Step | Hyp | Ref
| Expression |
1 | | ovnhoi.x |
. . 3
⊢ (𝜑 → 𝑋 ∈ Fin) |
2 | | ovnhoi.c |
. . . . 5
⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) |
3 | 2 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
4 | | nfv 1921 |
. . . . 5
⊢
Ⅎ𝑘𝜑 |
5 | | ovnhoi.a |
. . . . . 6
⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
6 | 5 | ffvelrnda 6861 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
7 | | ovnhoi.b |
. . . . . . 7
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
8 | 7 | ffvelrnda 6861 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
9 | 8 | rexrd 10769 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈
ℝ*) |
10 | 4, 6, 9 | hoissrrn2 43678 |
. . . 4
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ (ℝ ↑m 𝑋)) |
11 | 3, 10 | eqsstrd 3915 |
. . 3
⊢ (𝜑 → 𝐼 ⊆ (ℝ ↑m 𝑋)) |
12 | 1, 11 | ovnxrcl 43669 |
. 2
⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ∈
ℝ*) |
13 | | icossxr 12906 |
. . 3
⊢
(0[,)+∞) ⊆ ℝ* |
14 | | ovnhoi.l |
. . . 4
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
15 | 14, 1, 5, 7 | hoidmvcl 43682 |
. . 3
⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈ (0[,)+∞)) |
16 | 13, 15 | sseldi 3875 |
. 2
⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ∈
ℝ*) |
17 | | fveq2 6674 |
. . . . . . . 8
⊢ (𝑋 = ∅ →
(voln*‘𝑋) =
(voln*‘∅)) |
18 | 17 | fveq1d 6676 |
. . . . . . 7
⊢ (𝑋 = ∅ →
((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼)) |
19 | 18 | adantl 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = ((voln*‘∅)‘𝐼)) |
20 | | ixpeq1 8518 |
. . . . . . . . . . 11
⊢ (𝑋 = ∅ → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ ∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
21 | | ixp0x 8536 |
. . . . . . . . . . . 12
⊢ X𝑘 ∈
∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅} |
22 | 21 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑋 = ∅ → X𝑘 ∈
∅ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅}) |
23 | 20, 22 | eqtrd 2773 |
. . . . . . . . . 10
⊢ (𝑋 = ∅ → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅}) |
24 | 23 | adantl 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 = ∅) → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = {∅}) |
25 | 2 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
26 | | reex 10706 |
. . . . . . . . . . 11
⊢ ℝ
∈ V |
27 | | mapdm0 8452 |
. . . . . . . . . . 11
⊢ (ℝ
∈ V → (ℝ ↑m ∅) =
{∅}) |
28 | 26, 27 | ax-mp 5 |
. . . . . . . . . 10
⊢ (ℝ
↑m ∅) = {∅} |
29 | 28 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 = ∅) → (ℝ
↑m ∅) = {∅}) |
30 | 24, 25, 29 | 3eqtr4d 2783 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 = (ℝ ↑m
∅)) |
31 | | eqimss 3933 |
. . . . . . . 8
⊢ (𝐼 = (ℝ ↑m
∅) → 𝐼 ⊆
(ℝ ↑m ∅)) |
32 | 30, 31 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐼 ⊆ (ℝ ↑m
∅)) |
33 | 32 | ovn0val 43650 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 = ∅) →
((voln*‘∅)‘𝐼) = 0) |
34 | 19, 33 | eqtrd 2773 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = 0) |
35 | | 0red 10722 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 ∈
ℝ) |
36 | 34, 35 | eqeltrd 2833 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ∈ ℝ) |
37 | | eqidd 2739 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 = ∅) → 0 = 0) |
38 | | fveq2 6674 |
. . . . . . . 8
⊢ (𝑋 = ∅ → (𝐿‘𝑋) = (𝐿‘∅)) |
39 | 38 | oveqd 7187 |
. . . . . . 7
⊢ (𝑋 = ∅ → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵)) |
40 | 39 | adantl 485 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = (𝐴(𝐿‘∅)𝐵)) |
41 | 5 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ) |
42 | | simpr 488 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝑋 = ∅) |
43 | 42 | feq2d 6490 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴:𝑋⟶ℝ ↔ 𝐴:∅⟶ℝ)) |
44 | 41, 43 | mpbid 235 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐴:∅⟶ℝ) |
