Proof of Theorem ovolsplit
Step | Hyp | Ref
| Expression |
1 | | inundif 4368 |
. . . . 5
⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴 |
2 | 1 | eqcomi 2747 |
. . . 4
⊢ 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) |
3 | 2 | a1i 11 |
. . 3
⊢ (𝜑 → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) |
4 | 3 | fveq2d 6678 |
. 2
⊢ (𝜑 → (vol*‘𝐴) = (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)))) |
5 | | ovolsplit.1 |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ⊆ ℝ) |
6 | 5 | ssinss1d 42134 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ ℝ) |
7 | 5 | ssdifssd 4033 |
. . . . . . . 8
⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ ℝ) |
8 | 6, 7 | unssd 4076 |
. . . . . . 7
⊢ (𝜑 → ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ ℝ) |
9 | | ovolcl 24230 |
. . . . . . 7
⊢ (((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) ⊆ ℝ →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ∈
ℝ*) |
10 | 8, 9 | syl 17 |
. . . . . 6
⊢ (𝜑 → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ∈
ℝ*) |
11 | | pnfge 12608 |
. . . . . 6
⊢
((vol*‘((𝐴
∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ∈ ℝ* →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ +∞) |
12 | 10, 11 | syl 17 |
. . . . 5
⊢ (𝜑 → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ +∞) |
13 | 12 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ +∞) |
14 | | oveq1 7177 |
. . . . . 6
⊢
((vol*‘(𝐴
∩ 𝐵)) = +∞ →
((vol*‘(𝐴 ∩ 𝐵)) +𝑒
(vol*‘(𝐴 ∖
𝐵))) = (+∞
+𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
15 | 14 | adantl 485 |
. . . . 5
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵))) = (+∞ +𝑒
(vol*‘(𝐴 ∖
𝐵)))) |
16 | | ovolcl 24230 |
. . . . . . . 8
⊢ ((𝐴 ∖ 𝐵) ⊆ ℝ → (vol*‘(𝐴 ∖ 𝐵)) ∈
ℝ*) |
17 | 7, 16 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (vol*‘(𝐴 ∖ 𝐵)) ∈
ℝ*) |
18 | 17 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘(𝐴 ∖ 𝐵)) ∈
ℝ*) |
19 | | reex 10706 |
. . . . . . . . . . . . . 14
⊢ ℝ
∈ V |
20 | 19 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ℝ ∈
V) |
21 | 20, 5 | ssexd 5192 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ V) |
22 | 21 | difexd 5197 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V) |
23 | | elpwg 4491 |
. . . . . . . . . . 11
⊢ ((𝐴 ∖ 𝐵) ∈ V → ((𝐴 ∖ 𝐵) ∈ 𝒫 ℝ ↔ (𝐴 ∖ 𝐵) ⊆ ℝ)) |
24 | 22, 23 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴 ∖ 𝐵) ∈ 𝒫 ℝ ↔ (𝐴 ∖ 𝐵) ⊆ ℝ)) |
25 | 7, 24 | mpbird 260 |
. . . . . . . . 9
⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝒫 ℝ) |
26 | | ovolf 24234 |
. . . . . . . . . 10
⊢
vol*:𝒫 ℝ⟶(0[,]+∞) |
27 | 26 | ffvelrni 6860 |
. . . . . . . . 9
⊢ ((𝐴 ∖ 𝐵) ∈ 𝒫 ℝ →
(vol*‘(𝐴 ∖
𝐵)) ∈
(0[,]+∞)) |
28 | 25, 27 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (vol*‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞)) |
29 | 28 | xrge0nemnfd 42409 |
. . . . . . 7
⊢ (𝜑 → (vol*‘(𝐴 ∖ 𝐵)) ≠ -∞) |
