Step | Hyp | Ref
| Expression |
1 | | ltso 10913 |
. . 3
⊢ < Or
ℝ |
2 | 1 | a1i 11 |
. 2
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))) → < Or
ℝ) |
3 | | difss 4046 |
. . . 4
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 |
4 | | ovolsscl 24383 |
. . . 4
⊢ (((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘(𝐴 ∖
𝐵)) ∈
ℝ) |
5 | 3, 4 | mp3an1 1450 |
. . 3
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
6 | 5 | 3ad2ant1 1135 |
. 2
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))) →
(vol*‘(𝐴 ∖
𝐵)) ∈
ℝ) |
7 | | vex 3412 |
. . . . . 6
⊢ 𝑢 ∈ V |
8 | | eqeq1 2741 |
. . . . . . . 8
⊢ (𝑦 = 𝑢 → (𝑦 = (vol‘𝑏) ↔ 𝑢 = (vol‘𝑏))) |
9 | 8 | anbi2d 632 |
. . . . . . 7
⊢ (𝑦 = 𝑢 → ((𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)))) |
10 | 9 | rexbidv 3216 |
. . . . . 6
⊢ (𝑦 = 𝑢 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)))) |
11 | 7, 10 | elab 3587 |
. . . . 5
⊢ (𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))} ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏))) |
12 | | simprl 771 |
. . . . . . . . 9
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏))) → 𝑏 ⊆ (𝐴 ∖ 𝐵)) |
13 | | ssdifss 4050 |
. . . . . . . . 9
⊢ (𝐴 ⊆ ℝ → (𝐴 ∖ 𝐵) ⊆ ℝ) |
14 | | ovolss 24382 |
. . . . . . . . 9
⊢ ((𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (𝐴 ∖ 𝐵) ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘(𝐴 ∖ 𝐵))) |
15 | 12, 13, 14 | syl2anr 600 |
. . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)))) → (vol*‘𝑏) ≤ (vol*‘(𝐴 ∖ 𝐵))) |
16 | | uniretop 23660 |
. . . . . . . . . . . . 13
⊢ ℝ =
∪ (topGen‘ran (,)) |
17 | 16 | cldss 21926 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → 𝑏 ⊆ ℝ) |
18 | | ovolcl 24375 |
. . . . . . . . . . . 12
⊢ (𝑏 ⊆ ℝ →
(vol*‘𝑏) ∈
ℝ*) |
19 | 17, 18 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → (vol*‘𝑏) ∈
ℝ*) |
20 | | ovolcl 24375 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∖ 𝐵) ⊆ ℝ → (vol*‘(𝐴 ∖ 𝐵)) ∈
ℝ*) |
21 | 13, 20 | syl 17 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ ℝ →
(vol*‘(𝐴 ∖
𝐵)) ∈
ℝ*) |
22 | | xrlenlt 10898 |
. . . . . . . . . . 11
⊢
(((vol*‘𝑏)
∈ ℝ* ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ*) →
((vol*‘𝑏) ≤
(vol*‘(𝐴 ∖
𝐵)) ↔ ¬
(vol*‘(𝐴 ∖
𝐵)) < (vol*‘𝑏))) |
23 | 19, 21, 22 | syl2anr 600 |
. . . . . . . . . 10
⊢ ((𝐴 ⊆ ℝ ∧ 𝑏 ∈
(Clsd‘(topGen‘ran (,)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴 ∖ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∖ 𝐵)) < (vol*‘𝑏))) |
24 | 23 | adantrr 717 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴 ∖ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∖ 𝐵)) < (vol*‘𝑏))) |
25 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (vol‘𝑏) → 𝑢 = (vol‘𝑏)) |
26 | | dfss4 4173 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 ⊆ ℝ ↔ (ℝ
∖ (ℝ ∖ 𝑏)) = 𝑏) |
27 | 17, 26 | sylib 221 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖
𝑏)) = 𝑏) |
28 | | rembl 24437 |
. . . . . . . . . . . . . . . . 17
⊢ ℝ
∈ dom vol |
29 | 16 | cldopn 21928 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ (topGen‘ran
(,))) |
30 | | opnmbl 24499 |
. . . . . . . . . . . . . . . . . 18
⊢ ((ℝ
∖ 𝑏) ∈
(topGen‘ran (,)) → (ℝ ∖ 𝑏) ∈ dom vol) |
31 | 29, 30 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ dom vol) |
32 | | difmbl 24440 |
. . . . . . . . . . . . . . . . 17
⊢ ((ℝ
∈ dom vol ∧ (ℝ ∖ 𝑏) ∈ dom vol) → (ℝ ∖
(ℝ ∖ 𝑏)) ∈
dom vol) |
33 | 28, 31, 32 | sylancr 590 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖
𝑏)) ∈ dom
vol) |
34 | 27, 33 | eqeltrrd 2839 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → 𝑏 ∈ dom vol) |
35 | | mblvol 24427 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈ dom vol →
(vol‘𝑏) =
(vol*‘𝑏)) |
36 | 34, 35 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → (vol‘𝑏) = (vol*‘𝑏)) |
37 | 25, 36 | sylan9eqr 2800 |
. . . . . . . . . . . . 13
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → 𝑢 = (vol*‘𝑏)) |
38 | 37 | breq2d 5065 |
. . . . . . . . . . . 12
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → ((vol*‘(𝐴 ∖ 𝐵)) < 𝑢 ↔ (vol*‘(𝐴 ∖ 𝐵)) < (vol*‘𝑏))) |
39 | 38 | notbid 321 |
. . . . . . . . . . 11
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → (¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴 ∖ 𝐵)) < (vol*‘𝑏))) |
40 | 39 | adantrl 716 |
. . . . . . . . . 10
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏))) → (¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴 ∖ 𝐵)) < (vol*‘𝑏))) |
41 | 40 | adantl 485 |
. . . . . . . . 9
⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)))) → (¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴 ∖ 𝐵)) < (vol*‘𝑏))) |
42 | 24, 41 | bitr4d 285 |
. . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴 ∖ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢)) |
43 | 15, 42 | mpbid 235 |
. . . . . . 7
⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢) |
44 | 43 | rexlimdvaa 3204 |
. . . . . 6
⊢ (𝐴 ⊆ ℝ →
(∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)) → ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢)) |
45 | 44 | imp 410 |
. . . . 5
⊢ ((𝐴 ⊆ ℝ ∧
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏))) → ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢) |
46 | 11, 45 | sylan2b 597 |
. . . 4
⊢ ((𝐴 ⊆ ℝ ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢) |
47 | 46 | adantlr 715 |
. . 3
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ 𝑢 ∈
{𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢) |
48 | 47 | 3ad2antl1 1187 |
. 2
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))) ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢) |
49 | | simplr 769 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(vol*‘𝐴) ∈
ℝ) |
50 | | resubcl 11142 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ 𝑢 ∈ ℝ)
→ ((vol*‘(𝐴
∖ 𝐵)) − 𝑢) ∈
ℝ) |
51 | 50 | adantrr 717 |
. . . . . . . . . . . . . . . . . 18
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ (𝑢 ∈ ℝ
∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘(𝐴 ∖
𝐵)) − 𝑢) ∈
ℝ) |
52 | | posdif 11325 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑢 ∈ ℝ ∧
(vol*‘(𝐴 ∖
𝐵)) ∈ ℝ) →
(𝑢 < (vol*‘(𝐴 ∖ 𝐵)) ↔ 0 < ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢))) |
53 | 52 | ancoms 462 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ 𝑢 ∈ ℝ)
→ (𝑢 <
(vol*‘(𝐴 ∖
𝐵)) ↔ 0 <
((vol*‘(𝐴 ∖
𝐵)) − 𝑢))) |
54 | 53 | biimpd 232 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ 𝑢 ∈ ℝ)
→ (𝑢 <
(vol*‘(𝐴 ∖
𝐵)) → 0 <
((vol*‘(𝐴 ∖
𝐵)) − 𝑢))) |
55 | 54 | impr 458 |
. . . . . . . . . . . . . . . . . 18
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ (𝑢 ∈ ℝ
∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → 0 <
((vol*‘(𝐴 ∖
𝐵)) − 𝑢)) |
56 | 51, 55 | elrpd 12625 |
. . . . . . . . . . . . . . . . 17
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ (𝑢 ∈ ℝ
∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘(𝐴 ∖
𝐵)) − 𝑢) ∈
ℝ+) |
57 | | 3nn 11909 |
. . . . . . . . . . . . . . . . . 18
⊢ 3 ∈
ℕ |
58 | | nnrp 12597 |
. . . . . . . . . . . . . . . . . 18
⊢ (3 ∈
ℕ → 3 ∈ ℝ+) |
59 | 57, 58 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ 3 ∈
ℝ+ |
60 | | rpdivcl 12611 |
. . . . . . . . . . . . . . . . 17
⊢
((((vol*‘(𝐴
∖ 𝐵)) − 𝑢) ∈ ℝ+
∧ 3 ∈ ℝ+) → (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈
ℝ+) |
61 | 56, 59, 60 | sylancl 589 |
. . . . . . . . . . . . . . . 16
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ (𝑢 ∈ ℝ
∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3) ∈
ℝ+) |
62 | 5, 61 | sylan 583 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3) ∈
ℝ+) |
63 | 49, 62 | ltsubrpd 12660 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘𝐴) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < (vol*‘𝐴)) |
64 | 63 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) ∧
(vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) →
((vol*‘𝐴) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < (vol*‘𝐴)) |
65 | | simpr 488 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) ∧
(vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) →
(vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) |
66 | 64, 65 | breqtrd 5079 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) ∧
(vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) →
((vol*‘𝐴) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) |
67 | | reex 10820 |
. . . . . . . . . . . . . . . . . 18
⊢ ℝ
∈ V |
68 | 67 | ssex 5214 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V) |
69 | 68 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → 𝐴 ∈
V) |
70 | | sseq1 3926 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐴 → (𝑣 ⊆ ℝ ↔ 𝐴 ⊆ ℝ)) |
71 | | fveq2 6717 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 = 𝐴 → (vol*‘𝑣) = (vol*‘𝐴)) |
72 | 71 | eleq1d 2822 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐴 → ((vol*‘𝑣) ∈ ℝ ↔ (vol*‘𝐴) ∈
ℝ)) |
73 | 70, 72 | anbi12d 634 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐴 → ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ↔ (𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ))) |
74 | | sseq2 3927 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑣 = 𝐴 → (𝑏 ⊆ 𝑣 ↔ 𝑏 ⊆ 𝐴)) |
75 | 74 | anbi1d 633 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 = 𝐴 → ((𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)))) |
76 | 75 | rexbidv 3216 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 = 𝐴 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)))) |
77 | 76 | abbidv 2807 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 = 𝐴 → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} = {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}) |
78 | 77 | sseq1d 3932 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐴 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ↔ {𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ)) |