45 | 7 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ) |
46 | 42 | feq2d 6490 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐵:𝑋⟶ℝ ↔ 𝐵:∅⟶ℝ)) |
47 | 45, 46 | mpbid 235 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑋 = ∅) → 𝐵:∅⟶ℝ) |
48 | 14, 44, 47 | hoidmv0val 43683 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘∅)𝐵) = 0) |
49 | 40, 48 | eqtrd 2773 |
. . . . 5
⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = 0) |
50 | 37, 34, 49 | 3eqtr4d 2783 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵)) |
51 | 36, 50 | eqled 10821 |
. . 3
⊢ ((𝜑 ∧ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵)) |
52 | | eqid 2738 |
. . . . . 6
⊢ {𝑧 ∈ ℝ*
∣ ∃𝑖 ∈
(((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} |
53 | | eqeq1 2742 |
. . . . . . . . 9
⊢ (𝑛 = 𝑗 → (𝑛 = 1 ↔ 𝑗 = 1)) |
54 | 53 | ifbid 4437 |
. . . . . . . 8
⊢ (𝑛 = 𝑗 → if(𝑛 = 1, 〈(𝐴‘𝑘), (𝐵‘𝑘)〉, 〈0, 0〉) = if(𝑗 = 1, 〈(𝐴‘𝑘), (𝐵‘𝑘)〉, 〈0, 0〉)) |
55 | 54 | mpteq2dv 5126 |
. . . . . . 7
⊢ (𝑛 = 𝑗 → (𝑘 ∈ 𝑋 ↦ if(𝑛 = 1, 〈(𝐴‘𝑘), (𝐵‘𝑘)〉, 〈0, 0〉)) = (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, 〈(𝐴‘𝑘), (𝐵‘𝑘)〉, 〈0, 0〉))) |
56 | 55 | cbvmptv 5133 |
. . . . . 6
⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑛 = 1, 〈(𝐴‘𝑘), (𝐵‘𝑘)〉, 〈0, 0〉))) = (𝑗 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ if(𝑗 = 1, 〈(𝐴‘𝑘), (𝐵‘𝑘)〉, 〈0, 0〉))) |
57 | 1, 5, 7, 2, 52, 56 | ovnhoilem1 43701 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
58 | 57 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
59 | 1 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ∈ Fin) |
60 | | neqne 2942 |
. . . . . . 7
⊢ (¬
𝑋 = ∅ → 𝑋 ≠ ∅) |
61 | 60 | adantl 485 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅) |
62 | 5 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐴:𝑋⟶ℝ) |
63 | 7 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → 𝐵:𝑋⟶ℝ) |
64 | 14, 59, 61, 62, 63 | hoidmvn0val 43684 |
. . . . 5
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
65 | 64 | eqcomd 2744 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ∏𝑘 ∈ 𝑋 (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = (𝐴(𝐿‘𝑋)𝐵)) |
66 | 58, 65 | breqtrd 5056 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵)) |
67 | 51, 66 | pm2.61dan 813 |
. 2
⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) ≤ (𝐴(𝐿‘𝑋)𝐵)) |
68 | 49, 35 | eqeltrd 2833 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ∈ ℝ) |
69 | 50 | eqcomd 2744 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) = ((voln*‘𝑋)‘𝐼)) |
70 | 68, 69 | eqled 10821 |
. . 3
⊢ ((𝜑 ∧ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼)) |
71 | | fveq1 6673 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑐 → (𝑎‘𝑘) = (𝑐‘𝑘)) |
72 | 71 | fvoveq1d 7192 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝑐 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘)))) |
73 | 72 | prodeq2ad 42695 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑐 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘)))) |
74 | 73 | ifeq2d 4434 |
. . . . . . . . 9
⊢ (𝑎 = 𝑐 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))))) |
75 | | fveq1 6673 |
. . . . . . . . . . . . 13
⊢ (𝑏 = 𝑑 → (𝑏‘𝑘) = (𝑑‘𝑘)) |
76 | 75 | oveq2d 7186 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝑑 → ((𝑐‘𝑘)[,)(𝑏‘𝑘)) = ((𝑐‘𝑘)[,)(𝑑‘𝑘))) |
77 | 76 | fveq2d 6678 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑑 → (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))) |
78 | 77 | prodeq2ad 42695 |
. . . . . . . . . 10
⊢ (𝑏 = 𝑑 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))) |
79 | 78 | ifeq2d 4434 |
. . . . . . . . 9
⊢ (𝑏 = 𝑑 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))) |
80 | 74, 79 | cbvmpov 7263 |
. . . . . . . 8
⊢ (𝑎 ∈ (ℝ
↑m 𝑥),
𝑏 ∈ (ℝ
↑m 𝑥)
