Step | Hyp | Ref
| Expression |
1 | | voliunlem.3 |
. . . . 5
⊢ (𝜑 → 𝐹:ℕ⟶dom vol) |
2 | | frn 6053 |
. . . . 5
⊢ (𝐹:ℕ⟶dom vol →
ran 𝐹 ⊆ dom
vol) |
3 | 1, 2 | syl 17 |
. . . 4
⊢ (𝜑 → ran 𝐹 ⊆ dom vol) |
4 | | mblss 23299 |
. . . . . 6
⊢ (𝑥 ∈ dom vol → 𝑥 ⊆
ℝ) |
5 | | selpw 4165 |
. . . . . 6
⊢ (𝑥 ∈ 𝒫 ℝ ↔
𝑥 ⊆
ℝ) |
6 | 4, 5 | sylibr 224 |
. . . . 5
⊢ (𝑥 ∈ dom vol → 𝑥 ∈ 𝒫
ℝ) |
7 | 6 | ssriv 3607 |
. . . 4
⊢ dom vol
⊆ 𝒫 ℝ |
8 | 3, 7 | syl6ss 3615 |
. . 3
⊢ (𝜑 → ran 𝐹 ⊆ 𝒫 ℝ) |
9 | | sspwuni 4611 |
. . 3
⊢ (ran
𝐹 ⊆ 𝒫 ℝ
↔ ∪ ran 𝐹 ⊆ ℝ) |
10 | 8, 9 | sylib 208 |
. 2
⊢ (𝜑 → ∪ ran 𝐹 ⊆ ℝ) |
11 | | elpwi 4168 |
. . . 4
⊢ (𝑥 ∈ 𝒫 ℝ →
𝑥 ⊆
ℝ) |
12 | | inundif 4046 |
. . . . . . . 8
⊢ ((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran
𝐹)) = 𝑥 |
13 | 12 | fveq2i 6194 |
. . . . . . 7
⊢
(vol*‘((𝑥
∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran
𝐹))) = (vol*‘𝑥) |
14 | | inss1 3833 |
. . . . . . . . 9
⊢ (𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥 |
15 | | simp2 1062 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝑥 ⊆
ℝ) |
16 | 14, 15 | syl5ss 3614 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ ran 𝐹) ⊆ ℝ) |
17 | | ovolsscl 23254 |
. . . . . . . . . 10
⊢ (((𝑥 ∩ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ) |
18 | 14, 17 | mp3an1 1411 |
. . . . . . . . 9
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈
ℝ) |
19 | 18 | 3adant1 1079 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ) |
20 | | difss 3737 |
. . . . . . . . 9
⊢ (𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥 |
21 | 20, 15 | syl5ss 3614 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∖ ∪ ran 𝐹) ⊆ ℝ) |
22 | | ovolsscl 23254 |
. . . . . . . . . 10
⊢ (((𝑥 ∖ ∪ ran 𝐹) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∖
∪ ran 𝐹)) ∈ ℝ) |
23 | 20, 22 | mp3an1 1411 |
. . . . . . . . 9
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∖ ∪ ran
𝐹)) ∈
ℝ) |
24 | 23 | 3adant1 1079 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∖
∪ ran 𝐹)) ∈ ℝ) |
25 | | ovolun 23267 |
. . . . . . . 8
⊢ ((((𝑥 ∩ ∪ ran 𝐹) ⊆ ℝ ∧ (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈ ℝ) ∧ ((𝑥 ∖ ∪ ran 𝐹) ⊆ ℝ ∧ (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ)) →
(vol*‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran
𝐹))) ≤
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
26 | 16, 19, 21, 24, 25 | syl22anc 1327 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘((𝑥 ∩ ∪ ran 𝐹) ∪ (𝑥 ∖ ∪ ran
𝐹))) ≤
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
27 | 13, 26 | syl5eqbrr 4689 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘𝑥) ≤
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
28 | 19 | rexrd 10089 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ ∪ ran 𝐹)) ∈
ℝ*) |
29 | | nnuz 11723 |
. . . . . . . . . . . 12
⊢ ℕ =
(ℤ≥‘1) |
30 | | 1zzd 11408 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 1
∈ ℤ) |
31 | | fveq2 6191 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘)) |
32 | 31 | ineq2d 3814 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ (𝐹‘𝑘))) |
33 | 32 | fveq2d 6195 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ (𝐹‘𝑘)))) |
34 | | voliunlem.6 |
. . . . . . . . . . . . . . 15
⊢ 𝐻 = (𝑛 ∈ ℕ ↦ (vol*‘(𝑥 ∩ (𝐹‘𝑛)))) |
35 | | fvex 6201 |
. . . . . . . . . . . . . . 15
⊢
(vol*‘(𝑥 ∩
(𝐹‘𝑘))) ∈ V |
36 | 33, 34, 35 | fvmpt 6282 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ → (𝐻‘𝑘) = (vol*‘(𝑥 ∩ (𝐹‘𝑘)))) |
37 | 36 | adantl 482 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (vol*‘(𝑥 ∩ (𝐹‘𝑘)))) |
38 | | inss1 3833 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥 |
39 | | ovolsscl 23254 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 ∩ (𝐹‘𝑘)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ) |
40 | 38, 39 | mp3an1 1411 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ) |
41 | 40 | 3adant1 1079 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ) |
42 | 41 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑘))) ∈ ℝ) |
