Step | Hyp | Ref
| Expression |
1 | | ltso 9997 |
. . 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 3699 |
. . . 4
⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 |
4 | | ovolsscl 23061 |
. . . 4
⊢ (((𝐴 ∖ 𝐵) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘(𝐴 ∖
𝐵)) ∈
ℝ) |
5 | 3, 4 | mp3an1 1403 |
. . 3
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
6 | 5 | 3ad2ant1 1075 |
. 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 3176 |
. . . . . 6
⊢ 𝑢 ∈ V |
8 | | eqeq1 2614 |
. . . . . . . 8
⊢ (𝑦 = 𝑢 → (𝑦 = (vol‘𝑏) ↔ 𝑢 = (vol‘𝑏))) |
9 | 8 | anbi2d 736 |
. . . . . . 7
⊢ (𝑦 = 𝑢 → ((𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)))) |
10 | 9 | rexbidv 3034 |
. . . . . 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 3319 |
. . . . 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 790 |
. . . . . . . . 9
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏))) → 𝑏 ⊆ (𝐴 ∖ 𝐵)) |
13 | | ssdifss 3703 |
. . . . . . . . 9
⊢ (𝐴 ⊆ ℝ → (𝐴 ∖ 𝐵) ⊆ ℝ) |
14 | | ovolss 23060 |
. . . . . . . . 9
⊢ ((𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (𝐴 ∖ 𝐵) ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘(𝐴 ∖ 𝐵))) |
15 | 12, 13, 14 | syl2anr 494 |
. . . . . . . 8
⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)))) → (vol*‘𝑏) ≤ (vol*‘(𝐴 ∖ 𝐵))) |
16 | | uniretop 22376 |
. . . . . . . . . . . . 13
⊢ ℝ =
∪ (topGen‘ran (,)) |
17 | 16 | cldss 20643 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → 𝑏 ⊆ ℝ) |
18 | | ovolcl 23053 |
. . . . . . . . . . . 12
⊢ (𝑏 ⊆ ℝ →
(vol*‘𝑏) ∈
ℝ*) |
19 | 17, 18 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → (vol*‘𝑏) ∈
ℝ*) |
20 | | ovolcl 23053 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∖ 𝐵) ⊆ ℝ → (vol*‘(𝐴 ∖ 𝐵)) ∈
ℝ*) |
21 | 13, 20 | syl 17 |
. . . . . . . . . . 11
⊢ (𝐴 ⊆ ℝ →
(vol*‘(𝐴 ∖
𝐵)) ∈
ℝ*) |
22 | | xrlenlt 9982 |
. . . . . . . . . . 11
⊢
(((vol*‘𝑏)
∈ ℝ* ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ*) →
((vol*‘𝑏) ≤
(vol*‘(𝐴 ∖
𝐵)) ↔ ¬
(vol*‘(𝐴 ∖
𝐵)) < (vol*‘𝑏))) |
23 | 19, 21, 22 | syl2anr 494 |
. . . . . . . . . 10
⊢ ((𝐴 ⊆ ℝ ∧ 𝑏 ∈
(Clsd‘(topGen‘ran (,)))) → ((vol*‘𝑏) ≤ (vol*‘(𝐴 ∖ 𝐵)) ↔ ¬ (vol*‘(𝐴 ∖ 𝐵)) < (vol*‘𝑏))) |
24 | 23 | adantrr 749 |
. . . . . . . . 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 3820 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 ⊆ ℝ ↔ (ℝ
∖ (ℝ ∖ 𝑏)) = 𝑏) |
27 | 17, 26 | sylib 207 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖
𝑏)) = 𝑏) |
28 | | rembl 23115 |
. . . . . . . . . . . . . . . . 17
⊢ ℝ
∈ dom vol |
29 | 16 | cldopn 20645 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ (topGen‘ran
(,))) |
30 | | opnmbl 23176 |
. . . . . . . . . . . . . . . . . 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 23118 |
. . . . . . . . . . . . . . . . 17
⊢ ((ℝ
∈ dom vol ∧ (ℝ ∖ 𝑏) ∈ dom vol) → (ℝ ∖
(ℝ ∖ 𝑏)) ∈
dom vol) |
33 | 28, 31, 32 | sylancr 694 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖
𝑏)) ∈ dom
vol) |
34 | 27, 33 | eqeltrrd 2689 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) → 𝑏 ∈ dom vol) |
35 | | mblvol 23105 |
. . . . . . . . . . . . . . 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 2666 |
. . . . . . . . . . . . 13
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → 𝑢 = (vol*‘𝑏)) |
38 | 37 | breq2d 4595 |
. . . . . . . . . . . 12
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → ((vol*‘(𝐴 ∖ 𝐵)) < 𝑢 ↔ (vol*‘(𝐴 ∖ 𝐵)) < (vol*‘𝑏))) |
39 | 38 | notbid 307 |
. . . . . . . . . . 11
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑢 = (vol‘𝑏)) → (¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴 ∖ 𝐵)) < (vol*‘𝑏))) |
40 | 39 | adantrl 748 |
. . . . . . . . . 10
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏))) → (¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢 ↔ ¬ (vol*‘(𝐴 ∖ 𝐵)) < (vol*‘𝑏))) |
41 | 40 | adantl 481 |
. . . . . . . . 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 270 |
. . . . . . . 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 221 |
. . . . . . 7
⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)))) → ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢) |
44 | 43 | rexlimdvaa 3014 |
. . . . . 6
⊢ (𝐴 ⊆ ℝ →
(∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏)) → ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢)) |
45 | 44 | imp 444 |
. . . . 5
⊢ ((𝐴 ⊆ ℝ ∧
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑢 = (vol‘𝑏))) → ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢) |
46 | 11, 45 | sylan2b 491 |
. . . 4
⊢ ((𝐴 ⊆ ℝ ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘(𝐴 ∖ 𝐵)) < 𝑢) |
47 | 46 | adantlr 747 |
. . 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 1216 |
. 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 788 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(vol*‘𝐴) ∈
ℝ) |
50 | | resubcl 10224 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ 𝑢 ∈ ℝ)
→ ((vol*‘(𝐴
∖ 𝐵)) − 𝑢) ∈
ℝ) |
51 | 50 | adantrr 749 |
. . . . . . . . . . . . . . . . . 18
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ (𝑢 ∈ ℝ
∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘(𝐴 ∖
𝐵)) − 𝑢) ∈
ℝ) |
52 | | posdif 10400 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑢 ∈ ℝ ∧
(vol*‘(𝐴 ∖
𝐵)) ∈ ℝ) →
(𝑢 < (vol*‘(𝐴 ∖ 𝐵)) ↔ 0 < ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢))) |
53 | 52 | ancoms 468 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ 𝑢 ∈ ℝ)
→ (𝑢 <
(vol*‘(𝐴 ∖
𝐵)) ↔ 0 <
((vol*‘(𝐴 ∖
𝐵)) − 𝑢))) |
54 | 53 | biimpd 218 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ 𝑢 ∈ ℝ)
→ (𝑢 <
(vol*‘(𝐴 ∖
𝐵)) → 0 <
((vol*‘(𝐴 ∖
𝐵)) − 𝑢))) |
55 | 54 | impr 647 |
. . . . . . . . . . . . . . . . . 18
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ (𝑢 ∈ ℝ
∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → 0 <
((vol*‘(𝐴 ∖
𝐵)) − 𝑢)) |
56 | 51, 55 | elrpd 11745 |
. . . . . . . . . . . . . . . . 17
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ (𝑢 ∈ ℝ
∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘(𝐴 ∖
𝐵)) − 𝑢) ∈
ℝ+) |
57 | | 3nn 11063 |
. . . . . . . . . . . . . . . . . 18
⊢ 3 ∈
ℕ |
58 | | nnrp 11718 |
. . . . . . . . . . . . . . . . . 18
⊢ (3 ∈
ℕ → 3 ∈ ℝ+) |
59 | 57, 58 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ 3 ∈
ℝ+ |
60 | | rpdivcl 11732 |
. . . . . . . . . . . . . . . . 17
⊢
((((vol*‘(𝐴
∖ 𝐵)) − 𝑢) ∈ ℝ+
∧ 3 ∈ ℝ+) → (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈
ℝ+) |
61 | 56, 59, 60 | sylancl 693 |
. . . . . . . . . . . . . . . 16
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈ ℝ
∧ (𝑢 ∈ ℝ
∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3) ∈
ℝ+) |
62 | 5, 61 | sylan 487 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3) ∈
ℝ+) |
63 | 49, 62 | ltsubrpd 11780 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘𝐴) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) < (vol*‘𝐴)) |
64 | 63 | adantr 480 |
. . . . . . . . . . . . 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 476 |
. . . . . . . . . . . . 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 4609 |
. . . . . . . . . . . 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 9906 |
. . . . . . . . . . . . . . . . . 18
⊢ ℝ
∈ V |
68 | 67 | ssex 4730 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ⊆ ℝ → 𝐴 ∈ V) |
69 | 68 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → 𝐴 ∈
V) |
70 | | sseq1 3589 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐴 → (𝑣 ⊆ ℝ ↔ 𝐴 ⊆ ℝ)) |
71 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 = 𝐴 → (vol*‘𝑣) = (vol*‘𝐴)) |
72 | 71 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐴 → ((vol*‘𝑣) ∈ ℝ ↔ (vol*‘𝐴) ∈
ℝ)) |
73 | 70, 72 | anbi12d 743 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐴 → ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ↔ (𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ))) |
74 | | sseq2 3590 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑣 = 𝐴 → (𝑏 ⊆ 𝑣 ↔ 𝑏 ⊆ 𝐴)) |
75 | 74 | anbi1d 737 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 = 𝐴 → ((𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)))) |
76 | 75 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 = 𝐴 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)))) |
77 | 76 | abbidv 2728 |
. . . . . . . . . . . . . . . . . . . 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 3595 |