30 | 29 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘(𝐴 ∖ 𝐵)) ≠ -∞) |
31 | | xaddpnf2 12703 |
. . . . . 6
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈
ℝ* ∧ (vol*‘(𝐴 ∖ 𝐵)) ≠ -∞) → (+∞
+𝑒 (vol*‘(𝐴 ∖ 𝐵))) = +∞) |
32 | 18, 30, 31 | syl2anc 587 |
. . . . 5
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (+∞
+𝑒 (vol*‘(𝐴 ∖ 𝐵))) = +∞) |
33 | 15, 32 | eqtr2d 2774 |
. . . 4
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → +∞ =
((vol*‘(𝐴 ∩ 𝐵)) +𝑒
(vol*‘(𝐴 ∖
𝐵)))) |
34 | 13, 33 | breqtrd 5056 |
. . 3
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
35 | | simpl 486 |
. . . 4
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → 𝜑) |
36 | 20, 6 | sselpwd 5194 |
. . . . . . 7
⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ 𝒫 ℝ) |
37 | 26 | ffvelrni 6860 |
. . . . . . 7
⊢ ((𝐴 ∩ 𝐵) ∈ 𝒫 ℝ →
(vol*‘(𝐴 ∩ 𝐵)) ∈
(0[,]+∞)) |
38 | 36, 37 | syl 17 |
. . . . . 6
⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) ∈ (0[,]+∞)) |
39 | 38 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘(𝐴 ∩ 𝐵)) ∈ (0[,]+∞)) |
40 | | neqne 2942 |
. . . . . 6
⊢ (¬
(vol*‘(𝐴 ∩ 𝐵)) = +∞ →
(vol*‘(𝐴 ∩ 𝐵)) ≠
+∞) |
41 | 40 | adantl 485 |
. . . . 5
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘(𝐴 ∩ 𝐵)) ≠ +∞) |
42 | | ge0xrre 42609 |
. . . . 5
⊢
(((vol*‘(𝐴
∩ 𝐵)) ∈
(0[,]+∞) ∧ (vol*‘(𝐴 ∩ 𝐵)) ≠ +∞) → (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) |
43 | 39, 41, 42 | syl2anc 587 |
. . . 4
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) |
44 | 12 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ +∞) |
45 | | oveq2 7178 |
. . . . . . . . 9
⊢
((vol*‘(𝐴
∖ 𝐵)) = +∞
→ ((vol*‘(𝐴
∩ 𝐵))
+𝑒 (vol*‘(𝐴 ∖ 𝐵))) = ((vol*‘(𝐴 ∩ 𝐵)) +𝑒
+∞)) |
46 | 45 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵))) = ((vol*‘(𝐴 ∩ 𝐵)) +𝑒
+∞)) |
47 | | ovolcl 24230 |
. . . . . . . . . . 11
⊢ ((𝐴 ∩ 𝐵) ⊆ ℝ → (vol*‘(𝐴 ∩ 𝐵)) ∈
ℝ*) |
48 | 6, 47 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) ∈
ℝ*) |
49 | 38 | xrge0nemnfd 42409 |
. . . . . . . . . 10
⊢ (𝜑 → (vol*‘(𝐴 ∩ 𝐵)) ≠ -∞) |
50 | | xaddpnf1 12702 |
. . . . . . . . . 10
⊢
(((vol*‘(𝐴
∩ 𝐵)) ∈
ℝ* ∧ (vol*‘(𝐴 ∩ 𝐵)) ≠ -∞) → ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 +∞) =
+∞) |
51 | 48, 49, 50 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 +∞) =
+∞) |
52 | 51 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 +∞) =
+∞) |
53 | 46, 52 | eqtr2d 2774 |
. . . . . . 7
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → +∞ =
((vol*‘(𝐴 ∩ 𝐵)) +𝑒
(vol*‘(𝐴 ∖
𝐵)))) |
54 | 44, 53 | breqtrd 5056 |
. . . . . 6
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
55 | 54 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
56 | | simpll 767 |
. . . . . 6
⊢ (((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ ¬
(vol*‘(𝐴 ∖
𝐵)) = +∞) →
𝜑) |
57 | | simplr 769 |
. . . . . 6
⊢ (((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ ¬
(vol*‘(𝐴 ∖
𝐵)) = +∞) →
(vol*‘(𝐴 ∩ 𝐵)) ∈