↦ if(𝑥 = ∅, 0,
∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑥), 𝑑 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))) |
81 | 80 | a1i 11 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑥), 𝑑 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))))) |
82 | | oveq2 7178 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → (ℝ ↑m 𝑥) = (ℝ ↑m
𝑦)) |
83 | | eqeq1 2742 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) |
84 | | prodeq1 15355 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))) = ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))) |
85 | 83, 84 | ifbieq2d 4440 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))) = if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))) |
86 | 82, 82, 85 | mpoeq123dv 7243 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝑐 ∈ (ℝ ↑m 𝑥), 𝑑 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))))) |
87 | 81, 86 | eqtrd 2773 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) = (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))))) |
88 | 87 | cbvmptv 5133 |
. . . . 5
⊢ (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ
↑m 𝑥),
𝑏 ∈ (ℝ
↑m 𝑥)
↦ if(𝑥 = ∅, 0,
∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))))) |
89 | 14, 88 | eqtri 2761 |
. . . 4
⊢ 𝐿 = (𝑦 ∈ Fin ↦ (𝑐 ∈ (ℝ ↑m 𝑦), 𝑑 ∈ (ℝ ↑m 𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑐‘𝑘)[,)(𝑑‘𝑘)))))) |
90 | | eqeq1 2742 |
. . . . . . . 8
⊢ (𝑤 = 𝑧 → (𝑤 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) ↔ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))))) |
91 | 90 | anbi2d 632 |
. . . . . . 7
⊢ (𝑤 = 𝑧 → ((𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑤 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ (𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))))) |
92 | 91 | rexbidv 3207 |
. . . . . 6
⊢ (𝑤 = 𝑧 → (∃ℎ ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ)(𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑤 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ ∃ℎ ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ)(𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))))) |
93 | | simpl 486 |
. . . . . . . . . . . . . . 15
⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ℎ = 𝑖) |
94 | 93 | fveq1d 6676 |
. . . . . . . . . . . . . 14
⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (ℎ‘𝑗) = (𝑖‘𝑗)) |
95 | 94 | coeq2d 5705 |
. . . . . . . . . . . . 13
⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗))) |
96 | 95 | fveq1d 6676 |
. . . . . . . . . . . 12
⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘)) |
97 | 96 | ixpeq2dv 8523 |
. . . . . . . . . . 11
⊢ ((ℎ = 𝑖 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈
𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) |
98 | 97 | iuneq2dv 4905 |
. . . . . . . . . 10
⊢ (ℎ = 𝑖 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) = ∪ 𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘)) |
99 | 98 | sseq2d 3909 |
. . . . . . . . 9
⊢ (ℎ = 𝑖 → (𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ↔ 𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘))) |
100 | | simpl 486 |
. . . . . . . . . . . . . . . . 17
⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ℎ = 𝑖) |
101 | 100 | fveq1d 6676 |
. . . . . . . . . . . . . . . 16
⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (ℎ‘𝑗) = (𝑖‘𝑗)) |
102 | 101 | coeq2d 5705 |
. . . . . . . . . . . . . . 15
⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → ([,) ∘ (ℎ‘𝑗)) = ([,) ∘ (𝑖‘𝑗))) |
103 | 102 | fveq1d 6676 |
. . . . . . . . . . . . . 14
⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (ℎ‘𝑗))‘𝑘) = (([,) ∘ (𝑖‘𝑗))‘𝑘)) |
104 | 103 | fveq2d 6678 |
. . . . . . . . . . . . 13
⊢ ((ℎ = 𝑖 ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) |