. . . . . . . . . . . . . . . . . . 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 2841 |
. . . . . . . . . . . . . . . . . . 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 3121 |
. . . . . . . . . . . . . . . . . . . 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 3034 |
. . . . . . . . . . . . . . . . . . 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 1391 |
. . . . . . . . . . . . . . . . . 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 333 |
. . . . . . . . . . . . . . . . 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 476 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) → 𝑦 = (vol‘𝑏)) |
85 | 84, 36 | sylan9eqr 2666 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))) → 𝑦 = (vol*‘𝑏)) |
86 | 85 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)))) → 𝑦 = (vol*‘𝑏)) |
87 | | simprl 790 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))) → 𝑏 ⊆ 𝑣) |
88 | | ovolsscl 23061 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑏 ⊆ 𝑣 ∧ 𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) →
(vol*‘𝑏) ∈
ℝ) |
89 | 88 | 3expb 1258 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑏 ⊆ 𝑣 ∧ (𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ)) →
(vol*‘𝑏) ∈
ℝ) |
90 | 89 | ancoms 468 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) ∧ 𝑏 ⊆
𝑣) → (vol*‘𝑏) ∈
ℝ) |
91 | 87, 90 | sylan2 490 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)))) → (vol*‘𝑏) ∈ ℝ) |
92 | 86, 91 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)))) → 𝑦 ∈ ℝ) |
93 | 92 | rexlimdvaa 3014 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) → (∃𝑏
∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) → 𝑦 ∈ ℝ)) |
94 | 93 | abssdv 3639 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) → {𝑦 ∣
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ⊆ ℝ) |
95 | | retop 22375 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(topGen‘ran (,)) ∈ Top |
96 | | 0cld 20652 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((topGen‘ran (,)) ∈ Top → ∅ ∈
(Clsd‘(topGen‘ran (,)))) |
97 | 95, 96 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ∅
∈ (Clsd‘(topGen‘ran (,))) |
98 | | 0ss 3924 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ∅
⊆ 𝑣 |
99 | | 0mbl 23114 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ∅
∈ dom vol |
100 | | mblvol 23105 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (∅
∈ dom vol → (vol‘∅) =
(vol*‘∅)) |
101 | 99, 100 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(vol‘∅) = (vol*‘∅) |
102 | | ovol0 23068 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(vol*‘∅) = 0 |
103 | 101, 102 | eqtr2i 2633 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 =
(vol‘∅) |
104 | 98, 103 | pm3.2i 470 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (∅
⊆ 𝑣 ∧ 0 =
(vol‘∅)) |
105 | | sseq1 3589 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 = ∅ → (𝑏 ⊆ 𝑣 ↔ ∅ ⊆ 𝑣)) |
106 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑏 = ∅ →
(vol‘𝑏) =
(vol‘∅)) |
107 | 106 | eqeq2d 2620 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 = ∅ → (0 =
(vol‘𝑏) ↔ 0 =
(vol‘∅))) |
108 | 105, 107 | anbi12d 743 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑏 = ∅ → ((𝑏 ⊆ 𝑣 ∧ 0 = (vol‘𝑏)) ↔ (∅ ⊆ 𝑣 ∧ 0 =
(vol‘∅)))) |
109 | 108 | rspcev 3282 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((∅
∈ (Clsd‘(topGen‘ran (,))) ∧ (∅ ⊆ 𝑣 ∧ 0 = (vol‘∅)))
→ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 0 = (vol‘𝑏))) |
110 | 97, 104, 109 | mp2an 704 |
. . . . . . . . . . . . . . . . . . . 20
⊢
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 0 = (vol‘𝑏)) |
111 | | c0ex 9913 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 0 ∈
V |
112 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 0 → (𝑦 = (vol‘𝑏) ↔ 0 = (vol‘𝑏))) |
113 | 112 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 0 → ((𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝑣 ∧ 0 = (vol‘𝑏)))) |
114 | 113 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . 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 3273 |
. . . . . . . . . . . . . . . . . . . 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 3908 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ({𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ↔ ∃𝑦∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))) |
118 | 117 | biimpri 217 |
. . . . . . . . . . . . . . . . . . 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 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)) → 𝑧 = (vol‘𝑏)) |
121 | 120, 36 | sylan9eqr 2666 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏))) → 𝑧 = (vol*‘𝑏)) |
122 | 121 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑣 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)))) → 𝑧 = (vol*‘𝑏)) |
123 | | simprl 790 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏))) → 𝑏 ⊆ 𝑣) |
124 | | ovolss 23060 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑏 ⊆ 𝑣 ∧ 𝑣 ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘𝑣)) |
125 | 124 | ancoms 468 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑣 ⊆ ℝ ∧ 𝑏 ⊆ 𝑣) → (vol*‘𝑏) ≤ (vol*‘𝑣)) |
126 | 123, 125 | sylan2 490 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑣 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)))) → (vol*‘𝑏) ≤ (vol*‘𝑣)) |
127 | 122, 126 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑣 ⊆ ℝ ∧ (𝑏 ∈
(Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)))) → 𝑧 ≤ (vol*‘𝑣)) |
128 | 127 | rexlimdvaa 3014 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 ⊆ ℝ →
(∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣))) |
129 | 128 | alrimiv 1842 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 ⊆ ℝ →
∀𝑧(∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)) → 𝑧 ≤ (vol*‘𝑣))) |
130 | | eqeq1 2614 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = 𝑧 → (𝑦 = (vol‘𝑏) ↔ 𝑧 = (vol‘𝑏))) |
131 | 130 | anbi2d 736 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = 𝑧 → ((𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)))) |
132 | 131 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = 𝑧 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑧 = (vol‘𝑏)))) |
133 | 132 | ralab 3334 |
. . . . . . . . . . . . . . . . . . . . 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 223 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 ⊆ ℝ →
∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣)) |
135 | | breq2 4587 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = (vol*‘𝑣) → (𝑧 ≤ 𝑥 ↔ 𝑧 ≤ (vol*‘𝑣))) |
136 | 135 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = (vol*‘𝑣) → (∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥 ↔ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ (vol*‘𝑣))) |
137 | 136 | rspcev 3282 |
. . . . . . . . . . . . . . . . . . . 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) |
138 | 134, 137 | sylan2 490 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((vol*‘𝑣)
∈ ℝ ∧ 𝑣
⊆ ℝ) → ∃𝑥 ∈ ℝ ∀𝑧 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥) |
139 | 138 | ancoms 468 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 ⊆ ℝ ∧
(vol*‘𝑣) ∈
ℝ) → ∃𝑥
∈ ℝ ∀𝑧
∈ {𝑦 ∣
∃𝑏 ∈
(Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))}𝑧 ≤ 𝑥) |
140 | 94, 119, 139 | 3jca 1235 |
. . . . . . . . . . . . . . . . 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)) |
141 | 83, 140 | vtoclg 3239 |
. . . . . . . . . . . . . . . 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))) |
142 | 69, 141 | mpcom 37 |
. . . . . . . . . . . . . . 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)) |
143 | 142 | adantr 480 |
. . . . . . . . . . . . . 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)) |
144 | 62 | rpred 11748 |
. . . . . . . . . . . . . . 15
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3) ∈
ℝ) |
145 | 49, 144 | resubcld 10337 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘𝐴) −
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) ∈
ℝ) |
146 | | suprlub 10864 |
. . . . . . . . . . . . . 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)) |
147 | 143, 145,
146 | syl2anc 691 |
. . . . . . . . . . . . 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)) |
148 | 147 | adantr 480 |
. . . . . . . . . . . 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)) |
149 | 66, 148 | mpbid 221 |