ℝ) |
58 | 28 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘(𝐴 ∖ 𝐵)) ∈ (0[,]+∞)) |
59 | | neqne 2942 |
. . . . . . . . 9
⊢ (¬
(vol*‘(𝐴 ∖
𝐵)) = +∞ →
(vol*‘(𝐴 ∖
𝐵)) ≠
+∞) |
60 | 59 | adantl 485 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘(𝐴 ∖ 𝐵)) ≠ +∞) |
61 | | ge0xrre 42609 |
. . . . . . . 8
⊢
(((vol*‘(𝐴
∖ 𝐵)) ∈
(0[,]+∞) ∧ (vol*‘(𝐴 ∖ 𝐵)) ≠ +∞) → (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
62 | 58, 60, 61 | syl2anc 587 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∖ 𝐵)) = +∞) → (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
63 | 62 | adantlr 715 |
. . . . . 6
⊢ (((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ ¬
(vol*‘(𝐴 ∖
𝐵)) = +∞) →
(vol*‘(𝐴 ∖
𝐵)) ∈
ℝ) |
64 | 6 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) → (𝐴 ∩ 𝐵) ⊆ ℝ) |
65 | | simp2 1138 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) → (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) |
66 | 7 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) → (𝐴 ∖ 𝐵) ⊆ ℝ) |
67 | | simp3 1139 |
. . . . . . . 8
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) → (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) |
68 | | ovolun 24251 |
. . . . . . . 8
⊢ ((((𝐴 ∩ 𝐵) ⊆ ℝ ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ ((𝐴 ∖ 𝐵) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ)) →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) + (vol*‘(𝐴 ∖ 𝐵)))) |
69 | 64, 65, 66, 67, 68 | syl22anc 838 |
. . . . . . 7
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) + (vol*‘(𝐴 ∖ 𝐵)))) |
70 | | rexadd 12708 |
. . . . . . . . 9
⊢
(((vol*‘(𝐴
∩ 𝐵)) ∈ ℝ
∧ (vol*‘(𝐴
∖ 𝐵)) ∈ ℝ)
→ ((vol*‘(𝐴
∩ 𝐵))
+𝑒 (vol*‘(𝐴 ∖ 𝐵))) = ((vol*‘(𝐴 ∩ 𝐵)) + (vol*‘(𝐴 ∖ 𝐵)))) |
71 | 70 | eqcomd 2744 |
. . . . . . . 8
⊢
(((vol*‘(𝐴
∩ 𝐵)) ∈ ℝ
∧ (vol*‘(𝐴
∖ 𝐵)) ∈ ℝ)
→ ((vol*‘(𝐴
∩ 𝐵)) +
(vol*‘(𝐴 ∖
𝐵))) = ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
72 | 71 | 3adant1 1131 |
. . . . . . 7
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) →
((vol*‘(𝐴 ∩ 𝐵)) + (vol*‘(𝐴 ∖ 𝐵))) = ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
73 | 69, 72 | breqtrd 5056 |
. . . . . 6
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ ∧ (vol*‘(𝐴 ∖ 𝐵)) ∈ ℝ) →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
74 | 56, 57, 63, 73 | syl3anc 1372 |
. . . . 5
⊢ (((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) ∧ ¬
(vol*‘(𝐴 ∖
𝐵)) = +∞) →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
75 | 55, 74 | pm2.61dan 813 |
. . . 4
⊢ ((𝜑 ∧ (vol*‘(𝐴 ∩ 𝐵)) ∈ ℝ) →
(vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
76 | 35, 43, 75 | syl2anc 587 |
. . 3
⊢ ((𝜑 ∧ ¬ (vol*‘(𝐴 ∩ 𝐵)) = +∞) → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
77 | 34, 76 | pm2.61dan 813 |
. 2
⊢ (𝜑 → (vol*‘((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |
78 | 4, 77 | eqbrtrd 5052 |
1
⊢ (𝜑 → (vol*‘𝐴) ≤ ((vol*‘(𝐴 ∩ 𝐵)) +𝑒 (vol*‘(𝐴 ∖ 𝐵)))) |