
93 | 90, 91, 92 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛)) |
94 | | iunxun 5022 |
. . . . . . . . . . . . . 14
⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪
𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛)) |
95 | | ovex 7300 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 + 1) ∈ V |
96 | | fveq2 6766 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = (𝑘 + 1) → (𝐹‘𝑛) = (𝐹‘(𝑘 + 1))) |
97 | 95, 96 | iunxsn 5019 |
. . . . . . . . . . . . . . 15
⊢ ∪ 𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛) = (𝐹‘(𝑘 + 1)) |
98 | 97 | uneq2i 4093 |
. . . . . . . . . . . . . 14
⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ ∪
𝑛 ∈ {(𝑘 + 1)} (𝐹‘𝑛)) = (∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) |
99 | 94, 98 | eqtri 2766 |
. . . . . . . . . . . . 13
⊢ ∪ 𝑛 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})(𝐹‘𝑛) = (∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) |
100 | 93, 99 | eqtrdi 2794 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) = (∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1)))) |
101 | 100 | eleq1d 2823 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ↔ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) ∈ dom vol)) |
102 | 87, 101 | sylibrd 258 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol → ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol)) |
103 | | oveq1 7274 |
. . . . . . . . . . 11
⊢
((vol*‘(𝐸
∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘) → ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))) |
104 | | inss1 4162 |
. . . . . . . . . . . . . . 15
⊢ (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ 𝐸 |
105 | 104, 26 | sstrid 3931 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ ℝ) |
106 | | ovolsscl 24660 |
. . . . . . . . . . . . . . 15
⊢ (((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) ∈ ℝ) |
107 | 104, 2, 4, 106 | mp3an2ani 1467 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) ∈ ℝ) |
108 | | mblsplit 24706 |
. . . . . . . . . . . . . 14
⊢ (((𝐹‘(𝑘 + 1)) ∈ dom vol ∧ (𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ⊆ ℝ ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) ∈ ℝ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = ((vol*‘((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1)))))) |
109 | 84, 105, 107, 108 | syl3anc 1370 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = ((vol*‘((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1)))))) |
110 | | in32 4155 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1))) = ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) |
111 | | inss2 4163 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ (𝐹‘(𝑘 + 1)) |
112 | 82 | adantl 482 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈ ℕ) |
113 | 112, 89 | eleqtrdi 2849 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝑘 + 1) ∈
(ℤ≥‘1)) |
114 | | eluzfz2 13274 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 + 1) ∈
(ℤ≥‘1) → (𝑘 + 1) ∈ (1...(𝑘 + 1))) |
115 | 96 | ssiun2s 4977 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 + 1) ∈ (1...(𝑘 + 1)) → (𝐹‘(𝑘 + 1)) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) |
116 | 113, 114,
115 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) |
117 | 111, 116 | sstrid 3931 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) |
118 | | df-ss 3903 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ↔ ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1)))) |
119 | 117, 118 | sylib 217 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ (𝐹‘(𝑘 + 1))) ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1)))) |
120 | 110, 119 | eqtrid 2790 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ (𝐹‘(𝑘 + 1)))) |
121 | 120 | fveq2d 6770 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) |
122 | | indif2 4204 |
. . . . . . . . . . . . . . . 16
⊢ (𝐸 ∩ (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1)))) = ((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))) |
123 | | uncom 4086 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∪ (𝐹‘(𝑘 + 1))) = ((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) |
124 | 100, 123 | eqtr2di 2795 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) |
125 | | voliunlem.5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖)) |
126 | 125 | ad2antrr 723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → Disj 𝑖 ∈ ℕ (𝐹‘𝑖)) |
127 | 112 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ∈ ℕ) |
128 | 13 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℕ) |
129 | 128 | nnred 11998 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ∈ ℝ) |
130 | | elfzle2 13270 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 ∈ (1...𝑘) → 𝑛 ≤ 𝑘) |
131 | 130 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 ≤ 𝑘) |
132 | 88 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑘 ∈ ℕ) |
133 | | nnleltp1 12385 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑛 ∈ ℕ ∧ 𝑘 ∈ ℕ) → (𝑛 ≤ 𝑘 ↔ 𝑛 < (𝑘 + 1))) |
134 | 128, 132,
133 | syl2anc 584 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑛 ≤ 𝑘 ↔ 𝑛 < (𝑘 + 1))) |
135 | 131, 134 | mpbid 231 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → 𝑛 < (𝑘 + 1)) |
136 | 129, 135 | gtned 11120 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → (𝑘 + 1) ≠ 𝑛) |
137 | | fveq2 6766 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 = (𝑘 + 1) → (𝐹‘𝑖) = (𝐹‘(𝑘 + 1))) |
138 | | fveq2 6766 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑖 = 𝑛 → (𝐹‘𝑖) = (𝐹‘𝑛)) |
139 | 137, 138 | disji2 5055 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((Disj 𝑖
∈ ℕ (𝐹‘𝑖) ∧ ((𝑘 + 1) ∈ ℕ ∧ 𝑛 ∈ ℕ) ∧ (𝑘 + 1) ≠ 𝑛) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∅) |
140 | 126, 127,
128, 136, 139 | syl121anc 1374 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ (1...𝑘)) → ((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∅) |
141 | 140 | iuneq2dv 4948 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ∪
𝑛 ∈ (1...𝑘)∅) |
142 | | iunin2 4999 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ∪ 𝑛 ∈ (1...𝑘)((𝐹‘(𝑘 + 1)) ∩ (𝐹‘𝑛)) = ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) |