79 | 77 | neeq1d 3000 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐴 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅)) |
80 | 77 | raleqdv 3325 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 = 𝐴 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) |
81 | 80 | rexbidv 3216 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐴 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) |
82 | 78, 79, 81 | 3anbi123d 1438 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐴 → (({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥) ↔ ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥))) |
83 | 73, 82 | imbi12d 348 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = 𝐴 → (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) ↔ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)))) |
84 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) → 𝑦 = (vol‘𝑏)) |
85 | 84, 36 | sylan9eqr 2800 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))) → 𝑦 = (vol*‘𝑏)) |
86 | 85 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)))) → 𝑦 = (vol*‘𝑏)) |
87 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))) → 𝑏 ⊆ 𝑣) |
88 | | ovolsscl 24383 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑏 ⊆ 𝑣 ∧ 𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) →
(vol*‘𝑏) ∈
ℝ) |
89 | 88 | 3expb 1122 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑏 ⊆ 𝑣 ∧ (𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ)) →
(vol*‘𝑏) ∈
ℝ) |
90 | 89 | ancoms 462 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) ∧ 𝑏 ⊆
𝑣) → (vol*‘𝑏) ∈
ℝ) |
91 | 87, 90 | sylan2 596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)))) → (vol*‘𝑏) ∈ ℝ) |
92 | 86, 91 | eqeltrd 2838 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)))) → 𝑦 ∈ ℝ) |
93 | 92 | rexlimdvaa 3204 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) → (∃𝑏
∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) → 𝑦 ∈ ℝ)) |
94 | 93 | abssdv 3982 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) → {𝑦 ∣
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ) |
95 | | retop 23659 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(topGen‘ran (,)) ∈ Top |
96 | | 0cld 21935 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((topGen‘ran (,)) ∈ Top → ∅ ∈
(Clsd‘(topGen‘ran (,)))) |
97 | 95, 96 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ∅
∈ (Clsd‘(topGen‘ran (,))) |
98 | | 0ss 4311 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ∅
⊆ 𝑣 |
99 | | 0mbl 24436 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ∅
∈ dom vol |
100 | | mblvol 24427 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (∅
∈ dom vol → (vol‘∅) =
(vol*‘∅)) |
101 | 99, 100 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(vol‘∅) = (vol*‘∅) |
102 | | ovol0 24390 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(vol*‘∅) = 0 |
103 | 101, 102 | eqtr2i 2766 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 =
(vol‘∅) |
104 | 98, 103 | pm3.2i 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (∅
⊆ 𝑣 ∧ 0 =
(vol‘∅)) |
105 | | sseq1 3926 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 = ∅ → (𝑏 ⊆ 𝑣 ↔ ∅ ⊆ 𝑣)) |
106 | | fveq2 6717 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑏 = ∅ →
(vol‘𝑏) =
(vol‘∅)) |
107 | 106 | eqeq2d 2748 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 = ∅ → (0 =
(vol‘𝑏) ↔ 0 =
(vol‘∅))) |
108 | 105, 107 | anbi12d 634 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑏 = ∅ → ((𝑏 ⊆ 𝑣 ∧ 0 = (vol‘𝑏)) ↔ (∅ ⊆ 𝑣 ∧ 0 =
(vol‘∅)))) |
109 | 108 | rspcev 3537 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((∅
∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅)))
→ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 0 = (vol‘𝑏))) |
110 | 97, 104, 109 | mp2an 692 |
. . . . . . . . . . . . . . . . . . . 20
⊢
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 0 = (vol‘𝑏)) |
111 | | c0ex 10827 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 0 ∈
V |
112 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 0 → (𝑦 = (vol‘𝑏) ↔ 0 = (vol‘𝑏))) |
113 | 112 | anbi2d 632 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 0 → ((𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝑣 ∧ 0 = (vol‘𝑏)))) |
114 | 113 | rexbidv 3216 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = 0 → (∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 0 = (vol‘𝑏)))) |
115 | 111, 114 | spcev 3521 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 0 = (vol‘𝑏)) → ∃𝑦∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))) |
116 | 110, 115 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . 19
⊢
∃𝑦∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) |
117 | | abn0 4295 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ↔ ∃𝑦∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))) |
118 | 117 | biimpri 231 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∃𝑦∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅) |
119 | 116, 118 | mp1i 13 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) → {𝑦 ∣
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅) |
120 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)) → 𝑧 = (vol‘𝑏)) |
121 | 120, 36 | sylan9eqr 2800 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏))) → 𝑧 = (vol*‘𝑏)) |
122 | 121 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑣 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)))) → 𝑧 = (vol*‘𝑏)) |
123 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏))) → 𝑏 ⊆ 𝑣) |
124 | | ovolss 24382 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑏 ⊆ 𝑣 ∧ 𝑣 ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘𝑣)) |
125 | 124 | ancoms 462 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑣 ⊆ ℝ ∧ 𝑏 ⊆ 𝑣) → (vol*‘𝑏) ≤ (vol*‘𝑣)) |
126 | 123, 125 | sylan2 596 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑣 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)))) → (vol*‘𝑏) ≤ (vol*‘𝑣)) |
127 | 122, 126 | eqbrtrd 5075 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑣 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)))) → 𝑧 ≤ (vol*‘𝑣)) |
128 | 127 | rexlimdvaa 3204 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ⊆ ℝ →
(∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣))) |
129 | 128 | alrimiv 1935 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 ⊆ ℝ →
∀𝑧(∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣))) |
130 | | eqeq1 2741 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = 𝑧 → (𝑦 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑏))) |
131 | 130 | anbi2d 632 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 𝑧 → ((𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)))) |
132 | 131 | rexbidv 3216 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)))) |
133 | 132 | ralab 3606 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀𝑧 ∈
{𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣) ↔ ∀𝑧(∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣))) |
134 | 129, 133 | sylibr 237 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 ⊆ ℝ →
∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣)) |
135 | | brralrspcev 5113 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((vol*‘𝑣)
∈ ℝ ∧ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣)) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥) |
136 | 134, 135 | sylan2 596 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((vol*‘𝑣)
∈ ℝ ∧ 𝑣
⊆ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥) |
137 | 136 | ancoms 462 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) → ∃𝑥
∈ ℝ ∀𝑧
∈ {𝑦 ∣
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥) |
138 | 94, 119, 137 | 3jca 1130 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) → ({𝑦 ∣
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) |
139 | 83, 138 | vtoclg 3481 |
. . . . . . . . . . . . . . . 16
⊢ (𝐴 ∈ V → ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → ({𝑦 ∣
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥))) |
140 | 69, 139 | mpcom 38 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → ({𝑦 ∣
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) |
141 | 140 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → ({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) |
142 | 62 | rpred 12628 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3) ∈
ℝ) |
143 | 49, 142 | resubcld 11260 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘𝐴) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) ∈
ℝ) |
144 | | suprlub 11796 |
. . . . . . . . . . . . . 14
⊢ ((({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥) ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈ ℝ) →
(((vol*‘𝐴) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣)) |
145 | 141, 143,
144 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(((vol*‘𝐴) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣)) |
146 | 145 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) ∧
(vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) →
(((vol*‘𝐴) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣)) |
147 | 66, 146 | mpbid 235 |
. . . . . . . . . . 11
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) ∧
(vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣) |
148 | | eqeq1 2741 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑣 → (𝑦 = (vol‘𝑏) ↔ 𝑣 = (vol‘𝑏))) |
149 | 148 | anbi2d 632 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑣 → ((𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)))) |
150 | 149 | rexbidv 3216 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)))) |
151 | 150 | rexab 3608 |
. . . . . . . . . . . 12
⊢
(∃𝑣 ∈
{𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣)) |
152 | | breq2 5057 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = (vol‘𝑏) → (((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) |
153 | 152 | ad2antll 729 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏))) → (((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) |
154 | | sseq1 3926 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = 𝑏 → (𝑠 ⊆ 𝐴 ↔ 𝑏 ⊆ 𝐴)) |
155 | | fveq2 6717 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = 𝑏 → (vol‘𝑠) = (vol‘𝑏)) |
156 | 155 | breq2d 5065 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = 𝑏 → (((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠) ↔ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) |
157 | 154, 156 | anbi12d 634 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 = 𝑏 → ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ↔ (𝑏 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))) |
158 | 157 | rspcev 3537 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠))) |
159 | 158 | expr 460 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑏 ⊆ 𝐴) → (((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))) |
160 | 159 | adantrr 717 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏))) → (((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))) |
161 | 153, 160 | sylbid 243 |
. . . . . . . . . . . . . . 15
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏))) → (((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))) |
162 | 161 | rexlimiva 3200 |
. . . . . . . . . . . . . 14
⊢
(∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) → (((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))) |
163 | 162 | imp 410 |
. . . . . . . . . . . . 13
⊢
((∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠))) |
164 | 163 | exlimiv 1938 |
. . . . . . . . . . . 12
⊢
(∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣) → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠))) |
165 | 151, 164 | sylbi 220 |
. . . . . . . . . . 11
⊢
(∃𝑣 ∈
{𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠))) |
166 | 147, 165 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) ∧
(vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠))) |
167 | 166 | ex 416 |
. . . . . . . . 9
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘𝐴) =
sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) → ∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))) |
168 | 167 | adantlr 715 |
. . . . . . . 8
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) → ∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)))) |
169 | | simplrr 778 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (vol*‘𝐵) ∈ ℝ) |
170 | 62 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈
ℝ+) |
171 | 169, 170 | ltsubrpd 12660 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol*‘𝐵)) |
172 | 171 | adantr 484 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) →
((vol*‘𝐵) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < (vol*‘𝐵)) |
173 | | simpr 488 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) →
(vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) |
174 | 172, 173 | breqtrd 5079 |
. . . . . . . . . . 11
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) →
((vol*‘𝐵) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) |
175 | 67 | ssex 5214 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V) |
176 | 175 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ) → 𝐵 ∈
V) |
177 | | sseq1 3926 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐵 → (𝑣 ⊆ ℝ ↔ 𝐵 ⊆ ℝ)) |
178 | | fveq2 6717 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐵 → (vol*‘𝑣) = (vol*‘𝐵)) |
179 | 178 | eleq1d 2822 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐵 → ((vol*‘𝑣) ∈ ℝ ↔ (vol*‘𝐵) ∈
ℝ)) |
180 | 177, 179 | anbi12d 634 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = 𝐵 → ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ↔ (𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ))) |
181 | | sseq2 3927 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 = 𝐵 → (𝑏 ⊆ 𝑣 ↔ 𝑏 ⊆ 𝐵)) |
182 | 181 | anbi1d 633 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 = 𝐵 → ((𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏)))) |
183 | 182 | rexbidv 3216 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 = 𝐵 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏)))) |
184 | 183 | abbidv 2807 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐵 → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} = {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}) |
185 | 184 | sseq1d 3932 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐵 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ↔ {𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ)) |
186 | 184 | neeq1d 3000 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐵 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅)) |
187 | 184 | raleqdv 3325 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐵 → (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) |
188 | 187 | rexbidv 3216 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐵 → (∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) |
189 | 185, 186,
188 | 3anbi123d 1438 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = 𝐵 → (({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥) ↔ ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥))) |
190 | 180, 189 | imbi12d 348 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = 𝐵 → (((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) ↔ ((𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)))) |
191 | 190, 138 | vtoclg 3481 |
. . . . . . . . . . . . . . 15
⊢ (𝐵 ∈ V → ((𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ) → ({𝑦 ∣
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥))) |
192 | 176, 191 | mpcom 38 |
. . . . . . . . . . . . . 14
⊢ ((𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ) → ({𝑦 ∣
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) |
193 | 192 | ad2antlr 727 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥)) |
194 | 142 | adantlr 715 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈ ℝ) |
195 | 169, 194 | resubcld 11260 |
. . . . . . . . . . . . 13
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈ ℝ) |
196 | | suprlub 11796 |
. . . . . . . . . . . . 13
⊢ ((({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ ∧ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥) ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈ ℝ) →
(((vol*‘𝐵) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣)) |
197 | 193, 195,
196 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣)) |
198 | 197 | adantr 484 |
. . . . . . . . . . 11
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) →
(((vol*‘𝐵) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ↔ ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣)) |
199 | 174, 198 | mpbid 235 |
. . . . . . . . . 10
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣) |
200 | 148 | anbi2d 632 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑣 → ((𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏)))) |
201 | 200 | rexbidv 3216 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏)))) |
202 | 201 | rexab 3608 |
. . . . . . . . . . 11
⊢
(∃𝑣 ∈
{𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣)) |
203 | | breq2 5057 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = (vol‘𝑏) → (((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) |
204 | 203 | ad2antll 729 |
. . . . . . . . . . . . . . 15
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏))) → (((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) |
205 | | sseq1 3926 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 𝑏 → (𝑤 ⊆ 𝐵 ↔ 𝑏 ⊆ 𝐵)) |
206 | | fveq2 6717 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = 𝑏 → (vol‘𝑤) = (vol‘𝑏)) |
207 | 206 | breq2d 5065 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 𝑏 → (((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤) ↔ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) |
208 | 205, 207 | anbi12d 634 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 = 𝑏 → ((𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)) ↔ (𝑏 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏)))) |
209 | 208 | rspcev 3537 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) → ∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) |
210 | 209 | expr 460 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑏 ⊆ 𝐵) → (((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏) → ∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) |
211 | 210 | adantrr 717 |
. . . . . . . . . . . . . . 15
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏))) → (((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏) → ∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) |
212 | 204, 211 | sylbid 243 |
. . . . . . . . . . . . . 14
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏))) → (((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) |
213 | 212 | rexlimiva 3200 |
. . . . . . . . . . . . 13
⊢
(∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏)) → (((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) |
214 | 213 | imp 410 |
. . . . . . . . . . . 12
⊢
((∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣) → ∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) |
215 | 214 | exlimiv 1938 |
. . . . . . . . . . 11
⊢
(∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏)) ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣) → ∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) |
216 | 202, 215 | sylbi 220 |