. . . . . . . . . . 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) |
150 | | eqeq1 2614 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑣 → (𝑦 = (vol‘𝑏) ↔ 𝑣 = (vol‘𝑏))) |
151 | 150 | anbi2d 736 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑣 → ((𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)))) |
152 | 151 | rexbidv 3034 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)))) |
153 | 152 | rexab 3336 |
. . . . . . . . . . . 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)) |
154 | | breq2 4587 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = (vol‘𝑏) → (((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) |
155 | 154 | ad2antll 761 |
. . . . . . . . . . . . . . . 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))) |
156 | | sseq1 3589 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = 𝑏 → (𝑠 ⊆ 𝐴 ↔ 𝑏 ⊆ 𝐴)) |
157 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑠 = 𝑏 → (vol‘𝑠) = (vol‘𝑏)) |
158 | 157 | breq2d 4595 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = 𝑏 → (((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑠) ↔ ((vol*‘𝐴) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) |
159 | 156, 158 | anbi12d 743 |
. . . . . . . . . . . . . . . . . . 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)))) |
160 | 159 | rspcev 3282 |
. . . . . . . . . . . . . . . . . 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))) |
161 | 160 | expr 641 |
. . . . . . . . . . . . . . . . 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)))) |
162 | 161 | adantrr 749 |
. . . . . . . . . . . . . . . 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)))) |
163 | 155, 162 | sylbid 229 |
. . . . . . . . . . . . . . 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)))) |
164 | 163 | rexlimiva 3010 |
. . . . . . . . . . . . . 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)))) |
165 | 164 | imp 444 |
. . . . . . . . . . . . 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))) |
166 | 165 | exlimiv 1845 |
. . . . . . . . . . . 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))) |
167 | 153, 166 | sylbi 206 |
. . . . . . . . . . 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))) |
168 | 149, 167 | 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))) |
169 | 168 | ex 449 |
. . . . . . . . 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)))) |
170 | 169 | adantlr 747 |
. . . . . . . 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)))) |
171 | | simplrr 797 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (vol*‘𝐵) ∈ ℝ) |
172 | 62 | adantlr 747 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈
ℝ+) |
173 | 171, 172 | ltsubrpd 11780 |
. . . . . . . . . . . . 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)) |
174 | 173 | adantr 480 |
. . . . . . . . . . . 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)) |
175 | | simpr 476 |
. . . . . . . . . . . 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, < )) |
176 | 174, 175 | breqtrd 4609 |
. . . . . . . . . . 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, < )) |
177 | 67 | ssex 4730 |
. . . . . . . . . . . . . . . 16
⊢ (𝐵 ⊆ ℝ → 𝐵 ∈ V) |
178 | 177 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ) → 𝐵 ∈
V) |
179 | | sseq1 3589 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐵 → (𝑣 ⊆ ℝ ↔ 𝐵 ⊆ ℝ)) |
180 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐵 → (vol*‘𝑣) = (vol*‘𝐵)) |
181 | 180 | eleq1d 2672 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐵 → ((vol*‘𝑣) ∈ ℝ ↔ (vol*‘𝐵) ∈
ℝ)) |
182 | 179, 181 | anbi12d 743 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 = 𝐵 → ((𝑣 ⊆ ℝ ∧ (vol*‘𝑣) ∈ ℝ) ↔ (𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ))) |
183 | | sseq2 3590 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑣 = 𝐵 → (𝑏 ⊆ 𝑣 ↔ 𝑏 ⊆ 𝐵)) |
184 | 183 | anbi1d 737 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑣 = 𝐵 → ((𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏)))) |
185 | 184 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑣 = 𝐵 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏)))) |
186 | 185 | abbidv 2728 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 = 𝐵 → {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} = {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))}) |
187 | 186 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . 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)) |
188 | 186 | neeq1d 2841 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 = 𝐵 → ({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝑣 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅ ↔ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏))} ≠ ∅)) |
189 | 186 | raleqdv 3121 |
. . . . . . . . . . . . . . . . . . 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)) |
190 | 189 | rexbidv 3034 |
. . . . . . . . . . . . . . . . . 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)) |
191 | 187, 188,
190 | 3anbi123d 1391 |
. . . . . . . . . . . . . . . . 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))) |
192 | 182, 191 | imbi12d 333 |
. . . . . . . . . . . . . . . 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)))) |
193 | 192, 140 | vtoclg 3239 |
. . . . . . . . . . . . . . 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))) |
194 | 178, 193 | mpcom 37 |
. . . . . . . . . . . . . 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)) |
195 | 194 | ad2antlr 759 |
. . . . . . . . . . . . 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)) |
196 | 144 | adantlr 747 |
. . . . . . . . . . . . . 14
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈ ℝ) |
197 | 171, 196 | resubcld 10337 |
. . . . . . . . . . . . 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) |
198 | | suprlub 10864 |
. . . . . . . . . . . . 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)) |
199 | 195, 197,
198 | syl2anc 691 |
. . . . . . . . . . . 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)) |
200 | 199 | adantr 480 |
. . . . . . . . . . 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)) |
201 | 176, 200 | mpbid 221 |
. . . . . . . . . 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) |
202 | 150 | anbi2d 736 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑣 → ((𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏)))) |
203 | 202 | rexbidv 3034 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ 𝐵 ∧ 𝑣 = (vol‘𝑏)))) |
204 | 203 | rexab 3336 |
. . . . . . . . . . 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)) |
205 | | breq2 4587 |
. . . . . . . . . . . . . . . 16
⊢ (𝑣 = (vol‘𝑏) → (((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < 𝑣 ↔ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) |
206 | 205 | ad2antll 761 |
. . . . . . . . . . . . . . 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))) |
207 | | sseq1 3589 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 𝑏 → (𝑤 ⊆ 𝐵 ↔ 𝑏 ⊆ 𝐵)) |
208 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = 𝑏 → (vol‘𝑤) = (vol‘𝑏)) |
209 | 208 | breq2d 4595 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 𝑏 → (((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑤) ↔ ((vol*‘𝐵) − (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) < (vol‘𝑏))) |
210 | 207, 209 | anbi12d 743 |
. . . . . . . . . . . . . . . . . 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)))) |
211 | 210 | rspcev 3282 |
. . . . . . . . . . . . . . . . 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))) |
212 | 211 | expr 641 |
. . . . . . . . . . . . . . . 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)))) |
213 | 212 | adantrr 749 |
. . . . . . . . . . . . . . 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)))) |
214 | 206, 213 | sylbid 229 |
. . . . . . . . . . . . . 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)))) |
215 | 214 | rexlimiva 3010 |
. . . . . . . . . . . . 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)))) |
216 | 215 | imp 444 |
. . . . . . . . . . . 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))) |
217 | 216 | exlimiv 1845 |
. . . . . . . . . . 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))) |
218 | 204, 217 | sylbi 206 |
. . . . . . . . . 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))) |
219 | 201, 218 | 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))) |
220 | 219 | ex 449 |
. . . . . . . 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)))) |
221 | 170, 220 | anim12d 584 |
. . . . . . 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))))) |
222 | | reeanv 3086 |
. . . . . . 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)))) |
223 | 221, 222 | syl6ibr 241 |
. . . . . 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))))) |
224 | | eqid 2610 |
. . . . . . . . . . . . . 14
⊢ seq1( + ,
((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − )
∘ 𝑓)) |
225 | 224 | ovolgelb 23055 |
. . . . . . . . . . . . 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)))) |
226 | 225 | 3expa 1257 |
. . . . . . . . . . . 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)))) |
227 | 62, 226 | sylan2 490 |
. . . . . . . . . . 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)))) |
228 | 227 | ancoms 468 |
. . . . . . . . . 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)))) |
229 | 228 | an32s 842 |
. . . . . . . . 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)))) |
230 | | elmapi 7765 |
. . . . . . . . . . . 12
⊢ (𝑓 ∈ (( ≤ ∩ (ℝ
× ℝ)) ↑𝑚 ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ ×
ℝ))) |
231 | | ssid 3587 |
. . . . . . . . . . . . . . 15
⊢ ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran
((,) ∘ 𝑓) |
232 | 224 | ovollb 23054 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:ℕ⟶( ≤ ∩
(ℝ × ℝ)) ∧ ∪ ran ((,) ∘
𝑓) ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘
− ) ∘ 𝑓)),
ℝ*, < )) |
233 | 231, 232 | mpan2 703 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → (vol*‘∪ ran
((,) ∘ 𝑓)) ≤
sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, <
)) |
234 | 233 | adantl 481 |
. . . . . . . . . . . . 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*, <
)) |
235 | | eqid 2610 |
. . . . . . . . . . . . . . . 16
⊢ ((abs
∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓) |
236 | 235, 224 | ovolsf 23048 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘
𝑓)):ℕ⟶(0[,)+∞)) |
237 | | frn 5966 |
. . . . . . . . . . . . . . . 16
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞)) |
238 | | icossxr 12129 |
. . . . . . . . . . . . . . . 16
⊢
(0[,)+∞) ⊆ ℝ* |
239 | 237, 238 | syl6ss 3580 |
. . . . . . . . . . . . . . 15
⊢ (seq1( +
, ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆
ℝ*) |
240 | | supxrcl 12017 |
. . . . . . . . . . . . . . 15
⊢ (ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* → sup(ran
seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈
ℝ*) |
241 | 236, 239,
240 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝑓:ℕ⟶( ≤ ∩
(ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − )
∘ 𝑓)),
ℝ*, < ) ∈ ℝ*) |
242 | | simpr 476 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐵 ⊆ ℝ ∧
(vol*‘𝐵) ∈
ℝ) → (vol*‘𝐵) ∈ ℝ) |
243 | | readdcl 9898 |
. . . . . . . . . . . . . . . . 17
⊢
(((vol*‘𝐵)
∈ ℝ ∧ (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3) ∈ ℝ) →
((vol*‘𝐵) +
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) ∈
ℝ) |
244 | 242, 144,
243 | syl2anr 494 |
. . . . . . . . . . . . . . . 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) |
245 | 244 | rexrd 9968 |
. . . . . . . . . . . . . . 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*) |
246 | 245 | an32s 842 |
. . . . . . . . . . . . . 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*) |
247 | | rncoss 5307 |
. . . . . . . . . . . . . . . . . 18
⊢ ran ((,)
∘ 𝑓) ⊆ ran
(,) |
248 | 247 | unissi 4397 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran
(,) |
249 | | unirnioo 12144 |
. . . . . . . . . . . . . . . . 17
⊢ ℝ =
∪ ran (,) |
250 | 248, 249 | sseqtr4i 3601 |
. . . . . . . . . . . . . . . 16
⊢ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ |
251 | | ovolcl 23053 |
. . . . . . . . . . . . . . . 16
⊢ (∪ ran ((,) ∘ 𝑓) ⊆ ℝ → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈
ℝ*) |
252 | 250, 251 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢
(vol*‘∪ ran ((,) ∘ 𝑓)) ∈
ℝ* |
253 | | xrletr 11865 |
. . . . . . . . . . . . . . 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)))) |
254 | 252, 253 | mp3an1 1403 |
. . . . . . . . . . . . . 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)))) |
255 | 241, 246,
254 | syl2anr 494 |
. . . . . . . . . . . . 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)))) |
256 | 234, 255 | mpand 707 |
. . . . . . . . . . . 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)))) |
257 | 230, 256 | sylan2 490 |
. . . . . . . . . . 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)))) |
258 | 257 | anim2d 587 |
. . . . . . . . . 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))))) |
259 | 258 | reximdva 3000 |
. . . . . . . . 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))))) |
260 | 229, 259 | 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)))) |
261 | | rexex 2985 |
. . . . . . . 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)))) |
262 | 260, 261 | 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)))) |
263 | 16 | cldss 20643 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → 𝑠 ⊆ ℝ) |
264 | | indif2 3829 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑓))) = ((𝑠 ∩ ℝ) ∖ ∪ ran ((,) ∘ 𝑓)) |
265 | | df-ss 3554 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 ⊆ ℝ ↔ (𝑠 ∩ ℝ) = 𝑠) |
266 | 265 | biimpi 205 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 ⊆ ℝ → (𝑠 ∩ ℝ) = 𝑠) |
267 | 266 | difeq1d 3689 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 ⊆ ℝ → ((𝑠 ∩ ℝ) ∖ ∪ ran ((,) ∘ 𝑓)) = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) |
268 | 264, 267 | syl5eq 2656 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ⊆ ℝ → (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑓))) = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) |
269 | 263, 268 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑓))) = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) |
270 | | retopbas 22374 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ran (,)
∈ TopBases |
271 | | bastg 20581 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ran (,)
∈ TopBases → ran (,) ⊆ (topGen‘ran (,))) |
272 | 270, 271 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ran (,)
⊆ (topGen‘ran (,)) |
273 | 247, 272 | sstri 3577 |
. . . . . . . . . . . . . . . . . . 19
⊢ ran ((,)
∘ 𝑓) ⊆
(topGen‘ran (,)) |
274 | | uniopn 20527 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran
(,))) → ∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran
(,))) |
275 | 95, 273, 274 | mp2an 704 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran
(,)) |
276 | 16 | opncld 20647 |
. . . . . . . . . . . . . . . . . 18
⊢
(((topGen‘ran (,)) ∈ Top ∧ ∪
ran ((,) ∘ 𝑓) ∈
(topGen‘ran (,))) → (ℝ ∖ ∪
ran ((,) ∘ 𝑓)) ∈
(Clsd‘(topGen‘ran (,)))) |
277 | 95, 275, 276 | mp2an 704 |
. . . . . . . . . . . . . . . . 17
⊢ (ℝ
∖ ∪ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran
(,))) |
278 | | incld 20657 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ (ℝ ∖ ∪ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran
(,)))) → (𝑠 ∩
(ℝ ∖ ∪ ran ((,) ∘ 𝑓))) ∈
(Clsd‘(topGen‘ran (,)))) |
279 | 277, 278 | mpan2 703 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑓))) ∈ (Clsd‘(topGen‘ran
(,)))) |
280 | 269, 279 | eqeltrrd 2689 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ∈
(Clsd‘(topGen‘ran (,)))) |
281 | 280 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ∈ (Clsd‘(topGen‘ran
(,)))) |
282 | 281 | ad2antlr 759 |
. . . . . . . . . . . . 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 (,)))) |
283 | | simprll 798 |
. . . . . . . . . . . . . 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) |
284 | | simplll 794 |
. . . . . . . . . . . . . 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)) |
285 | 283, 284 | ssdif2d 3711 |
. . . . . . . . . . . . 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)) |
286 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) = 𝑏 → (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)) |
287 | 286 | eqcoms 2618 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) →
(vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) = (vol‘𝑏)) |