43 | 37, 42 | eqeltrd 2701 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) ∈ ℝ) |
44 | 29, 30, 43 | serfre 12830 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → seq1( +
, 𝐻):ℕ⟶ℝ) |
45 | | frn 6053 |
. . . . . . . . . . 11
⊢ (seq1( +
, 𝐻):ℕ⟶ℝ
→ ran seq1( + , 𝐻)
⊆ ℝ) |
46 | 44, 45 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran
seq1( + , 𝐻) ⊆
ℝ) |
47 | | ressxr 10083 |
. . . . . . . . . 10
⊢ ℝ
⊆ ℝ* |
48 | 46, 47 | syl6ss 3615 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → ran
seq1( + , 𝐻) ⊆
ℝ*) |
49 | | supxrcl 12145 |
. . . . . . . . 9
⊢ (ran
seq1( + , 𝐻) ⊆
ℝ* → sup(ran seq1( + , 𝐻), ℝ*, < ) ∈
ℝ*) |
50 | 48, 49 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran
seq1( + , 𝐻),
ℝ*, < ) ∈ ℝ*) |
51 | | simp3 1063 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘𝑥) ∈
ℝ) |
52 | 51, 24 | resubcld 10458 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
((vol*‘𝑥) −
(vol*‘(𝑥 ∖
∪ ran 𝐹))) ∈ ℝ) |
53 | 52 | rexrd 10089 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
((vol*‘𝑥) −
(vol*‘(𝑥 ∖
∪ ran 𝐹))) ∈
ℝ*) |
54 | | iunin2 4584 |
. . . . . . . . . . 11
⊢ ∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪
𝑛 ∈ ℕ (𝐹‘𝑛)) |
55 | | ffn 6045 |
. . . . . . . . . . . . . 14
⊢ (𝐹:ℕ⟶dom vol →
𝐹 Fn
ℕ) |
56 | | fniunfv 6505 |
. . . . . . . . . . . . . 14
⊢ (𝐹 Fn ℕ → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹) |
57 | 1, 55, 56 | 3syl 18 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹) |
58 | 57 | 3ad2ant1 1082 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ ran 𝐹) |
59 | 58 | ineq2d 3814 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ ∪ 𝑛 ∈ ℕ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹)) |
60 | 54, 59 | syl5eq 2668 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛)) = (𝑥 ∩ ∪ ran 𝐹)) |
61 | 60 | fveq2d 6195 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) = (vol*‘(𝑥 ∩ ∪ ran 𝐹))) |
62 | | eqid 2622 |
. . . . . . . . . 10
⊢ seq1( + ,
𝐻) = seq1( + , 𝐻) |
63 | | inss1 3833 |
. . . . . . . . . . . 12
⊢ (𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥 |
64 | 63, 15 | syl5ss 3614 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → (𝑥 ∩ (𝐹‘𝑛)) ⊆ ℝ) |
65 | 64 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝑥 ∩ (𝐹‘𝑛)) ⊆ ℝ) |
66 | | ovolsscl 23254 |
. . . . . . . . . . . . 13
⊢ (((𝑥 ∩ (𝐹‘𝑛)) ⊆ 𝑥 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ) |
67 | 63, 66 | mp3an1 1411 |
. . . . . . . . . . . 12
⊢ ((𝑥 ⊆ ℝ ∧
(vol*‘𝑥) ∈
ℝ) → (vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ) |
68 | 67 | 3adant1 1079 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ) |
69 | 68 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑛 ∈ ℕ) →
(vol*‘(𝑥 ∩ (𝐹‘𝑛))) ∈ ℝ) |
70 | 62, 34, 65, 69 | ovoliun 23273 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘∪ 𝑛 ∈ ℕ (𝑥 ∩ (𝐹‘𝑛))) ≤ sup(ran seq1( + , 𝐻), ℝ*, <
)) |
71 | 61, 70 | eqbrtrrd 4677 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ sup(ran seq1( + , 𝐻), ℝ*, <
)) |
72 | 1 | 3ad2ant1 1082 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → 𝐹:ℕ⟶dom
vol) |
73 | | voliunlem.5 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → Disj 𝑖 ∈ ℕ (𝐹‘𝑖)) |
74 | 73 | 3ad2ant1 1082 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
Disj 𝑖 ∈
ℕ (𝐹‘𝑖)) |
75 | 72, 74, 34, 15, 51 | voliunlem1 23318 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → ((seq1(
+ , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥)) |
76 | 44 | ffvelrnda 6359 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( +
, 𝐻)‘𝑘) ∈
ℝ) |
77 | 24 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) →
(vol*‘(𝑥 ∖
∪ ran 𝐹)) ∈ ℝ) |
78 | | simpl3 1066 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) →
(vol*‘𝑥) ∈
ℝ) |
79 | | leaddsub 10504 |
. . . . . . . . . . . . 13
⊢ (((seq1(
+ , 𝐻)‘𝑘) ∈ ℝ ∧
(vol*‘(𝑥 ∖
∪ ran 𝐹)) ∈ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(((seq1( + , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
80 | 76, 77, 78, 79 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (((seq1(