. . . . . . . . . 10
⊢
(∃𝑣 ∈
{𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 → ∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) |
217 | 199, 216 | syl 17 |
. . . . . . . . 9
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑤 ∈
(Clsd‘(topGen‘ran (,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) |
218 | 217 | ex 416 |
. . . . . . . 8
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) → ∃𝑤 ∈
(Clsd‘(topGen‘ran (,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) |
219 | 168, 218 | anim12d 612 |
. . . . . . 7
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → (∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ ∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))))) |
220 | | reeanv 3279 |
. . . . . . 7
⊢
(∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) ↔ (∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))(𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ ∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))(𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) |
221 | 219, 220 | syl6ibr 255 |
. . . . . 6
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑠 ∈
(Clsd‘(topGen‘ran (,)))∃𝑤 ∈ (Clsd‘(topGen‘ran
(,)))((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))))) |
222 | | eqid 2737 |
. . . . . . . . . . . . . 14
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − )
∘ 𝑓)) |
223 | 222 | ovolgelb 24377 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ ∧ (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈ ℝ+) →
∃𝑓 ∈ (( ≤
∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤
((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)))) |
224 | 223 | 3expa 1120 |
. . . . . . . . . . . 12
⊢ (((𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ) ∧ (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈ ℝ+) →
∃𝑓 ∈ (( ≤
∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤
((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)))) |
225 | 62, 224 | sylan2 596 |
. . . . . . . . . . 11
⊢ (((𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ) ∧ ((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵))))) → ∃𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤
((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)))) |
226 | 225 | ancoms 462 |
. . . . . . . . . 10
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) ∧ (𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ)) → ∃𝑓
∈ (( ≤ ∩ (ℝ × ℝ)) ↑m
ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
227 | 226 | an32s 652 |
. . . . . . . . 9
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ∃𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤
((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)))) |
228 | | elmapi 8530 |
. . . . . . . . . . . 12
⊢ (𝑓 ∈ (( ≤ ∩ (ℝ
× ℝ)) ↑m ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
229 | | ssid 3923 |
. . . . . . . . . . . . . . 15
⊢ ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran
((,) ∘ 𝑓) |
230 | 222 | ovollb 24376 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ ∪ ran ((,) ∘
𝑓) ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < )) |
231 | 229, 230 | mpan2 691 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → (vol*‘∪ ran
((,) ∘ 𝑓)) ≤
sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, <
)) |
232 | 231 | adantl 485 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) → (vol*‘∪ ran ((,) ∘
𝑓)) ≤ sup(ran seq1( + ,
((abs ∘ − ) ∘ 𝑓)), ℝ*, <
)) |
233 | | eqid 2737 |
. . . . . . . . . . . . . . . 16
⊢ ((abs
∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓) |
234 | 233, 222 | ovolsf 24369 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘
𝑓)):ℕ⟶(0[,)+∞)) |
235 | | frn 6552 |
. . . . . . . . . . . . . . . 16
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞)) |
236 | | icossxr 13020 |
. . . . . . . . . . . . . . . 16
⊢
(0[,)+∞) ⊆ ℝ* |
237 | 235, 236 | sstrdi 3913 |
. . . . . . . . . . . . . . 15
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆
ℝ*) |
238 | | supxrcl 12905 |
. . . . . . . . . . . . . . 15
⊢ (ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* → sup(ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈
ℝ*) |
239 | 234, 237,
238 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − )
∘ 𝑓)),
ℝ*, < ) ∈ ℝ*) |
240 | | simpr 488 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ) → (vol*‘𝐵) ∈ ℝ) |
241 | | readdcl 10812 |
. . . . . . . . . . . . . . . . 17
⊢
(((vol*‘𝐵)
∈ ℝ ∧ (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈ ℝ) →
((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) ∈
ℝ) |
242 | 240, 142,
241 | syl2anr 600 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) ∧ (𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ)) → ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈ ℝ) |
243 | 242 | rexrd 10883 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) ∧ (𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ)) → ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈
ℝ*) |
244 | 243 | an32s 652 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈
ℝ*) |
245 | | rncoss 5841 |
. . . . . . . . . . . . . . . . . 18
⊢ ran ((,)
∘ 𝑓) ⊆ ran
(,) |
246 | 245 | unissi 4828 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran
(,) |
247 | | unirnioo 13037 |
. . . . . . . . . . . . . . . . 17
⊢ ℝ =
∪ ran (,) |
248 | 246, 247 | sseqtrri 3938 |
. . . . . . . . . . . . . . . 16
⊢ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ |
249 | | ovolcl 24375 |
. . . . . . . . . . . . . . . 16
⊢ (∪ ran ((,) ∘ 𝑓) ⊆ ℝ → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈
ℝ*) |
250 | 248, 249 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
(vol*‘∪ ran ((,) ∘ 𝑓)) ∈
ℝ* |
251 | | xrletr 12748 |
. . . . . . . . . . . . . . 15
⊢
(((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ*
∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈
ℝ* ∧ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈ ℝ*) →
(((vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − )
∘ 𝑓)),
ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
252 | 250, 251 | mp3an1 1450 |
. . . . . . . . . . . . . 14
⊢ ((sup(ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈
ℝ* ∧ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈ ℝ*) →
(((vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − )
∘ 𝑓)),
ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
253 | 239, 244,
252 | syl2anr 600 |
. . . . . . . . . . . . 13
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) → (((vol*‘∪ ran ((,) ∘
𝑓)) ≤ sup(ran seq1( + ,
((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤
((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3))) →
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
254 | 232, 253 | mpand 695 |
. . . . . . . . . . . 12
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < )
≤ ((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
255 | 228, 254 | sylan2 596 |
. . . . . . . . . . 11
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → (sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < ) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
256 | 255 | anim2d 615 |
. . . . . . . . . 10
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ 𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)) → ((𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤
((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3))) → (𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))))) |
257 | 256 | reximdva 3193 |
. . . . . . . . 9
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (∃𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤
((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3))) → ∃𝑓 ∈ (( ≤ ∩ (ℝ
× ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))))) |
258 | 227, 257 | mpd 15 |
. . . . . . . 8
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ∃𝑓 ∈ (( ≤ ∩ (ℝ ×
ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
259 | | rexex 3162 |
. . . . . . . 8
⊢
(∃𝑓 ∈ ((
≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) → ∃𝑓(𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
260 | 258, 259 | syl 17 |
. . . . . . 7
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ∃𝑓(𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
261 | 16 | cldss 21926 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → 𝑠 ⊆ ℝ) |
262 | | indif2 4185 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑓))) = ((𝑠 ∩ ℝ) ∖ ∪ ran ((,) ∘ 𝑓)) |
263 | | df-ss 3883 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ⊆ ℝ ↔ (𝑠 ∩ ℝ) = 𝑠) |
264 | 263 | biimpi 219 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ⊆ ℝ → (𝑠 ∩ ℝ) = 𝑠) |
265 | 264 | difeq1d 4036 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ⊆ ℝ → ((𝑠 ∩ ℝ) ∖ ∪ ran ((,) ∘ 𝑓)) = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) |
266 | 262, 265 | syl5eq 2790 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ⊆ ℝ → (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑓))) = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) |
267 | 261, 266 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑓))) = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) |
268 | | retopbas 23658 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ran (,)
∈ TopBases |
269 | | bastg 21863 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ran (,)
∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) |
270 | 268, 269 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ran (,)
⊆ (topGen‘ran (,)) |
271 | 245, 270 | sstri 3910 |
. . . . . . . . . . . . . . . . . . 19
⊢ ran ((,)
∘ 𝑓) ⊆
(topGen‘ran (,)) |
272 | | uniopn 21794 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran
(,))) → ∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran
(,))) |