105 | 104 | prodeq2dv 15369 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑖 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) |
106 | 105 | mpteq2dv 5126 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑖 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) |
107 | 106 | fveq2d 6678 |
. . . . . . . . . 10
⊢ (ℎ = 𝑖 →
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) |
108 | 107 | eqeq2d 2749 |
. . . . . . . . 9
⊢ (ℎ = 𝑖 → (𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))) ↔ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) |
109 | 99, 108 | anbi12d 634 |
. . . . . . . 8
⊢ (ℎ = 𝑖 → ((𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ (𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))) |
110 | 109 | cbvrexvw 3350 |
. . . . . . 7
⊢
(∃ℎ ∈
(((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ)(𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) |
111 | 110 | a1i 11 |
. . . . . 6
⊢ (𝑤 = 𝑧 → (∃ℎ ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ)(𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ)(𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))) |
112 | 92, 111 | bitrd 282 |
. . . . 5
⊢ (𝑤 = 𝑧 → (∃ℎ ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ)(𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑤 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘))))) ↔ ∃𝑖 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ)(𝐼 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))))) |
113 | 112 | cbvrabv 3393 |
. . . 4
⊢ {𝑤 ∈ ℝ*
∣ ∃ℎ ∈
(((ℝ × ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (ℎ‘𝑗))‘𝑘) ∧ 𝑤 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (ℎ‘𝑗))‘𝑘)))))} = {𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑m 𝑋) ↑m ℕ)(𝐼 ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (([,) ∘ (𝑖‘𝑗))‘𝑘) ∧ 𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))} |
114 | | simpl 486 |
. . . . . . . . . 10
⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → 𝑗 = 𝑛) |
115 | 114 | fveq2d 6678 |
. . . . . . . . 9
⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (𝑖‘𝑗) = (𝑖‘𝑛)) |
116 | 115 | fveq1d 6676 |
. . . . . . . 8
⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → ((𝑖‘𝑗)‘𝑙) = ((𝑖‘𝑛)‘𝑙)) |
117 | 116 | fveq2d 6678 |
. . . . . . 7
⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (1st ‘((𝑖‘𝑗)‘𝑙)) = (1st ‘((𝑖‘𝑛)‘𝑙))) |
118 | 117 | mpteq2dva 5125 |
. . . . . 6
⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))) |
119 | 118 | cbvmptv 5133 |
. . . . 5
⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙)))) |
120 | 119 | mpteq2i 5122 |
. . . 4
⊢ (𝑖 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑗)‘𝑙))))) = (𝑖 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (1st ‘((𝑖‘𝑛)‘𝑙))))) |
121 | 116 | fveq2d 6678 |
. . . . . . 7
⊢ ((𝑗 = 𝑛 ∧ 𝑙 ∈ 𝑋) → (2nd ‘((𝑖‘𝑗)‘𝑙)) = (2nd ‘((𝑖‘𝑛)‘𝑙))) |
122 | 121 | mpteq2dva 5125 |
. . . . . 6
⊢ (𝑗 = 𝑛 → (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))) = (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))) |
123 | 122 | cbvmptv 5133 |
. . . . 5
⊢ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙)))) = (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙)))) |
124 | 123 | mpteq2i 5122 |
. . . 4
⊢ (𝑖 ∈ (((ℝ ×
ℝ) ↑m 𝑋) ↑m ℕ) ↦ (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑗)‘𝑙))))) = (𝑖 ∈ (((ℝ × ℝ)
↑m 𝑋)
↑m ℕ) ↦ (𝑛 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ (2nd ‘((𝑖‘𝑛)‘𝑙))))) |
125 | 59, 61, 62, 63, 2, 89, 113, 120, 124 | ovnhoilem2 43702 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝑋 = ∅) → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼)) |
126 | 70, 125 | pm2.61dan 813 |
. 2
⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤ ((voln*‘𝑋)‘𝐼)) |
127 | 12, 16, 67, 126 | xrletrid 12631 |
1
⊢ (𝜑 → ((voln*‘𝑋)‘𝐼) = (𝐴(𝐿‘𝑋)𝐵)) |