+ , 𝐻)‘𝑘) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))) ≤ (vol*‘𝑥) ↔ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
81 | 75, 80 | mpbid 222 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (seq1( +
, 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))) |
82 | 81 | ralrimiva 2966 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
∀𝑘 ∈ ℕ
(seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran 𝐹)))) |
83 | | ffn 6045 |
. . . . . . . . . . 11
⊢ (seq1( +
, 𝐻):ℕ⟶ℝ
→ seq1( + , 𝐻) Fn
ℕ) |
84 | | breq1 4656 |
. . . . . . . . . . . 12
⊢ (𝑧 = (seq1( + , 𝐻)‘𝑘) → (𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ↔ (seq1( + ,
𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
85 | 84 | ralrn 6362 |
. . . . . . . . . . 11
⊢ (seq1( +
, 𝐻) Fn ℕ →
(∀𝑧 ∈ ran seq1(
+ , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ↔ ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
86 | 44, 83, 85 | 3syl 18 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(∀𝑧 ∈ ran seq1(
+ , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ↔ ∀𝑘 ∈ ℕ (seq1( + , 𝐻)‘𝑘) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
87 | 82, 86 | mpbird 247 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
∀𝑧 ∈ ran seq1(
+ , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
88 | | supxrleub 12156 |
. . . . . . . . . 10
⊢ ((ran
seq1( + , 𝐻) ⊆
ℝ* ∧ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ∈
ℝ*) → (sup(ran seq1( + , 𝐻), ℝ*, < ) ≤
((vol*‘𝑥) −
(vol*‘(𝑥 ∖
∪ ran 𝐹))) ↔ ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
89 | 48, 53, 88 | syl2anc 693 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(sup(ran seq1( + , 𝐻),
ℝ*, < ) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ↔ ∀𝑧 ∈ ran seq1( + , 𝐻)𝑧 ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
90 | 87, 89 | mpbird 247 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) → sup(ran
seq1( + , 𝐻),
ℝ*, < ) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
91 | 28, 50, 53, 71, 90 | xrletrd 11993 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
92 | | leaddsub 10504 |
. . . . . . . 8
⊢
(((vol*‘(𝑥
∩ ∪ ran 𝐹)) ∈ ℝ ∧ (vol*‘(𝑥 ∖ ∪ ran 𝐹)) ∈ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(((vol*‘(𝑥 ∩
∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ≤ (vol*‘𝑥) ↔ (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
93 | 19, 24, 51, 92 | syl3anc 1326 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(((vol*‘(𝑥 ∩
∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ≤ (vol*‘𝑥) ↔ (vol*‘(𝑥 ∩ ∪ ran 𝐹)) ≤ ((vol*‘𝑥) − (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
94 | 91, 93 | mpbird 247 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ≤ (vol*‘𝑥)) |
95 | 19, 24 | readdcld 10069 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ∈
ℝ) |
96 | 51, 95 | letri3d 10179 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
((vol*‘𝑥) =
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ↔
((vol*‘𝑥) ≤
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ∧
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))) ≤ (vol*‘𝑥)))) |
97 | 27, 94, 96 | mpbir2and 957 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ ∧ (vol*‘𝑥) ∈ ℝ) →
(vol*‘𝑥) =
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹)))) |
98 | 97 | 3expia 1267 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ⊆ ℝ) → ((vol*‘𝑥) ∈ ℝ →
(vol*‘𝑥) =
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
99 | 11, 98 | sylan2 491 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ℝ) →
((vol*‘𝑥) ∈
ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹))))) |
100 | 99 | ralrimiva 2966 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ 𝒫 ℝ((vol*‘𝑥) ∈ ℝ →
(vol*‘𝑥) =
((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran
𝐹))))) |
101 | | ismbl 23294 |
. 2
⊢ (∪ ran 𝐹 ∈ dom vol ↔ (∪ ran 𝐹 ⊆ ℝ ∧ ∀𝑥 ∈ 𝒫
ℝ((vol*‘𝑥)
∈ ℝ → (vol*‘𝑥) = ((vol*‘(𝑥 ∩ ∪ ran 𝐹)) + (vol*‘(𝑥 ∖ ∪ ran 𝐹)))))) |
102 | 10, 100, 101 | sylanbrc 698 |
1
⊢ (𝜑 → ∪ ran 𝐹 ∈ dom vol) |