Step | Hyp | Ref
| Expression |
1 | | vonioolem2.x |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ Fin) |
2 | 1 | vonmea 40788 |
. . . 4
⊢ (𝜑 → (voln‘𝑋) ∈ Meas) |
3 | | 1zzd 11408 |
. . . 4
⊢ (𝜑 → 1 ∈
ℤ) |
4 | | nnuz 11723 |
. . . 4
⊢ ℕ =
(ℤ≥‘1) |
5 | 1 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin) |
6 | | eqid 2622 |
. . . . . 6
⊢ dom
(voln‘𝑋) = dom
(voln‘𝑋) |
7 | | vonioolem2.a |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
8 | 7 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ) |
9 | 8 | ffvelrnda 6359 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
10 | | nnrecre 11057 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ) |
11 | 10 | ad2antlr 763 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ) |
12 | 9, 11 | readdcld 10069 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ) |
13 | | eqid 2622 |
. . . . . . . 8
⊢ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) |
14 | 12, 13 | fmptd 6385 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ) |
15 | | vonioolem2.c |
. . . . . . . . . 10
⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) |
16 | 15 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))) |
17 | 1 | mptexd 6487 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V) |
18 | 17 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V) |
19 | 16, 18 | fvmpt2d 6293 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) |
20 | 19 | feq1d 6030 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)) |
21 | 14, 20 | mpbird 247 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ) |
22 | | vonioolem2.b |
. . . . . . 7
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
23 | 22 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ) |
24 | 5, 6, 21, 23 | hoimbl 40845 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋)) |
25 | | vonioolem2.d |
. . . . 5
⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) |
26 | 24, 25 | fmptd 6385 |
. . . 4
⊢ (𝜑 → 𝐷:ℕ⟶dom (voln‘𝑋)) |
27 | | nfv 1843 |
. . . . . 6
⊢
Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ) |
28 | | oveq2 6658 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑚 → (1 / 𝑛) = (1 / 𝑚)) |
29 | 28 | oveq2d 6666 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑚 → ((𝐴‘𝑘) + (1 / 𝑛)) = ((𝐴‘𝑘) + (1 / 𝑚))) |
30 | 29 | mpteq2dv 4745 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑚 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚)))) |
31 | 30 | cbvmptv 4750 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚)))) |
32 | 15, 31 | eqtri 2644 |
. . . . . . . . . . . 12
⊢ 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚)))) |
33 | 32 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐶 = (𝑚 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))))) |
34 | | oveq2 6658 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝑛 + 1) → (1 / 𝑚) = (1 / (𝑛 + 1))) |
35 | 34 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑚 = (𝑛 + 1) → ((𝐴‘𝑘) + (1 / 𝑚)) = ((𝐴‘𝑘) + (1 / (𝑛 + 1)))) |
36 | 35 | mpteq2dv 4745 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝑛 + 1) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1))))) |
37 | 36 | adantl 482 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 = (𝑛 + 1)) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑚))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1))))) |
38 | | simpr 477 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ) |
39 | 38 | peano2nnd 11037 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑛 + 1) ∈ ℕ) |
40 | 5 | mptexd 6487 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1)))) ∈ V) |
41 | 33, 37, 39, 40 | fvmptd 6288 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘(𝑛 + 1)) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / (𝑛 + 1))))) |
42 | | ovexd 6680 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ∈ V) |
43 | 41, 42 | fvmpt2d 6293 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) = ((𝐴‘𝑘) + (1 / (𝑛 + 1)))) |
44 | | 1red 10055 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 1 ∈
ℝ) |
45 | | nnre 11027 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ) |
46 | 45, 44 | readdcld 10069 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℝ) |
47 | | peano2nn 11032 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℕ) |
48 | | nnne0 11053 |
. . . . . . . . . . . . 13
⊢ ((𝑛 + 1) ∈ ℕ →
(𝑛 + 1) ≠
0) |
49 | 47, 48 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ≠ 0) |
50 | 44, 46, 49 | redivcld 10853 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → (1 /
(𝑛 + 1)) ∈
ℝ) |