143 | | iun0 4990 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ∪ 𝑛 ∈ (1...𝑘)∅ = ∅ |
144 | 141, 142,
143 | 3eqtr3g 2801 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∅) |
145 | | uneqdifeq 4423 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐹‘(𝑘 + 1)) ⊆ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∧ ((𝐹‘(𝑘 + 1)) ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∅) → (((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ↔ (∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1))) = ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) |
146 | 116, 144,
145 | syl2anc 584 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (((𝐹‘(𝑘 + 1)) ∪ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) = ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ↔ (∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1))) = ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) |
147 | 124, 146 | mpbid 231 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1))) = ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)) |
148 | 147 | ineq2d 4146 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐸 ∩ (∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∖ (𝐹‘(𝑘 + 1)))) = (𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) |
149 | 122, 148 | eqtr3id 2792 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))) = (𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) |
150 | 149 | fveq2d 6770 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1)))) = (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) |
151 | 121, 150 | oveq12d 7285 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol*‘((𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘((𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛)) ∖ (𝐹‘(𝑘 + 1))))) = ((vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))) |
152 | | inss1 4162 |
. . . . . . . . . . . . . . . 16
⊢ (𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸 |
153 | | ovolsscl 24660 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐸 ∩ (𝐹‘(𝑘 + 1))) ⊆ 𝐸 ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ) |
154 | 152, 2, 4, 153 | mp3an2ani 1467 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℝ) |
155 | 154 | recnd 11013 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) ∈ ℂ) |
156 | 12 | recnd 11013 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) ∈ ℂ) |
157 | 155, 156 | addcomd 11187 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))) + (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) = ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))) |
158 | 109, 151,
157 | 3eqtrd 2782 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))) |
159 | | seqp1 13746 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈
(ℤ≥‘1) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1)))) |
160 | 90, 159 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1)))) |
161 | 96 | ineq2d 4146 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = (𝑘 + 1) → (𝐸 ∩ (𝐹‘𝑛)) = (𝐸 ∩ (𝐹‘(𝑘 + 1)))) |
162 | 161 | fveq2d 6770 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = (𝑘 + 1) → (vol*‘(𝐸 ∩ (𝐹‘𝑛))) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) |
163 | | fvex 6779 |
. . . . . . . . . . . . . . . 16
⊢
(vol*‘(𝐸 ∩
(𝐹‘(𝑘 + 1)))) ∈
V |
164 | 162, 72, 163 | fvmpt 6867 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 + 1) ∈ ℕ →
(𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) |
165 | 112, 164 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘(𝑘 + 1)) = (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) |
166 | 165 | oveq2d 7283 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (𝐻‘(𝑘 + 1))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))) |
167 | 160, 166 | eqtrd 2778 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘(𝑘 + 1)) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1)))))) |
168 | 158, 167 | eqeq12d 2754 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)) ↔ ((vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))) = ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∩ (𝐹‘(𝑘 + 1))))))) |
169 | 103, 168 | syl5ibr 245 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘) → (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))) |
170 | 102, 169 | anim12d 609 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1))))) |
171 | 170 | expcom 414 |
. . . . . . . 8
⊢ (𝑘 ∈ ℕ → (𝜑 → ((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)) → (∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))))) |
172 | 171 | a2d 29 |
. . . . . . 7
⊢ (𝑘 ∈ ℕ → ((𝜑 → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))) → (𝜑 → (∪
𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...(𝑘 + 1))(𝐹‘𝑛))) = (seq1( + , 𝐻)‘(𝑘 + 1)))))) |
173 | 39, 48, 57, 48, 81, 172 | nnind 12001 |
. . . . . 6
⊢ (𝑘 ∈ ℕ → (𝜑 → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)))) |
174 | 173 | impcom 408 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘))) |
175 | 174 | simprd 496 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) = (seq1( + , 𝐻)‘𝑘)) |
176 | 175 | eqcomd 2744 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (seq1( + , 𝐻)‘𝑘) = (vol*‘(𝐸 ∩ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛)))) |
177 | 176 | oveq1d 7282 |
. 2
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∖ ∪ ran
𝐹))) = ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪ ran
𝐹)))) |
178 | 174 | simpld 495 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol) |
179 | | mblsplit 24706 |
. . 3
⊢
((∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛) ∈ dom vol ∧ 𝐸 ⊆ ℝ ∧ (vol*‘𝐸) ∈ ℝ) →
(vol*‘𝐸) =
((vol*‘(𝐸 ∩
∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))) |
180 | 178, 26, 4, 179 | syl3anc 1370 |
. 2
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (vol*‘𝐸) = ((vol*‘(𝐸 ∩ ∪ 𝑛 ∈ (1...𝑘)(𝐹‘𝑛))) + (vol*‘(𝐸 ∖ ∪
𝑛 ∈ (1...𝑘)(𝐹‘𝑛))))) |
181 | 30, 177, 180 | 3brtr4d 5105 |
1
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝐸 ∖ ∪ ran
𝐹))) ≤ (vol*‘𝐸)) |