288 | 287 | biantrud 527 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) →
(𝑏 ⊆ (𝐴 ∖ 𝐵) ↔ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)))) |
289 | | sseq1 3589 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) →
(𝑏 ⊆ (𝐴 ∖ 𝐵) ↔ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ⊆
(𝐴 ∖ 𝐵))) |
290 | 288, 289 | bitr3d 269 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) →
((𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)) ↔
(𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ⊆ (𝐴 ∖ 𝐵))) |
291 | 290 | rspcev 3282 |
. . . . . . . . . . . . 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))) |
292 | 282, 285,
291 | syl2anc 691 |
. . . . . . . . . . . 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))) |
293 | 292 | adantlll 750 |
. . . . . . . . . . 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))) |
294 | | difss 3699 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
(𝐴 ∖ 𝐵) |
295 | 294, 3 | sstri 3577 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 |
296 | | ovolsscl 23061 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘((𝐴 ∖
𝐵) ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))) ∈ ℝ) |
297 | 295, 296 | mp3an1 1403 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
298 | 297 | ad5antr 766 |
. . . . . . . . . . . . 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) |
299 | 5 | ad5antr 766 |
. . . . . . . . . . . . 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) |
300 | | simpl 472 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵))) → 𝑢 ∈ ℝ) |
301 | 300 | ad4antlr 765 |
. . . . . . . . . . . . 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) |
302 | | difdif2 3843 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∖ 𝐵) ∖ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓))) |
303 | 302 | fveq2i 6106 |
. . . . . . . . . . . . . 14
⊢
(vol*‘((𝐴
∖ 𝐵) ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))) = (vol*‘(((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) |
304 | | difss 3699 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∖ 𝐵) ∖ 𝑠) ⊆ (𝐴 ∖ 𝐵) |
305 | 304, 3 | sstri 3577 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∖ 𝐵) ∖ 𝑠) ⊆ 𝐴 |
306 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)) ⊆
(𝐴 ∖ 𝐵) |
307 | 306, 3 | sstri 3577 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)) ⊆
𝐴 |
308 | 305, 307 | unssi 3750 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 |
309 | | ovolsscl 23061 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘(((𝐴 ∖
𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
310 | 308, 309 | mp3an1 1403 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(((𝐴 ∖ 𝐵) ∖ 𝑠) ∪ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
311 | 310 | ad5antr 766 |
. . . . . . . . . . . . . . 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) |
312 | | difss 3699 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐴 ∖ 𝑠) ⊆ 𝐴 |
313 | | ovolsscl 23061 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ∖ 𝑠) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘(𝐴 ∖
𝑠)) ∈
ℝ) |
314 | 312, 313 | mp3an1 1403 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(𝐴 ∖ 𝑠)) ∈ ℝ) |
315 | 314 | ad5antr 766 |
. . . . . . . . . . . . . . . 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) |
316 | 171, 196 | readdcld 9948 |
. . . . . . . . . . . . . . . . . . . 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) |
317 | 316, 252 | jctil 558 |
. . . . . . . . . . . . . . . . . . 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)) |
318 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐵 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐵) + (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3))) |
319 | | ovolge0 23056 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (∪ ran ((,) ∘ 𝑓) ⊆ ℝ → 0 ≤
(vol*‘∪ ran ((,) ∘ 𝑓))) |
320 | 250, 319 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ≤
(vol*‘∪ ran ((,) ∘ 𝑓)) |
321 | 318, 320 | jctil 558 |
. . . . . . . . . . . . . . . . . . 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)))) |
322 | | xrrege0 11879 |
. . . . . . . . . . . . . . . . . . 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) |
323 | 317, 321,
322 | syl2an 493 |
. . . . . . . . . . . . . . . . . 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) |
324 | | difss 3699 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ∪ ran
((,) ∘ 𝑓) |
325 | | ovolsscl 23061 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ) |
326 | 324, 250,
325 | mp3an12 1406 |
. . . . . . . . . . . . . . . . . 18
⊢
((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ →
(vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)) ∈ ℝ) |
327 | 323, 326 | 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) |
328 | 327 | ad2antrr 758 |
. . . . . . . . . . . . . . . 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) |
329 | 315, 328 | readdcld 9948 |
. . . . . . . . . . . . . . 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) |
330 | 5, 50 | sylan 487 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ 𝑢 ∈
ℝ) → ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) ∈ ℝ) |
331 | 330 | adantrr 749 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘(𝐴 ∖
𝐵)) − 𝑢) ∈
ℝ) |
332 | 331 | adantlr 747 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝐵 ⊆
ℝ ∧ (vol*‘𝐵) ∈ ℝ)) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘(𝐴 ∖ 𝐵)))) → ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) ∈ ℝ) |
333 | 332 | ad3antrrr 762 |
. . . . . . . . . . . . . . 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) |
334 | | ssdifss 3703 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐴 ⊆ ℝ → (𝐴 ∖ 𝑠) ⊆ ℝ) |
335 | 324, 250 | sstri 3577 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ |
336 | 334, 335 | jctir 559 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐴 ⊆ ℝ → ((𝐴 ∖ 𝑠) ⊆ ℝ ∧ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ)) |
337 | | unss 3749 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐴 ∖ 𝑠) ⊆ ℝ ∧ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤) ⊆ ℝ) ↔ ((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)) ⊆
ℝ) |
338 | 336, 337 | sylib 207 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴 ⊆ ℝ → ((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)) ⊆
ℝ) |
339 | | ovolcl 23053 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)) ⊆ ℝ →
(vol*‘((𝐴 ∖
𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈
ℝ*) |
340 | 338, 339 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ⊆ ℝ →
(vol*‘((𝐴 ∖
𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) ∈
ℝ*) |
341 | 340 | ad4antr 764 |
. . . . . . . . . . . . . . . . . 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*) |
342 | 314 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . 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) |
343 | 342, 327 | readdcld 9948 |
. . . . . . . . . . . . . . . . . 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) |
344 | | ovolge0 23056 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ∖ 𝑠) ∪ (∪ ran
((,) ∘ 𝑓) ∖
𝑤)) ⊆ ℝ →
0 ≤ (vol*‘((𝐴
∖ 𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
345 | 338, 344 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐴 ⊆ ℝ → 0 ≤
(vol*‘((𝐴 ∖
𝑠) ∪ (∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
346 | 345 | ad4antr 764 |
. . . . . . . . . . . . . . . . . 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)))) |
347 | 334 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (𝐴 ∖
𝑠) ⊆
ℝ) |
348 | 347, 314 | jca 553 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → ((𝐴 ∖
𝑠) ⊆ ℝ ∧
(vol*‘(𝐴 ∖
𝑠)) ∈
ℝ)) |
349 | 348 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . 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)) |
350 | 327, 335 | jctil 558 |
. . . . . . . . . . . . . . . . . . 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)) |
351 | | ovolun 23074 |
. . . . . . . . . . . . . . . . . . 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)))) |
352 | 349, 350,
351 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 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)))) |
353 | | xrrege0 11879 |
. . . . . . . . . . . . . . . . . 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) |
354 | 341, 343,
346, 352, 353 | syl22anc 1319 |
