Proof of Theorem ovolioo
Step | Hyp | Ref
| Expression |
1 | | ioombl 23769 |
. . 3
⊢ (𝐴(,)𝐵) ∈ dom vol |
2 | | mblvol 23734 |
. . 3
⊢ ((𝐴(,)𝐵) ∈ dom vol → (vol‘(𝐴(,)𝐵)) = (vol*‘(𝐴(,)𝐵))) |
3 | 1, 2 | ax-mp 5 |
. 2
⊢
(vol‘(𝐴(,)𝐵)) = (vol*‘(𝐴(,)𝐵)) |
4 | | iccmbl 23770 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol) |
5 | | mblvol 23734 |
. . . . 5
⊢ ((𝐴[,]𝐵) ∈ dom vol → (vol‘(𝐴[,]𝐵)) = (vol*‘(𝐴[,]𝐵))) |
6 | 4, 5 | syl 17 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) →
(vol‘(𝐴[,]𝐵)) = (vol*‘(𝐴[,]𝐵))) |
7 | 6 | 3adant3 1123 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴[,]𝐵)) = (vol*‘(𝐴[,]𝐵))) |
8 | 1 | a1i 11 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴(,)𝐵) ∈ dom vol) |
9 | | prssi 4583 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → {𝐴, 𝐵} ⊆ ℝ) |
10 | 9 | 3adant3 1123 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → {𝐴, 𝐵} ⊆ ℝ) |
11 | | prfi 8523 |
. . . . . . 7
⊢ {𝐴, 𝐵} ∈ Fin |
12 | | ovolfi 23698 |
. . . . . . 7
⊢ (({𝐴, 𝐵} ∈ Fin ∧ {𝐴, 𝐵} ⊆ ℝ) → (vol*‘{𝐴, 𝐵}) = 0) |
13 | 11, 10, 12 | sylancr 581 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘{𝐴, 𝐵}) = 0) |
14 | | nulmbl 23739 |
. . . . . 6
⊢ (({𝐴, 𝐵} ⊆ ℝ ∧ (vol*‘{𝐴, 𝐵}) = 0) → {𝐴, 𝐵} ∈ dom vol) |
15 | 10, 13, 14 | syl2anc 579 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → {𝐴, 𝐵} ∈ dom vol) |
16 | | df-pr 4401 |
. . . . . . . 8
⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) |
17 | 16 | ineq2i 4034 |
. . . . . . 7
⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ((𝐴(,)𝐵) ∩ ({𝐴} ∪ {𝐵})) |
18 | | indi 4100 |
. . . . . . 7
⊢ ((𝐴(,)𝐵) ∩ ({𝐴} ∪ {𝐵})) = (((𝐴(,)𝐵) ∩ {𝐴}) ∪ ((𝐴(,)𝐵) ∩ {𝐵})) |
19 | 17, 18 | eqtri 2802 |
. . . . . 6
⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = (((𝐴(,)𝐵) ∩ {𝐴}) ∪ ((𝐴(,)𝐵) ∩ {𝐵})) |
20 | | simp1 1127 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ) |
21 | 20 | ltnrd 10510 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ¬ 𝐴 < 𝐴) |
22 | | eliooord 12545 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐴 ∧ 𝐴 < 𝐵)) |
23 | 22 | simpld 490 |
. . . . . . . . . 10
⊢ (𝐴 ∈ (𝐴(,)𝐵) → 𝐴 < 𝐴) |
24 | 21, 23 | nsyl 138 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ¬ 𝐴 ∈ (𝐴(,)𝐵)) |
25 | | disjsn 4478 |
. . . . . . . . 9
⊢ (((𝐴(,)𝐵) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ (𝐴(,)𝐵)) |
26 | 24, 25 | sylibr 226 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∩ {𝐴}) = ∅) |
27 | | simp2 1128 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ) |
28 | 27 | ltnrd 10510 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ¬ 𝐵 < 𝐵) |
29 | | eliooord 12545 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐵 ∧ 𝐵 < 𝐵)) |
30 | 29 | simprd 491 |
. . . . . . . . . 10
⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 < 𝐵) |
31 | 28, 30 | nsyl 138 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ¬ 𝐵 ∈ (𝐴(,)𝐵)) |
32 | | disjsn 4478 |
. . . . . . . . 9
⊢ (((𝐴(,)𝐵) ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ (𝐴(,)𝐵)) |
33 | 31, 32 | sylibr 226 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∩ {𝐵}) = ∅) |
34 | 26, 33 | uneq12d 3991 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (((𝐴(,)𝐵) ∩ {𝐴}) ∪ ((𝐴(,)𝐵) ∩ {𝐵})) = (∅ ∪
∅)) |
35 | | un0 4193 |
. . . . . . 7
⊢ (∅
∪ ∅) = ∅ |