273 | 95, 271, 272 | mp2an 692 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran
(,)) |
274 | 16 | opncld 21930 |
. . . . . . . . . . . . . . . . . 18
⊢
(((topGen‘ran (,)) ∈ Top ∧ ∪
ran ((,) ∘ 𝑓) ∈
(topGen‘ran (,))) → (ℝ ∖ ∪
ran ((,) ∘ 𝑓)) ∈
(Clsd‘(topGen‘ran (,)))) |
275 | 95, 273, 274 | mp2an 692 |
. . . . . . . . . . . . . . . . 17
⊢ (ℝ
∖ ∪ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran
(,))) |
276 | | incld 21940 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ (ℝ ∖ ∪ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran
(,)))) → (𝑠 ∩
(ℝ ∖ ∪ ran ((,) ∘ 𝑓))) ∈
(Clsd‘(topGen‘ran (,)))) |
277 | 275, 276 | mpan2 691 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑓))) ∈ (Clsd‘(topGen‘ran
(,)))) |
278 | 267, 277 | eqeltrrd 2839 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ∈
(Clsd‘(topGen‘ran (,)))) |
279 | 278 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran
(,)))) |
280 | 279 | ad2antlr 727 |
. . . . . . . . . . . . 13
⊢ ((((𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ∈
(Clsd‘(topGen‘ran (,)))) |
281 | | simprll 779 |
. . . . . . . . . . . . . 14
⊢ ((((𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑠 ⊆ 𝐴) |
282 | | simplll 775 |
. . . . . . . . . . . . . 14
⊢ ((((𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝐵 ⊆ ∪ ran
((,) ∘ 𝑓)) |
283 | 281, 282 | ssdif2d 4058 |
. . . . . . . . . . . . 13
⊢ ((((𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ⊆
(𝐴 ∖ 𝐵)) |
284 | | fveq2 6717 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) = 𝑏 → (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)) |
285 | 284 | eqcoms 2745 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) →
(vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) = (vol‘𝑏)) |
286 | 285 | biantrud 535 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) →
(𝑏 ⊆ (𝐴 ∖ 𝐵) ↔ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)))) |
287 | | sseq1 3926 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) →
(𝑏 ⊆ (𝐴 ∖ 𝐵) ↔ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ⊆
(𝐴 ∖ 𝐵))) |
288 | 286, 287 | bitr3d 284 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) →
((𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)) ↔
(𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ⊆ (𝐴 ∖ 𝐵))) |
289 | 288 | rspcev 3537 |
. . . . . . . . . . . . 13
⊢ (((𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran (,)))
∧ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ⊆ (𝐴 ∖ 𝐵)) → ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏))) |
290 | 280, 283,
289 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((((𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏))) |
291 | 290 | adantlll 718 |
. . . . . . . . . . 11
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏))) |
292 | | difss 4046 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
(𝐴 ∖ 𝐵) |
293 | 292, 3 | sstri 3910 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 |
294 | | ovolsscl 24383 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘((𝐴 ∖
𝐵) ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))) ∈ ℝ) |
295 | 293, 294 | mp3an1 1450 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
296 | 295 | ad5antr 734 |
. . . . . . . . . . . . 13
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
297 | 5 | ad5antr 734 |
. . . . . . . . . . . . 13
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
298 | | simpl 486 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵))) → 𝑢 ∈ ℝ) |
299 | 298 | ad4antlr 733 |
. . . . . . . . . . . . 13
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑢 ∈ ℝ) |
300 | | difdif2 4201 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓))) |
301 | 300 | fveq2i 6720 |
. . . . . . . . . . . . . 14
⊢
(vol*‘((𝐴
∖ 𝐵) ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))) = (vol*‘(((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) |
302 | | difss 4046 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∖ 𝐵) ∖ 𝑠) ⊆ (𝐴 ∖ 𝐵) |
303 | 302, 3 | sstri 3910 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∖ 𝐵) ∖ 𝑠) ⊆ 𝐴 |
304 | | inss1 4143 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)) ⊆
(𝐴 ∖ 𝐵) |
305 | 304, 3 | sstri 3910 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)) ⊆
𝐴 |
306 | 303, 305 | unssi 4099 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 |
307 | | ovolsscl 24383 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘(((𝐴 ∖
𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
308 | 306, 307 | mp3an1 1450 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
309 | 308 | ad5antr 734 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
310 | | difss 4046 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∖ 𝑠) ⊆ 𝐴 |
311 | | ovolsscl 24383 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∖ 𝑠) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘(𝐴 ∖
𝑠)) ∈
ℝ) |
312 | 310, 311 | mp3an1 1450 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(𝐴 ∖ 𝑠)) ∈ ℝ) |
313 | 312 | ad5antr 734 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴 ∖ 𝑠)) ∈ ℝ) |
314 | 169, 194 | readdcld 10862 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈ ℝ) |
315 | 314, 250 | jctil 523 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧
((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) ∈
ℝ)) |
316 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) |
317 | | ovolge0 24378 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (∪ ran ((,) ∘ 𝑓) ⊆ ℝ → 0 ≤
(vol*‘∪ ran ((,) ∘ 𝑓))) |
318 | 248, 317 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ≤
(vol*‘∪ ran ((,) ∘ 𝑓)) |
319 | 316, 318 | jctil 523 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) → (0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
320 | | xrrege0 12764 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ*
∧ ((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) ∈ ℝ) ∧ (0
≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) |
321 | 315, 319,
320 | syl2an 599 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) |
322 | | difss 4046 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ∪ ran
((,) ∘ 𝑓) |
323 | | ovolsscl 24383 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ) |
324 | 322, 248,
323 | mp3an12 1453 |
. . . . . . . . . . . . . . . . . 18
⊢
((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ →
(vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ) |
325 | 321, 324 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ) |
326 | 325 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ) |
327 | 313, 326 | readdcld 10862 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ) |
328 | 5, 50 | sylan 583 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ 𝑢 ∈
ℝ) → ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) ∈ ℝ) |
329 | 328 | adantrr 717 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘(𝐴 ∖
𝐵)) − 𝑢) ∈
ℝ) |
330 | 329 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) ∈ ℝ) |
331 | 330 | ad3antrrr 730 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) ∈ ℝ) |
332 | | ssdifss 4050 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ⊆ ℝ → (𝐴 ∖ 𝑠) ⊆ ℝ) |
333 | 322, 248 | sstri 3910 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ |
334 | | unss 4098 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ∖ 𝑠) ⊆ ℝ ∧ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ) ↔ ((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)) ⊆
ℝ) |
335 | 332, 333,
334 | sylanblc 592 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ⊆ ℝ → ((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)) ⊆
ℝ) |
336 | | ovolcl 24375 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)) ⊆ ℝ →
(vol*‘((𝐴 ∖
𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈
ℝ*) |
337 | 335, 336 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ⊆ ℝ →
(vol*‘((𝐴 ∖
𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈
ℝ*) |
338 | 337 | ad4antr 732 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤))) ∈
ℝ*) |
339 | 312 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘(𝐴 ∖ 𝑠)) ∈ ℝ) |
340 | 339, 325 | readdcld 10862 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ) |
341 | | ovolge0 24378 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)) ⊆ ℝ →
0 ≤ (vol*‘((𝐴
∖ 𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
342 | 335, 341 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ⊆ ℝ → 0 ≤
(vol*‘((𝐴 ∖
𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
343 | 342 | ad4antr 732 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → 0 ≤ (vol*‘((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)))) |
344 | 332 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (𝐴 ∖
𝑠) ⊆
ℝ) |
345 | 344, 312 | jca 515 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → ((𝐴 ∖
𝑠) ⊆ ℝ ∧
(vol*‘(𝐴 ∖
𝑠)) ∈
ℝ)) |
346 | 345 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → ((𝐴 ∖ 𝑠) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ 𝑠)) ∈ ℝ)) |
347 | 325, 333 | jctil 523 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → ((∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ ∧ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)) |
348 | | ovolun 24396 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝐴 ∖ 𝑠) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ 𝑠)) ∈ ℝ) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ ∧ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)) →
(vol*‘((𝐴 ∖
𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) ≤ ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
349 | 346, 347,
348 | syl2anc 587 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤))) ≤
((vol*‘(𝐴 ∖
𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
350 | | xrrege0 12764 |