. . . . . . . . . . . . . . . . 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) |
355 | 354 | ad2antrr 758 |
. . . . . . . . . . . . . . . 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) |
356 | | ssdif 3707 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐴 → ((𝐴 ∖ 𝐵) ∖ 𝑠) ⊆ (𝐴 ∖ 𝑠)) |
357 | 3, 356 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∖ 𝐵) ∖ 𝑠) ⊆ (𝐴 ∖ 𝑠) |
358 | | incom 3767 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)) = (∪ ran ((,) ∘ 𝑓) ∩ (𝐴 ∖ 𝐵)) |
359 | | indif2 3829 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (∪ ran ((,) ∘ 𝑓) ∩ (𝐴 ∖ 𝐵)) = ((∪ ran ((,)
∘ 𝑓) ∩ 𝐴) ∖ 𝐵) |
360 | 358, 359 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∖ 𝐵) ∩ ∪ ran
((,) ∘ 𝑓)) = ((∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ∖ 𝐵) |
361 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ ∪ ran
((,) ∘ 𝑓) |
362 | 361 | 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)) |
363 | | simprrl 800 |
. . . . . . . . . . . . . . . . . . . 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) |
364 | 362, 363 | ssdif2d 3711 |
. . . . . . . . . . . . . . . . . . 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)) |
365 | 360, 364 | syl5eqss 3612 |
. . . . . . . . . . . . . . . . . 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)) |
366 | | unss12 3747 |
. . . . . . . . . . . . . . . . . 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))) |
367 | 357, 365,
366 | sylancr 694 |
. . . . . . . . . . . . . . . . 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))) |
368 | 338 | ad6antr 768 |
. . . . . . . . . . . . . . . . 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) |
369 | | ovolss 23060 |
. . . . . . . . . . . . . . . . 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)))) |
370 | 367, 368,
369 | syl2anc 691 |
. . . . . . . . . . . . . . . 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)))) |
371 | 334 | ad6antr 768 |
. . . . . . . . . . . . . . . . 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) |
372 | 328, 335 | jctil 558 |
. . . . . . . . . . . . . . . . 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)) |
373 | 371, 315,
372, 351 | syl21anc 1317 |
. . . . . . . . . . . . . . . 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)))) |
374 | 311, 355,
329, 370, 373 | letrd 10073 |
. . . . . . . . . . . . . . 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)))) |
375 | 196 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . 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) |
376 | 196, 196 | readdcld 9948 |
. . . . . . . . . . . . . . . . . 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) |
377 | 376 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . 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) |
378 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑏 = 𝑠 → (𝑏 ∈ dom vol ↔ 𝑠 ∈ dom vol)) |
379 | 378, 34 | vtoclga 3245 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → 𝑠 ∈ dom vol) |
380 | | mblvol 23105 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ∈ dom vol →
(vol‘𝑠) =
(vol*‘𝑠)) |
381 | 379, 380 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (vol‘𝑠) = (vol*‘𝑠)) |
382 | 381 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ (vol‘𝑠) =
(vol*‘𝑠)) |
383 | | sseqin2 3779 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑠 ⊆ 𝐴 ↔ (𝐴 ∩ 𝑠) = 𝑠) |
384 | 383 | biimpi 205 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑠 ⊆ 𝐴 → (𝐴 ∩ 𝑠) = 𝑠) |
385 | 384 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 ⊆ 𝐴 → 𝑠 = (𝐴 ∩ 𝑠)) |
386 | 385 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑠 ⊆ 𝐴 → (vol*‘𝑠) = (vol*‘(𝐴 ∩ 𝑠))) |
387 | 386 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . 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))) |
388 | 382, 387 | sylan9eq 2664 |
. . . . . . . . . . . . . . . . . . . . 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))) |
389 | 388 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . 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)))) |
390 | 389 | adantll 746 |
. . . . . . . . . . . . . . . . . . 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)))) |
391 | 379 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ 𝑠 ∈ dom
vol) |
392 | | simplll 794 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
393 | | mblsplit 23107 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘𝐴) = ((vol*‘(𝐴 ∩ 𝑠)) + (vol*‘(𝐴 ∖ 𝑠)))) |
394 | 393 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → ((vol*‘(𝐴 ∩ 𝑠)) + (vol*‘(𝐴 ∖ 𝑠))) = (vol*‘𝐴)) |
395 | 394 | 3expb 1258 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈ dom vol ∧ (𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ)) → ((vol*‘(𝐴 ∩ 𝑠)) + (vol*‘(𝐴 ∖ 𝑠))) = (vol*‘𝐴)) |
396 | 391, 392,
395 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . . 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)) |
397 | 396 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 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)) |
398 | | simp-6r 807 |
. . . . . . . . . . . . . . . . . . . . . 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) |
399 | 398 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . 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) |
400 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝐴 ∩ 𝑠) ⊆ 𝐴 |
401 | | ovolsscl 23061 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝐴 ∩ 𝑠) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘(𝐴 ∩ 𝑠)) ∈
ℝ) |
402 | 400, 401 | mp3an1 1403 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(𝐴 ∩ 𝑠)) ∈ ℝ) |
403 | 402 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(𝐴 ∩ 𝑠)) ∈ ℂ) |
404 | 403 | ad5antr 766 |
. . . . . . . . . . . . . . . . . . . . 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) |
405 | 314 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘(𝐴 ∖ 𝑠)) ∈ ℂ) |
406 | 405 | ad5antr 766 |
. . . . . . . . . . . . . . . . . . . . 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) |
407 | 399, 404,
406 | subaddd 10289 |
. . . . . . . . . . . . . . . . . . . 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))) |
408 | 397, 407 | mpbird 246 |
. . . . . . . . . . . . . . . . . . 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))) |
409 | 390, 408 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . 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))) |
410 | 382 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . 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)) |
411 | | simpll 786 |
. . . . . . . . . . . . . . . . . . . . 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) |
412 | | simp-4l 802 |
. . . . . . . . . . . . . . . . . . . . 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)) |
413 | | ovolsscl 23061 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘𝑠) ∈
ℝ) |
414 | 413 | 3expb 1258 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑠 ⊆ 𝐴 ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) →
(vol*‘𝑠) ∈
ℝ) |
415 | 411, 412,
414 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . 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) |
416 | 410, 415 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . 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) |
417 | | simprlr 799 |
. . . . . . . . . . . . . . . . . . 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)) |
418 | 398, 375,
416, 417 | ltsub23d 10511 |
. . . . . . . . . . . . . . . . . 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)) |
419 | 409, 418 | eqbrtrrd 4607 |
. . . . . . . . . . . . . . . . 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)) |
420 | 323 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . 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) |
421 | 420 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . 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) |
422 | 242 | ad5antlr 767 |
. . . . . . . . . . . . . . . . . . . . 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) |
423 | 422 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . 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) |
424 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑏 = 𝑤 → (𝑏 ∈ dom vol ↔ 𝑤 ∈ dom vol)) |
425 | 424, 34 | vtoclga 3245 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑤 ∈
(Clsd‘(topGen‘ran (,))) → 𝑤 ∈ dom vol) |
426 | | mblvol 23105 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑤 ∈ dom vol →
(vol‘𝑤) =
(vol*‘𝑤)) |
427 | 425, 426 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑤 ∈
(Clsd‘(topGen‘ran (,))) → (vol‘𝑤) = (vol*‘𝑤)) |
428 | 427 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ (vol‘𝑤) =
(vol*‘𝑤)) |
429 | 428 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
430 | | simprl 790 |
. . . . . . . . . . . . . . . . . . . . . . 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) |
431 | | simp-4r 803 |
. . . . . . . . . . . . . . . . . . . . . . 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)) |
432 | | ovolsscl 23061 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑤 ⊆ 𝐵 ∧ 𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ) →
(vol*‘𝑤) ∈
ℝ) |
433 | 432 | 3expb 1258 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑤 ⊆ 𝐵 ∧ (𝐵 ⊆ ℝ ∧ (vol*‘𝐵) ∈ ℝ)) →
(vol*‘𝑤) ∈
ℝ) |
434 | 430, 431,
433 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . . . 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) |
435 | 429, 434 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . 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) |
436 | 435 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . 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) |
437 | 421, 423,
436 | npncand 10295 |
. . . . . . . . . . . . . . . . . . 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))) |
438 | | simplrl 796 |
. . . . . . . . . . . . . . . . . . . . . . . 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)) |
439 | 430, 438 | sylan9ssr 3582 |
. . . . . . . . . . . . . . . . . . . . . . 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)) |
440 | | sseqin2 3779 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑤 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ (∪ ran
((,) ∘ 𝑓) ∩ 𝑤) = 𝑤) |
441 | 439, 440 | sylib 207 |
. . . . . . . . . . . . . . . . . . . . . 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) |
442 | 441 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 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)) |
443 | 429, 442 | eqtr4d 2647 |
. . . . . . . . . . . . . . . . . . . 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))) |
444 | 443 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . 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)))) |
445 | 425 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ 𝑤 ∈ dom
vol) |
446 | 323, 250 | jctil 558 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
447 | | mblsplit 23107 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑤 ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘∪ ran ((,) ∘ 𝑓)) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤)))) |
448 | 447 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑤 ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘∪ ran ((,) ∘ 𝑓))) |
449 | 448 | 3expb 1258 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑤 ∈ dom vol ∧ (∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)) →
((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑤))) = (vol*‘∪ ran ((,) ∘ 𝑓))) |
450 | 445, 446,
449 | syl2anr 494 |
. . . . . . . . . . . . . . . . . . . . 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))) |
451 | 450 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 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))) |
452 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (∪ ran ((,) ∘ 𝑓) ∩ 𝑤) ⊆ ∪ ran
((,) ∘ 𝑓) |
453 | | ovolsscl 23061 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((∪ ran ((,) ∘ 𝑓) ∩ 𝑤) ⊆ ∪ ran
((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ) |
454 | 452, 250,
453 | mp3an12 1406 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ →
(vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑤)) ∈ ℝ) |
455 | 323, 454 | 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) |
456 | 455 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . 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) |
457 | 327 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . . . 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) |
458 | 420, 456,
457 | subaddd 10289 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
459 | 458 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . 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)))) |
460 | 451, 459 | mpbird 246 |
. . . . . . . . . . . . . . . . . . 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))) |
461 | 437, 444,
460 | 3eqtrd 2648 |
. . . . . . . . . . . . . . . . . 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))) |
462 | 242 | ad3antlr 763 |
. . . . . . . . . . . . . . . . . . . . 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) |
463 | 323, 462 | resubcld 10337 |
. . . . . . . . . . . . . . . . . . . 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) |
464 | 463 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . 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) |
465 | 422, 435 | resubcld 10337 |
. . . . . . . . . . . . . . . . . . 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) |
466 | | simprr 792 |
. . . . . . . . . . . . . . . . . . . . 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))) |
467 | 196 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 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) |
468 | 323, 462,
467 | lesubadd2d 10505 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
469 | 466, 468 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . 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)) |
470 | 469 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . 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)) |
471 | | simprrr 801 |
. . . . . . . . . . . . . . . . . . . 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)) |
472 | 422, 375,
435, 471 | ltsub23d 10511 |
. . . . . . . . . . . . . . . . . . 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)) |
473 | 464, 465,
375, 375, 470, 472 | leltaddd 10528 |
. . . . . . . . . . . . . . . . . 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))) |
474 | 461, 473 | eqbrtrrd 4607 |
. . . . . . . . . . . . . . . . 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))) |
475 | 315, 328,
375, 377, 419, 474 | lt2addd 10529 |
. . . . . . . . . . . . . . . 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)))) |
476 | | df-3 10957 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 3 = (2 +
1) |
477 | | 2cn 10968 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 2 ∈
ℂ |
478 | | ax-1cn 9873 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 1 ∈
ℂ |
479 | 477, 478 | addcomi 10106 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (2 + 1) =
(1 + 2) |
480 | 476, 479 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 3 = (1 +
2) |
481 | 480 | oveq1i 6559 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (3
· (((vol*‘(𝐴
∖ 𝐵)) − 𝑢) / 3)) = ((1 + 2) ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) |
482 | 62 | rpcnd 11750 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3) ∈
ℂ) |
483 | | adddir 9910 |
. . . . . . . . . . . . . . . . . . . . . . 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)))) |
484 | 478, 477,
483 | mp3an12 1406 |
. . . . . . . . . . . . . . . . . . . . . 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)))) |
485 | 482, 484 | 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)))) |
486 | 482 | mulid2d 9937 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → (1 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) = (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) |
487 | 482 | 2timesd 11152 |
. . . . . . . . . . . . . . . . . . . . . 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))) |
488 | 486, 487 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . 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)))) |
489 | 485, 488 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . 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)))) |
490 | 481, 489 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . 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)))) |
491 | 331 | recnd 9947 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) →