51 | 50 | ad2antlr 763 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ∈ ℝ) |
52 | 9, 51 | readdcld 10069 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ∈ ℝ) |
53 | 43, 52 | eqeltrd 2701 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ) |
54 | 53 | rexrd 10089 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ∈
ℝ*) |
55 | | ressxr 10083 |
. . . . . . . . 9
⊢ ℝ
⊆ ℝ* |
56 | 22 | ffvelrnda 6359 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
57 | 55, 56 | sseldi 3601 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈
ℝ*) |
58 | 57 | adantlr 751 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈
ℝ*) |
59 | 45 | ltp1d 10954 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → 𝑛 < (𝑛 + 1)) |
60 | | nnrp 11842 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
61 | 47 | nnrpd 11870 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ → (𝑛 + 1) ∈
ℝ+) |
62 | 60, 61 | ltrecd 11890 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ → (𝑛 < (𝑛 + 1) ↔ (1 / (𝑛 + 1)) < (1 / 𝑛))) |
63 | 59, 62 | mpbid 222 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ → (1 /
(𝑛 + 1)) < (1 / 𝑛)) |
64 | 50, 10, 63 | ltled 10185 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
(𝑛 + 1)) ≤ (1 / 𝑛)) |
65 | 64 | ad2antlr 763 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / (𝑛 + 1)) ≤ (1 / 𝑛)) |
66 | 51, 11, 9, 65 | leadd2dd 10642 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐴‘𝑘) + (1 / 𝑛))) |
67 | | ovexd 6680 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ V) |
68 | 19, 67 | fvmpt2d 6293 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛))) |
69 | 43, 68 | breq12d 4666 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘) ↔ ((𝐴‘𝑘) + (1 / (𝑛 + 1))) ≤ ((𝐴‘𝑘) + (1 / 𝑛)))) |
70 | 66, 69 | mpbird 247 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘)) |
71 | 56 | adantlr 751 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
72 | | eqidd 2623 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) = (𝐵‘𝑘)) |
73 | 71, 72 | eqled 10140 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ≤ (𝐵‘𝑘)) |
74 | | icossico 12243 |
. . . . . . 7
⊢
(((((𝐶‘(𝑛 + 1))‘𝑘) ∈ ℝ* ∧ (𝐵‘𝑘) ∈ ℝ*) ∧ (((𝐶‘(𝑛 + 1))‘𝑘) ≤ ((𝐶‘𝑛)‘𝑘) ∧ (𝐵‘𝑘) ≤ (𝐵‘𝑘))) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) |
75 | 54, 58, 70, 73, 74 | syl22anc 1327 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) |
76 | 27, 75 | ixpssixp 39269 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) |
77 | 25 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))) |
78 | 24 | elexd 3214 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V) |
79 | 77, 78 | fvmpt2d 6293 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) |
80 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → (𝐶‘𝑛) = (𝐶‘𝑚)) |
81 | 80 | fveq1d 6193 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) |
82 | 81 | oveq1d 6665 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑚 → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘))) |
83 | 82 | ixpeq2dv 7924 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 → X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘))) |
84 | 83 | cbvmptv 4750 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = (𝑚 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘))) |
85 | 25, 84 | eqtri 2644 |
. . . . . . . 8
⊢ 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘))) |
86 | 85 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐷 = (𝑚 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)))) |
87 | | fveq2 6191 |
. . . . . . . . . . 11
⊢ (𝑚 = (𝑛 + 1) → (𝐶‘𝑚) = (𝐶‘(𝑛 + 1))) |
88 | 87 | fveq1d 6193 |
. . . . . . . . . 10
⊢ (𝑚 = (𝑛 + 1) → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘(𝑛 + 1))‘𝑘)) |
89 | 88 | oveq1d 6665 |
. . . . . . . . 9
⊢ (𝑚 = (𝑛 + 1) → (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) |
90 | 89 | ixpeq2dv 7924 |
. . . . . . . 8
⊢ (𝑚 = (𝑛 + 1) → X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) |
91 | 90 | adantl 482 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑚 = (𝑛 + 1)) → X𝑘 ∈ 𝑋 (((𝐶‘𝑚)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) |
92 | | ovex 6678 |
. . . . . . . . . 10
⊢ (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V |
93 | 92 | rgenw 2924 |
. . . . . . . . 9
⊢
∀𝑘 ∈
𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V |
94 | | ixpexg 7932 |
. . . . . . . . 9
⊢
(∀𝑘 ∈
𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V → X𝑘 ∈
𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V) |
95 | 93, 94 | ax-mp 5 |
. . . . . . . 8
⊢ X𝑘 ∈
𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V |