36 | 34, 35 | syl6eq 2830 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (((𝐴(,)𝐵) ∩ {𝐴}) ∪ ((𝐴(,)𝐵) ∩ {𝐵})) = ∅) |
37 | 19, 36 | syl5eq 2826 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅) |
38 | | ioossicc 12571 |
. . . . . . . 8
⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) |
39 | 38 | a1i 11 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
40 | | iccssre 12567 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) |
41 | 40 | 3adant3 1123 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) ⊆ ℝ) |
42 | | ovolicc 23727 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)) |
43 | 27, 20 | resubcld 10803 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (𝐵 − 𝐴) ∈ ℝ) |
44 | 42, 43 | eqeltrd 2859 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) ∈ ℝ) |
45 | | ovolsscl 23690 |
. . . . . . 7
⊢ (((𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) ∧ (𝐴[,]𝐵) ⊆ ℝ ∧ (vol*‘(𝐴[,]𝐵)) ∈ ℝ) → (vol*‘(𝐴(,)𝐵)) ∈ ℝ) |
46 | 39, 41, 44, 45 | syl3anc 1439 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴(,)𝐵)) ∈ ℝ) |
47 | 3, 46 | syl5eqel 2863 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) ∈ ℝ) |
48 | | mblvol 23734 |
. . . . . . . 8
⊢ ({𝐴, 𝐵} ∈ dom vol → (vol‘{𝐴, 𝐵}) = (vol*‘{𝐴, 𝐵})) |
49 | 15, 48 | syl 17 |
. . . . . . 7
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘{𝐴, 𝐵}) = (vol*‘{𝐴, 𝐵})) |
50 | 49, 13 | eqtrd 2814 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘{𝐴, 𝐵}) = 0) |
51 | | 0re 10378 |
. . . . . 6
⊢ 0 ∈
ℝ |
52 | 50, 51 | syl6eqel 2867 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘{𝐴, 𝐵}) ∈ ℝ) |
53 | | volun 23749 |
. . . . 5
⊢ ((((𝐴(,)𝐵) ∈ dom vol ∧ {𝐴, 𝐵} ∈ dom vol ∧ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅) ∧ ((vol‘(𝐴(,)𝐵)) ∈ ℝ ∧ (vol‘{𝐴, 𝐵}) ∈ ℝ)) →
(vol‘((𝐴(,)𝐵) ∪ {𝐴, 𝐵})) = ((vol‘(𝐴(,)𝐵)) + (vol‘{𝐴, 𝐵}))) |
54 | 8, 15, 37, 47, 52, 53 | syl32anc 1446 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘((𝐴(,)𝐵) ∪ {𝐴, 𝐵})) = ((vol‘(𝐴(,)𝐵)) + (vol‘{𝐴, 𝐵}))) |
55 | | rexr 10422 |
. . . . . 6
⊢ (𝐴 ∈ ℝ → 𝐴 ∈
ℝ*) |
56 | | rexr 10422 |
. . . . . 6
⊢ (𝐵 ∈ ℝ → 𝐵 ∈
ℝ*) |
57 | | id 22 |
. . . . . 6
⊢ (𝐴 ≤ 𝐵 → 𝐴 ≤ 𝐵) |
58 | | prunioo 12618 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ*
∧ 𝐵 ∈
ℝ* ∧ 𝐴
≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) |
59 | 55, 56, 57, 58 | syl3an 1160 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) |
60 | 59 | fveq2d 6450 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘((𝐴(,)𝐵) ∪ {𝐴, 𝐵})) = (vol‘(𝐴[,]𝐵))) |
61 | 50 | oveq2d 6938 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ((vol‘(𝐴(,)𝐵)) + (vol‘{𝐴, 𝐵})) = ((vol‘(𝐴(,)𝐵)) + 0)) |
62 | 47 | recnd 10405 |
. . . . . 6
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) ∈ ℂ) |
63 | 62 | addid1d 10576 |
. . . . 5
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ((vol‘(𝐴(,)𝐵)) + 0) = (vol‘(𝐴(,)𝐵))) |
64 | 61, 63 | eqtrd 2814 |
. . . 4
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → ((vol‘(𝐴(,)𝐵)) + (vol‘{𝐴, 𝐵})) = (vol‘(𝐴(,)𝐵))) |
65 | 54, 60, 64 | 3eqtr3d 2822 |
. . 3
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴[,]𝐵)) = (vol‘(𝐴(,)𝐵))) |
66 | 7, 65, 42 | 3eqtr3d 2822 |
. 2
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol‘(𝐴(,)𝐵)) = (𝐵 − 𝐴)) |
67 | 3, 66 | syl5eqr 2828 |
1
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴(,)𝐵)) = (𝐵 − 𝐴)) |