((vol*‘(𝐴 ∖
𝐵)) − 𝑢) ∈
ℂ) |
492 | | 3cn 10972 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 3 ∈
ℂ |
493 | | 3ne0 10992 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 3 ≠
0 |
494 | | divcan2 10572 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((vol*‘(𝐴
∖ 𝐵)) − 𝑢) ∈ ℂ ∧ 3 ∈
ℂ ∧ 3 ≠ 0) → (3 · (((vol*‘(𝐴 ∖ 𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢)) |
495 | 492, 493,
494 | mp3an23 1408 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((vol*‘(𝐴
∖ 𝐵)) − 𝑢) ∈ ℂ → (3
· (((vol*‘(𝐴
∖ 𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢)) |
496 | 491, 495 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) ∧ (𝑢 ∈
ℝ ∧ 𝑢 <
(vol*‘(𝐴 ∖
𝐵)))) → (3 ·
(((vol*‘(𝐴 ∖
𝐵)) − 𝑢) / 3)) = ((vol*‘(𝐴 ∖ 𝐵)) − 𝑢)) |
497 | 490, 496 | eqtr3d 2646 |
. . . . . . . . . . . . . . . . . 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)) |
498 | 497 | adantlr 747 |
. . . . . . . . . . . . . . . . 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)) |
499 | 498 | ad3antrrr 762 |
. . . . . . . . . . . . . . . 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)) |
500 | 475, 499 | breqtrd 4609 |
. . . . . . . . . . . . . . 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)) |
501 | 311, 329,
333, 374, 500 | lelttrd 10074 |
. . . . . . . . . . . . . 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)) |
502 | 303, 501 | syl5eqbr 4618 |
. . . . . . . . . . . . 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)) |
503 | 298, 299,
301, 502 | ltsub13d 10512 |
. . . . . . . . . . . 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)))))) |
504 | 285 | adantlll 750 |
. . . . . . . . . . . . . . 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)) |
505 | | sseqin2 3779 |
. . . . . . . . . . . . . . 15
⊢ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ⊆ (𝐴 ∖ 𝐵) ↔ ((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) = (𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) |
506 | 504, 505 | sylib 207 |
. . . . . . . . . . . . . 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))) |
507 | 506 | fveq2d 6107 |
. . . . . . . . . . . . 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)))) |
508 | | opnmbl 23176 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) → ∪ ran ((,) ∘ 𝑓) ∈ dom vol) |
509 | 275, 508 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ ran ((,) ∘ 𝑓) ∈ dom vol |
510 | | difmbl 23118 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑠 ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑓) ∈ dom vol) → (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ∈ dom
vol) |
511 | 379, 509,
510 | sylancl 693 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)) ∈ dom
vol) |
512 | 511 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ∈ dom vol) |
513 | 512 | ad2antlr 759 |
. . . . . . . . . . . . . . 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) |
514 | 13 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (𝐴 ∖
𝐵) ⊆
ℝ) |
515 | 514, 5 | jca 553 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → ((𝐴 ∖
𝐵) ⊆ ℝ ∧
(vol*‘(𝐴 ∖
𝐵)) ∈
ℝ)) |
516 | 515 | ad5antr 766 |
. . . . . . . . . . . . . . 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)) |
517 | | mblsplit 23107 |
. . . . . . . . . . . . . . . . 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)))))) |
518 | 517 | 3expb 1258 |
. . . . . . . . . . . . . . . 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)))))) |
519 | 518 | eqcomd 2616 |
. . . . . . . . . . . . . . 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))) |
520 | 513, 516,
519 | syl2anc 691 |
. . . . . . . . . . . . . 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))) |
521 | 299 | recnd 9947 |
. . . . . . . . . . . . . . 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) |
522 | 298 | recnd 9947 |
. . . . . . . . . . . . . . 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) |
523 | | inss1 3795 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
(𝐴 ∖ 𝐵) |
524 | 523, 3 | sstri 3577 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 |
525 | | ovolsscl 23061 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) ⊆
𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) →
(vol*‘((𝐴 ∖
𝐵) ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑓)))) ∈ ℝ) |
526 | 524, 525 | mp3an1 1403 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ⊆ ℝ ∧
(vol*‘𝐴) ∈
ℝ) → (vol*‘((𝐴 ∖ 𝐵) ∩ (𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) ∈
ℝ) |
527 | 526 | ad5antr 766 |
. . . . . . . . . . . . . . . 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) |
528 | 527 | recnd 9947 |
. . . . . . . . . . . . . . 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) |
529 | 521, 522,
528 | subadd2d 10290 |
. . . . . . . . . . . . . 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)))) |
530 | 520, 529 | mpbird 246 |
. . . . . . . . . . . . 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))))) |
531 | | mblvol 23105 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑓)) ∈ dom vol → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) |
532 | 510, 531 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑠 ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑓) ∈ dom vol) → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) |
533 | 379, 509,
532 | sylancl 693 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 ∈
(Clsd‘(topGen‘ran (,))) → (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol*‘(𝑠 ∖
∪ ran ((,) ∘ 𝑓)))) |
534 | 533 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑠 ∈
(Clsd‘(topGen‘ran (,))) ∧ 𝑤 ∈ (Clsd‘(topGen‘ran (,))))
→ (vol‘(𝑠
∖ ∪ ran ((,) ∘ 𝑓))) = (vol*‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓)))) |
535 | 534 | ad2antlr 759 |
. . . . . . . . . . . . 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)))) |
536 | 507, 530,
535 | 3eqtr4rd 2655 |
. . . . . . . . . . . 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)))))) |
537 | 503, 536 | breqtrrd 4611 |
. . . . . . . . . . 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)))) |
538 | | fvex 6113 |
. . . . . . . . . . . 12
⊢
(vol‘(𝑠
∖ ∪ ran ((,) ∘ 𝑓))) ∈ V |
539 | | eqeq1 2614 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) → (𝑣 = (vol‘𝑏) ↔ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏))) |
540 | 539 | anbi2d 736 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) → ((𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))) =
(vol‘𝑏)))) |
541 | 540 | rexbidv 3034 |
. . . . . . . . . . . . 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)))) |
542 | | breq2 4587 |
. . . . . . . . . . . . 13
⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑓))) → (𝑢 < 𝑣 ↔ 𝑢 < (vol‘(𝑠 ∖ ∪ ran
((,) ∘ 𝑓))))) |
543 | 541, 542 | anbi12d 743 |
. . . . . . . . . . . 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)))))) |
544 | 538, 543 | spcev 3273 |
. . . . . . . . . . 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)) |
545 | 293, 537,
544 | syl2anc 691 |
. . . . . . . . . 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)) |
546 | 150 | anbi2d 736 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑣 → ((𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)))) |
547 | 546 | rexbidv 3034 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran
(,)))(𝑏 ⊆ (𝐴 ∖ 𝐵) ∧ 𝑣 = (vol‘𝑏)))) |
548 | 547 | rexab 3336 |
. . . . . . . . . 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)) |
549 | 545, 548 | sylibr 223 |
. . . . . . . . 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) |
550 | 549 | ex 449 |
. . . . . . . 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)) |
551 | 550 | rexlimdvva 3020 |
. . . . . . 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)) |
552 | 262, 551 | exlimddv 1850 |
. . . . . 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)) |
553 | 223, 552 | syld 46 |
. . . . 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)) |
554 | 553 | exp31 628 |
. . . 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)))) |
555 | 554 | com34 89 |
. . 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)))) |
556 | 555 | 3imp1 1272 |
. 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) |
557 | 2, 6, 48, 556 | eqsupd 8246 |
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))) |