96 | 95 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)) ∈ V) |
97 | 86, 91, 39, 96 | fvmptd 6288 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘(𝑛 + 1)) = X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘))) |
98 | 79, 97 | sseq12d 3634 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛) ⊆ (𝐷‘(𝑛 + 1)) ↔ X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 (((𝐶‘(𝑛 + 1))‘𝑘)[,)(𝐵‘𝑘)))) |
99 | 76, 98 | mpbird 247 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ⊆ (𝐷‘(𝑛 + 1))) |
100 | 1, 6, 7, 22 | hoimbl 40845 |
. . . . 5
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋)) |
101 | | nfv 1843 |
. . . . . 6
⊢
Ⅎ𝑘𝜑 |
102 | 7 | ffvelrnda 6359 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
103 | 101, 1, 102, 56 | vonhoire 40886 |
. . . . 5
⊢ (𝜑 → ((voln‘𝑋)‘X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∈ ℝ) |
104 | | vonioolem2.i |
. . . . . . 7
⊢ 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) |
105 | 104 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))) |
106 | | nftru 1730 |
. . . . . . . . 9
⊢
Ⅎ𝑘⊤ |
107 | | ioossico 12262 |
. . . . . . . . . 10
⊢ ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) |
108 | 107 | a1i 11 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ 𝑋) → ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
109 | 106, 108 | ixpssixp 39269 |
. . . . . . . 8
⊢ (⊤
→ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
110 | 109 | trud 1493 |
. . . . . . 7
⊢ X𝑘 ∈
𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) |
111 | 110 | a1i 11 |
. . . . . 6
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
112 | 105, 111 | eqsstrd 3639 |
. . . . 5
⊢ (𝜑 → 𝐼 ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
113 | 55 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ℝ ⊆
ℝ*) |
114 | 7, 113 | fssd 6057 |
. . . . . . 7
⊢ (𝜑 → 𝐴:𝑋⟶ℝ*) |
115 | 22, 113 | fssd 6057 |
. . . . . . 7
⊢ (𝜑 → 𝐵:𝑋⟶ℝ*) |
116 | 1, 6, 114, 115 | ioovonmbl 40891 |
. . . . . 6
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋)) |
117 | 104, 116 | syl5eqel 2705 |
. . . . 5
⊢ (𝜑 → 𝐼 ∈ dom (voln‘𝑋)) |
118 | 2, 100, 103, 112, 117 | meassre 40694 |
. . . 4
⊢ (𝜑 → ((voln‘𝑋)‘𝐼) ∈ ℝ) |
119 | 2 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (voln‘𝑋) ∈ Meas) |
120 | 79, 24 | eqeltrd 2701 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ∈ dom (voln‘𝑋)) |
121 | 117 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐼 ∈ dom (voln‘𝑋)) |
122 | 55, 102 | sseldi 3601 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈
ℝ*) |
123 | 122 | adantlr 751 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈
ℝ*) |
124 | 60 | rpreccld 11882 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ+) |
125 | 124 | ad2antlr 763 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈
ℝ+) |
126 | 9, 125 | ltaddrpd 11905 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < ((𝐴‘𝑘) + (1 / 𝑛))) |
127 | | icossioo 12264 |
. . . . . . . 8
⊢ ((((𝐴‘𝑘) ∈ ℝ* ∧ (𝐵‘𝑘) ∈ ℝ*) ∧ ((𝐴‘𝑘) < ((𝐴‘𝑘) + (1 / 𝑛)) ∧ (𝐵‘𝑘) ≤ (𝐵‘𝑘))) → (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)(,)(𝐵‘𝑘))) |
128 | 123, 58, 126, 73, 127 | syl22anc 1327 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ ((𝐴‘𝑘)(,)(𝐵‘𝑘))) |
129 | 27, 128 | ixpssixp 39269 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))) |
130 | 68 | oveq1d 6665 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘))) |
131 | 130 | ixpeq2dva 7923 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘))) |
132 | 79, 131 | eqtrd 2656 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘))) |
133 | 104 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐼 = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))) |
134 | 132, 133 | sseq12d 3634 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐷‘𝑛) ⊆ 𝐼 ↔ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) ⊆ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘)))) |
135 | 129, 134 | mpbird 247 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) ⊆ 𝐼) |
136 | 119, 6, 120, 121, 135 | meassle 40680 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((voln‘𝑋)‘(𝐷‘𝑛)) ≤ ((voln‘𝑋)‘𝐼)) |
137 | | eqid 2622 |
. . . 4
⊢ (𝑛 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) |
138 | 2, 3, 4, 26, 99, 118, 136, 137 | meaiuninc2 40699 |
. . 3
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘∪
𝑛 ∈ ℕ (𝐷‘𝑛))) |
139 | 101, 1, 102, 57 | iunhoiioo 40890 |
. . . . . . 7
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)(,)(𝐵‘𝑘))) |
140 | 132 | iuneq2dv 4542 |
. . . . . . 7
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐷‘𝑛) = ∪ 𝑛 ∈ ℕ X𝑘 ∈