. . . . . . . . . . . . . . . . . 18
⊢
((((vol*‘((𝐴
∖ 𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ* ∧
((vol*‘(𝐴 ∖
𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈ ℝ) ∧ (0 ≤
(vol*‘((𝐴 ∖
𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) ∧ (vol*‘((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤))) ≤
((vol*‘(𝐴 ∖
𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))))) → (vol*‘((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤))) ∈
ℝ) |
351 | 338, 340,
343, 349, 350 | syl22anc 839 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤))) ∈
ℝ) |
352 | 351 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤))) ∈
ℝ) |
353 | | ssdif 4054 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐴 → ((𝐴 ∖ 𝐵) ∖ 𝑠) ⊆ (𝐴 ∖ 𝑠)) |
354 | 3, 353 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∖ 𝐵) ∖ 𝑠) ⊆ (𝐴 ∖ 𝑠) |
355 | | incom 4115 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)) = (∪ ran ((,) ∘ 𝑓) ∩ (𝐴 ∖ 𝐵)) |
356 | | indif2 4185 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (∪ ran ((,) ∘ 𝑓) ∩ (𝐴 ∖ 𝐵)) = ((∪ ran ((,)
∘ 𝑓) ∩ 𝐴) ∖ 𝐵) |
357 | 355, 356 | eqtri 2765 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)) = ((∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∖ 𝐵) |
358 | | inss1 4143 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ∪ ran
((,) ∘ 𝑓) |
359 | 358 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (∪ ran
((,) ∘ 𝑓) ∩ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓)) |
360 | | simprrl 781 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑤 ⊆ 𝐵) |
361 | 359, 360 | ssdif2d 4058 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((∪
ran ((,) ∘ 𝑓) ∩
𝐴) ∖ 𝐵) ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) |
362 | 357, 361 | eqsstrid 3949 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)) ⊆
(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) |
363 | | unss12 4096 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐴 ∖ 𝐵) ∖ 𝑠) ⊆ (𝐴 ∖ 𝑠) ∧ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)) ⊆
(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) → (((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓))) ⊆
((𝐴 ∖ 𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) |
364 | 354, 362,
363 | sylancr 590 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓))) ⊆
((𝐴 ∖ 𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) |
365 | 335 | ad6antr 736 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)) ⊆
ℝ) |
366 | | ovolss 24382 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓))) ⊆
((𝐴 ∖ 𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∧ ((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)) ⊆ ℝ) →
(vol*‘(((𝐴 ∖
𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) ≤
(vol*‘((𝐴 ∖
𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
367 | 364, 365,
366 | syl2anc 587 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) ≤
(vol*‘((𝐴 ∖
𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
368 | 332 | ad6antr 736 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (𝐴 ∖ 𝑠) ⊆ ℝ) |
369 | 326, 333 | jctil 523 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((∪
ran ((,) ∘ 𝑓) ∖
𝑤) ⊆ ℝ ∧
(vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ)) |
370 | 368, 313,
369, 348 | syl21anc 838 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤))) ≤
((vol*‘(𝐴 ∖
𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
371 | 309, 352,
327, 367, 370 | letrd 10989 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) ≤
((vol*‘(𝐴 ∖
𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
372 | 194 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈ ℝ) |
373 | 194, 194 | readdcld 10862 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈ ℝ) |
374 | 373 | ad3antrrr 730 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) ∈ ℝ) |
375 | | eleq1w 2820 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑏 = 𝑠 → (𝑏 ∈ dom vol ↔ 𝑠 ∈ dom vol)) |
376 | 375, 34 | vtoclga 3489 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → 𝑠 ∈ dom vol) |
377 | | mblvol 24427 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ dom vol →
(vol‘𝑠) =
(vol*‘𝑠)) |
378 | 376, 377 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (vol‘𝑠) = (vol*‘𝑠)) |
379 | 378 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ (vol‘𝑠) =
(vol*‘𝑠)) |
380 | | sseqin2 4130 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑠 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑠) = 𝑠) |
381 | 380 | biimpi 219 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑠 ⊆ 𝐴 → (𝐴 ∩ 𝑠) = 𝑠) |
382 | 381 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ⊆ 𝐴 → 𝑠 = (𝐴 ∩ 𝑠)) |
383 | 382 | fveq2d 6721 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ⊆ 𝐴 → (vol*‘𝑠) = (vol*‘(𝐴 ∩ 𝑠))) |
384 | 383 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → (vol*‘𝑠) = (vol*‘(𝐴 ∩ 𝑠))) |
385 | 379, 384 | sylan9eq 2798 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑠) = (vol*‘(𝐴 ∩ 𝑠))) |
386 | 385 | oveq2d 7229 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (vol‘𝑠)) = ((vol*‘𝐴) − (vol*‘(𝐴 ∩ 𝑠)))) |
387 | 386 | adantll 714 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (vol‘𝑠)) = ((vol*‘𝐴) − (vol*‘(𝐴 ∩ 𝑠)))) |
388 | 376 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ 𝑠 ∈ dom
vol) |
389 | | simplll 775 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈
ℝ)) |
390 | | mblsplit 24429 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘𝐴) = ((vol*‘(𝐴 ∩ 𝑠)) + (vol*‘(𝐴 ∖ 𝑠)))) |
391 | 390 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → ((vol*‘(𝐴 ∩ 𝑠)) + (vol*‘(𝐴 ∖ 𝑠))) = (vol*‘𝐴)) |
392 | 391 | 3expb 1122 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ dom vol ∧ (𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ)) → ((vol*‘(𝐴 ∩ 𝑠)) + (vol*‘(𝐴 ∖ 𝑠))) = (vol*‘𝐴)) |
393 | 388, 389,
392 | syl2anr 600 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) → ((vol*‘(𝐴 ∩ 𝑠)) + (vol*‘(𝐴 ∖ 𝑠))) = (vol*‘𝐴)) |
394 | 393 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴 ∩ 𝑠)) + (vol*‘(𝐴 ∖ 𝑠))) = (vol*‘𝐴)) |
395 | | simp-6r 788 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝐴) ∈ ℝ) |
396 | 395 | recnd 10861 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝐴) ∈ ℂ) |
397 | | inss1 4143 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝐴 ∩ 𝑠) ⊆ 𝐴 |
398 | | ovolsscl 24383 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐴 ∩ 𝑠) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘(𝐴 ∩ 𝑠)) ∈
ℝ) |
399 | 397, 398 | mp3an1 1450 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(𝐴 ∩ 𝑠)) ∈ ℝ) |
400 | 399 | recnd 10861 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(𝐴 ∩ 𝑠)) ∈ ℂ) |
401 | 400 | ad5antr 734 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴 ∩ 𝑠)) ∈ ℂ) |
402 | 312 | recnd 10861 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(𝐴 ∖ 𝑠)) ∈ ℂ) |
403 | 402 | ad5antr 734 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴 ∖ 𝑠)) ∈ ℂ) |
404 | 396, 401,
403 | subaddd 11207 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘𝐴) − (vol*‘(𝐴 ∩ 𝑠))) = (vol*‘(𝐴 ∖ 𝑠)) ↔ ((vol*‘(𝐴 ∩ 𝑠)) + (vol*‘(𝐴 ∖ 𝑠))) = (vol*‘𝐴))) |
405 | 394, 404 | mpbird 260 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (vol*‘(𝐴 ∩ 𝑠))) = (vol*‘(𝐴 ∖ 𝑠))) |
406 | 387, 405 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (vol‘𝑠)) = (vol*‘(𝐴 ∖ 𝑠))) |
407 | 379 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑠) = (vol*‘𝑠)) |
408 | | simpll 767 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → 𝑠 ⊆ 𝐴) |
409 | | simp-4l 783 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈
ℝ)) |
410 | | ovolsscl 24383 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘𝑠) ∈
ℝ) |
411 | 410 | 3expb 1122 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ⊆ 𝐴 ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) →
(vol*‘𝑠) ∈
ℝ) |
412 | 408, 409,
411 | syl2anr 600 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝑠) ∈ ℝ) |
413 | 407, 412 | eqeltrd 2838 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑠) ∈ ℝ) |
414 | | simprlr 780 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) |
415 | 395, 372,
413, 414 | ltsub23d 11437 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐴) − (vol‘𝑠)) < (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) |
416 | 406, 415 | eqbrtrrd 5077 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴 ∖ 𝑠)) < (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) |
417 | 321 | recnd 10861 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℂ) |
418 | 417 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℂ) |
419 | 240 | ad5antlr 735 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝐵) ∈ ℝ) |
420 | 419 | recnd 10861 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝐵) ∈ ℂ) |
421 | | eleq1w 2820 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑏 = 𝑤 → (𝑏 ∈ dom vol ↔ 𝑤 ∈ dom vol)) |
422 | 421, 34 | vtoclga 3489 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑤 ∈
(Clsd‘(topGen‘ran (,))) → 𝑤 ∈ dom vol) |
423 | | mblvol 24427 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑤 ∈ dom vol →
(vol‘𝑤) =
(vol*‘𝑤)) |
424 | 422, 423 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑤 ∈
(Clsd‘(topGen‘ran (,))) → (vol‘𝑤) = (vol*‘𝑤)) |
425 | 424 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ (vol‘𝑤) =
(vol*‘𝑤)) |
426 | 425 | ad2antlr 727 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑤) = (vol*‘𝑤)) |
427 | | simprl 771 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → 𝑤 ⊆ 𝐵) |
428 | | simp-4r 784 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) → (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈
ℝ)) |
429 | | ovolsscl 24383 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑤 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) →
(vol*‘𝑤) ∈
ℝ) |
430 | 429 | 3expb 1122 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑤 ⊆ 𝐵 ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) →
(vol*‘𝑤) ∈
ℝ) |
431 | 427, 428,
430 | syl2anr 600 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘𝑤) ∈ ℝ) |
432 | 426, 431 | eqeltrd 2838 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑤) ∈ ℝ) |
433 | 432 | recnd 10861 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑤) ∈ ℂ) |
434 | 418, 420,
433 | npncand 11213 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) + ((vol*‘𝐵) − (vol‘𝑤))) = ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol‘𝑤))) |
435 | | simplrl 777 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) → 𝐵 ⊆ ∪ ran
((,) ∘ 𝑓)) |
436 | 427, 435 | sylan9ssr 3915 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑤 ⊆ ∪ ran
((,) ∘ 𝑓)) |
437 | | sseqin2 4130 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ (∪ ran
((,) ∘ 𝑓) ∩ 𝑤) = 𝑤) |
438 | 436, 437 | sylib 221 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (∪ ran
((,) ∘ 𝑓) ∩ 𝑤) = 𝑤) |
439 | 438 | fveq2d 6721 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) = (vol*‘𝑤)) |
440 | 426, 439 | eqtr4d 2780 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘𝑤) = (vol*‘(∪
ran ((,) ∘ 𝑓) ∩
𝑤))) |
441 | 440 | oveq2d 7229 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol‘𝑤)) = ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)))) |
442 | 422 | adantl 485 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ 𝑤 ∈ dom
vol) |
443 | 321, 248 | jctil 523 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)) |
444 | | mblsplit 24429 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑤 ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘∪ ran ((,) ∘ 𝑓)) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
445 | 444 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑤 ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘∪ ran ((,) ∘ 𝑓))) |
446 | 445 | 3expb 1122 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑤 ∈ dom vol ∧ (∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)) →
((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘∪ ran ((,) ∘ 𝑓))) |
447 | 442, 443,
446 | syl2anr 600 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) → ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘∪ ran ((,) ∘ 𝑓))) |
448 | 447 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘∪ ran ((,) ∘ 𝑓))) |
449 | | inss1 4143 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (∪ ran ((,) ∘ 𝑓) ∩ 𝑤) ⊆ ∪ ran
((,) ∘ 𝑓) |
450 | | ovolsscl 24383 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((∪ ran ((,) ∘ 𝑓) ∩ 𝑤) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ) |
451 | 449, 248,
450 | mp3an12 1453 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ →
(vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ) |
452 | 321, 451 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ) |
453 | 452 | recnd 10861 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℂ) |
454 | 325 | recnd 10861 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℂ) |
455 | 417, 453,
454 | subaddd 11207 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤))) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ↔ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘∪ ran ((,) ∘ 𝑓)))) |
456 | 455 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤))) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ↔ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘∪ ran ((,) ∘ 𝑓)))) |
457 | 448, 456 | mpbird 260 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤))) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) |
458 | 434, 441,
457 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) + ((vol*‘𝐵) − (vol‘𝑤))) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) |
459 | 240 | ad3antlr 731 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘𝐵) ∈
ℝ) |
460 | 321, 459 | resubcld 11260 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) ∈ ℝ) |
461 | 460 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) ∈ ℝ) |
462 | 419, 432 | resubcld 11260 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐵) − (vol‘𝑤)) ∈ ℝ) |
463 | | simprr 773 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) |
464 | 194 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈ ℝ) |
465 | 321, 459,
464 | lesubadd2d 11431 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) ≤ (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ↔ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
466 | 463, 465 | mpbird 260 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) ≤ (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) |
467 | 466 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) ≤ (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) |
468 | | simprrr 782 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)) |
469 | 419, 372,
432, 468 | ltsub23d 11437 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘𝐵) − (vol‘𝑤)) < (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) |
470 | 461, 462,
372, 372, 467, 469 | leltaddd 11454 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝐵)) + ((vol*‘𝐵) − (vol‘𝑤))) < ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) |
471 | 458, 470 | eqbrtrrd 5077 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) < ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) |
472 | 313, 326,
372, 374, 416, 471 | lt2addd 11455 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) < ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
473 | | df-3 11894 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 3 = (2 +
1) |
474 | | 2cn 11905 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 2 ∈
ℂ |
475 | | ax-1cn 10787 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 ∈
ℂ |
476 | 474, 475 | addcomi 11023 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (2 + 1) =
(1 + 2) |
477 | 473, 476 | eqtri 2765 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 3 = (1 +
2) |
478 | 477 | oveq1i 7223 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (3
· (((vol*‘(𝐴
∖ 𝐵)) − 𝑢) / 3)) = ((1 + 2) ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) |
479 | 62 | rpcnd 12630 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3) ∈
ℂ) |
480 | | adddir 10824 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((1
∈ ℂ ∧ 2 ∈ ℂ ∧ (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈ ℂ) → ((1 + 2)
· (((vol*‘(𝐴
∖ 𝐵)) − 𝑢) / 3)) = ((1 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) + (2 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)))) |
481 | 475, 474,
480 | mp3an12 1453 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((vol*‘(𝐴
∖ 𝐵)) − 𝑢) / 3) ∈ ℂ → ((1
+ 2) · (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) = ((1 · (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) + (2 · (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
482 | 479, 481 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → ((1 + 2)
· (((vol*‘(𝐴
∖ 𝐵)) − 𝑢) / 3)) = ((1 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) + (2 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)))) |
483 | 479 | mulid2d 10851 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → (1 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) = (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) |
484 | 479 | 2timesd 12073 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → (2 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) = ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) |
485 | 483, 484 | oveq12d 7231 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → ((1 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) + (2 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3))) = ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
486 | 482, 485 | eqtrd 2777 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → ((1 + 2)
· (((vol*‘(𝐴
∖ 𝐵)) − 𝑢) / 3)) = ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
487 | 478, 486 | syl5eq 2790 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → (3 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) = ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) |
488 | 329 | recnd 10861 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘(𝐴 ∖
𝐵)) − 𝑢) ∈
ℂ) |
489 | | 3cn 11911 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 3 ∈
ℂ |
490 | | 3ne0 11936 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 3 ≠
0 |
491 | | divcan2 11498 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((vol*‘(𝐴
∖ 𝐵)) − 𝑢) ∈ ℂ ∧ 3 ∈
ℂ ∧ 3 ≠ 0) → (3 · (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢)) |
492 | 489, 490,
491 | mp3an23 1455 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((vol*‘(𝐴
∖ 𝐵)) − 𝑢) ∈ ℂ → (3
· (((vol*‘(𝐴
∖ 𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢)) |
493 | 488, 492 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → (3 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢)) |
494 | 487, 493 | eqtr3d 2779 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) = ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢)) |
495 | 494 | adantlr 715 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) = ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢)) |
496 | 495 | ad3antrrr 730 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + ((((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) = ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢)) |
497 | 472, 496 | breqtrd 5079 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) < ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢)) |
498 | 309, 327,
331, 371, 497 | lelttrd 10990 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) <
((vol*‘(𝐴 ∖
𝐵)) − 𝑢)) |
499 | 301, 498 | eqbrtrid 5088 |
. . . . . . . . . . . . 13
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) <
((vol*‘(𝐴 ∖
𝐵)) − 𝑢)) |
500 | 296, 297,
299, 499 | ltsub13d 11438 |
. . . . . . . . . . . 12
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑢 < ((vol*‘(𝐴 ∖ 𝐵)) − (vol*‘((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))))) |
501 | 283 | adantlll 718 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ⊆
(𝐴 ∖ 𝐵)) |
502 | | sseqin2 4130 |
. . . . . . . . . . . . . . 15
⊢ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ⊆ (𝐴 ∖ 𝐵) ↔ ((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) = (𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) |
503 | 501, 502 | sylib 221 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) = (𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) |
504 | 503 | fveq2d 6721 |
. . . . . . . . . . . . 13
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) =
(vol*‘(𝑠 ∖
∪ ran ((,) ∘ 𝑓)))) |
505 | | opnmbl 24499 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) → ∪ ran ((,) ∘ 𝑓) ∈ dom vol) |
506 | 273, 505 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ ran ((,) ∘ 𝑓) ∈ dom vol |
507 | | difmbl 24440 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑠 ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑓) ∈ dom vol) → (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ∈ dom
vol) |
508 | 376, 506,
507 | sylancl 589 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ∈ dom
vol) |
509 | 508 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ∈ dom vol) |
510 | 509 | ad2antlr 727 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ∈ dom
vol) |
511 | 13 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (𝐴 ∖
𝐵) ⊆
ℝ) |
512 | 511, 5 | jca 515 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → ((𝐴 ∖
𝐵) ⊆ ℝ ∧
(vol*‘(𝐴 ∖
𝐵)) ∈
ℝ)) |
513 | 512 | ad5antr 734 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((𝐴 ∖ 𝐵) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ)) |
514 | | mblsplit 24429 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ∈ dom vol ∧ (𝐴 ∖ 𝐵) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) → (vol*‘(𝐴 ∖ 𝐵)) = ((vol*‘((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) +
(vol*‘((𝐴 ∖
𝐵) ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))))) |
515 | 514 | 3expb 1122 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ∈ dom vol ∧ ((𝐴 ∖ 𝐵) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ)) →
(vol*‘(𝐴 ∖
𝐵)) = ((vol*‘((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) +
(vol*‘((𝐴 ∖
𝐵) ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))))) |
516 | 515 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ (((𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ∈ dom vol ∧ ((𝐴 ∖ 𝐵) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ)) →
((vol*‘((𝐴 ∖
𝐵) ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))) + (vol*‘((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))))) =
(vol*‘(𝐴 ∖
𝐵))) |
517 | 510, 513,
516 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) +
(vol*‘((𝐴 ∖
𝐵) ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓))))) = (vol*‘(𝐴 ∖ 𝐵))) |
518 | 297 | recnd 10861 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘(𝐴 ∖ 𝐵)) ∈ ℂ) |
519 | 296 | recnd 10861 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) ∈
ℂ) |
520 | | inss1 4143 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
(𝐴 ∖ 𝐵) |
521 | 520, 3 | sstri 3910 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 |
522 | | ovolsscl 24383 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘((𝐴 ∖
𝐵) ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))) ∈ ℝ) |
523 | 521, 522 | mp3an1 1450 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
524 | 523 | ad5antr 734 |
. . . . . . . . . . . . . . . 16
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
525 | 524 | recnd 10861 |
. . . . . . . . . . . . . . 15
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol*‘((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) ∈
ℂ) |
526 | 518, 519,
525 | subadd2d 11208 |
. . . . . . . . . . . . . 14
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (((vol*‘(𝐴 ∖ 𝐵)) − (vol*‘((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))))) =
(vol*‘((𝐴 ∖
𝐵) ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))) ↔ ((vol*‘((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) +
(vol*‘((𝐴 ∖
𝐵) ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓))))) = (vol*‘(𝐴 ∖ 𝐵)))) |
527 | 517, 526 | mpbird 260 |
. . . . . . . . . . . . 13
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ((vol*‘(𝐴 ∖ 𝐵)) − (vol*‘((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))))) =
(vol*‘((𝐴 ∖
𝐵) ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓))))) |
528 | | mblvol 24427 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ∈ dom vol → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) |
529 | 507, 528 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑠 ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑓) ∈ dom vol) → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) |
530 | 376, 506,
529 | sylancl 589 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol*‘(𝑠 ∖
∪ ran ((,) ∘ 𝑓)))) |
531 | 530 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ (vol‘(𝑠
∖ ∪ ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) |
532 | 531 | ad2antlr 727 |
. . . . . . . . . . . . 13
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol*‘(𝑠 ∖
∪ ran ((,) ∘ 𝑓)))) |
533 | 504, 527,
532 | 3eqtr4rd 2788 |
. . . . . . . . . . . 12
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
((vol*‘(𝐴 ∖
𝐵)) −
(vol*‘((𝐴 ∖
𝐵) ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))))) |
534 | 500, 533 | breqtrrd 5081 |
. . . . . . . . . . 11
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → 𝑢 < (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) |
535 | | fvex 6730 |
. . . . . . . . . . . 12
⊢
(vol‘(𝑠
∖ ∪ ran ((,) ∘ 𝑓))) ∈ V |
536 | | eqeq1 2741 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) → (𝑣 = (vol‘𝑏) ↔ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏))) |
537 | 536 | anbi2d 632 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) → ((𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)))) |
538 | 537 | rexbidv 3216 |
. . . . . . . . . . . . 13
⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)))) |
539 | | breq2 5057 |
. . . . . . . . . . . . 13
⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) → (𝑢 < 𝑣 ↔ 𝑢 < (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))))) |
540 | 538, 539 | anbi12d 634 |
. . . . . . . . . . . 12
⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) → ((∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣) ↔ (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)) ∧ 𝑢 < (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))))) |
541 | 535, 540 | spcev 3521 |
. . . . . . . . . . 11
⊢
((∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)) ∧ 𝑢 < (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣)) |
542 | 291, 534,
541 | syl2anc 587 |
. . . . . . . . . 10
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣)) |
543 | 148 | anbi2d 632 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑣 → ((𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)))) |
544 | 543 | rexbidv 3216 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)))) |
545 | 544 | rexab 3608 |
. . . . . . . . . 10
⊢
(∃𝑣 ∈
{𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣 ↔ ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣)) |
546 | 542, 545 | sylibr 237 |
. . . . . . . . 9
⊢
(((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) ∧ ((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤)))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣) |
547 | 546 | ex 416 |
. . . . . . . 8
⊢
((((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,)))
∧ 𝑤 ∈
(Clsd‘(topGen‘ran (,))))) → (((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣)) |
548 | 547 | rexlimdvva 3213 |
. . . . . . 7
⊢
(((((𝐴 ⊆
ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) ∧ (𝐵 ⊆ ∪ ran
((,) ∘ 𝑓) ∧
(vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)))) → (∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))∃𝑤 ∈
(Clsd‘(topGen‘ran (,)))((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣)) |
549 | 260, 548 | exlimddv 1943 |
. . . . . 6
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (∃𝑠 ∈ (Clsd‘(topGen‘ran
(,)))∃𝑤 ∈
(Clsd‘(topGen‘ran (,)))((𝑠 ⊆ 𝐴 ∧ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠)) ∧ (𝑤 ⊆ 𝐵 ∧ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣)) |
550 | 221, 549 | syld 47 |
. . . . 5
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣)) |
551 | 550 | exp31 423 |
. . . 4
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → ((𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ) → ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵))) → (((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣)))) |
552 | 551 | com34 91 |
. . 3
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → ((𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ) → (((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < )) → ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣)))) |
553 | 552 | 3imp1 1349 |
. 2
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣) |
554 | 2, 6, 48, 553 | eqsupd 9073 |
1
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ) ∧ ((vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) ∧ (vol*‘𝐵) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))) → sup({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) = (vol*‘(𝐴 ∖ 𝐵))) |