𝑋 (((𝐴‘𝑘) + (1 / 𝑛))[,)(𝐵‘𝑘))) |
141 | 139, 140,
105 | 3eqtr4d 2666 |
. . . . . 6
⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐷‘𝑛) = 𝐼) |
142 | 141 | eqcomd 2628 |
. . . . 5
⊢ (𝜑 → 𝐼 = ∪ 𝑛 ∈ ℕ (𝐷‘𝑛)) |
143 | 142 | fveq2d 6195 |
. . . 4
⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ((voln‘𝑋)‘∪
𝑛 ∈ ℕ (𝐷‘𝑛))) |
144 | 143 | eqcomd 2628 |
. . 3
⊢ (𝜑 → ((voln‘𝑋)‘∪ 𝑛 ∈ ℕ (𝐷‘𝑛)) = ((voln‘𝑋)‘𝐼)) |
145 | 138, 144 | breqtrd 4679 |
. 2
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼)) |
146 | | fveq2 6191 |
. . . . . 6
⊢ (𝑛 = 𝑚 → (𝐷‘𝑛) = (𝐷‘𝑚)) |
147 | 146 | fveq2d 6195 |
. . . . 5
⊢ (𝑛 = 𝑚 → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘(𝐷‘𝑚))) |
148 | 147 | cbvmptv 4750 |
. . . 4
⊢ (𝑛 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) |
149 | 148 | a1i 11 |
. . 3
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) = (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚)))) |
150 | | vonioolem2.n |
. . . 4
⊢ (𝜑 → 𝑋 ≠ ∅) |
151 | | vonioolem2.t |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘)) |
152 | 148 | eqcomi 2631 |
. . . 4
⊢ (𝑚 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑚))) = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) |
153 | | eqcom 2629 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑚 ↔ 𝑚 = 𝑛) |
154 | 153 | imbi1i 339 |
. . . . . . . . 9
⊢ ((𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘))) |
155 | | eqcom 2629 |
. . . . . . . . . 10
⊢ (((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘) ↔ ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘)) |
156 | 155 | imbi2i 326 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑛 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘))) |
157 | 154, 156 | bitri 264 |
. . . . . . . 8
⊢ ((𝑛 = 𝑚 → ((𝐶‘𝑛)‘𝑘) = ((𝐶‘𝑚)‘𝑘)) ↔ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘))) |
158 | 81, 157 | mpbi 220 |
. . . . . . 7
⊢ (𝑚 = 𝑛 → ((𝐶‘𝑚)‘𝑘) = ((𝐶‘𝑛)‘𝑘)) |
159 | 158 | oveq2d 6666 |
. . . . . 6
⊢ (𝑚 = 𝑛 → ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘)) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
160 | 159 | prodeq2ad 39824 |
. . . . 5
⊢ (𝑚 = 𝑛 → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
161 | 160 | cbvmptv 4750 |
. . . 4
⊢ (𝑚 ∈ ℕ ↦
∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑚)‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
162 | | eqid 2622 |
. . . 4
⊢ inf(ran
(𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) |
163 | | eqid 2622 |
. . . 4
⊢
((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1) =
((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1) |
164 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝐵‘𝑗) = (𝐵‘𝑘)) |
165 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑘 → (𝐴‘𝑗) = (𝐴‘𝑘)) |
166 | 164, 165 | oveq12d 6668 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑘 → ((𝐵‘𝑗) − (𝐴‘𝑗)) = ((𝐵‘𝑘) − (𝐴‘𝑘))) |
167 | 166 | cbvmptv 4750 |
. . . . . . . . . 10
⊢ (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) |
168 | 167 | rneqi 5352 |
. . . . . . . . 9
⊢ ran
(𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))) = ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) |
169 | 168 | infeq1i 8384 |
. . . . . . . 8
⊢ inf(ran
(𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ) = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) |
170 | 169 | oveq2i 6661 |
. . . . . . 7
⊢ (1 /
inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < )) = (1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < )) |
171 | 170 | fveq2i 6194 |
. . . . . 6
⊢
(⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) = (⌊‘(1 /
inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) |
172 | 171 | oveq1i 6660 |
. . . . 5
⊢
((⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) + 1) =
((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1) |
173 | 172 | fveq2i 6194 |
. . . 4
⊢
(ℤ≥‘((⌊‘(1 / inf(ran (𝑗 ∈ 𝑋 ↦ ((𝐵‘𝑗) − (𝐴‘𝑗))), ℝ, < ))) + 1)) =
(ℤ≥‘((⌊‘(1 / inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ))) + 1)) |
174 | 1, 7, 22, 150, 151, 15, 25, 152, 161, 162, 163, 173 | vonioolem1 40894 |
. . 3
⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑚))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
175 | 149, 174 | eqbrtrd 4675 |
. 2
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
176 | | climuni 14283 |
. 2
⊢ (((𝑛 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ((voln‘𝑋)‘𝐼) ∧ (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
177 | 145, 175,
176 | syl2anc 693 |
1
⊢ (𝜑 → ((voln‘𝑋)‘𝐼) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |