Proof of Theorem vonioolem1
Step | Hyp | Ref
| Expression |
1 | | vonioolem1.r |
. . . . 5
⊢ 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
2 | 1 | a1i 11 |
. . . 4
⊢ (𝜑 → 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)))) |
3 | | vonioolem1.c |
. . . . . . . . . . 11
⊢ 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) |
4 | 3 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶 = (𝑛 ∈ ℕ ↦ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))))) |
5 | | vonioolem1.x |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑋 ∈ Fin) |
6 | 5 | mptexd 6991 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V) |
7 | 6 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V) |
8 | 4, 7 | fvmpt2d 6782 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) |
9 | | ovexd 7199 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ V) |
10 | 8, 9 | fvmpt2d 6782 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛))) |
11 | 10 | oveq2d 7180 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = ((𝐵‘𝑘) − ((𝐴‘𝑘) + (1 / 𝑛)))) |
12 | | vonioolem1.b |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
13 | 12 | ffvelrnda 6855 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
14 | 13 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
15 | 14 | recnd 10740 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℂ) |
16 | | vonioolem1.a |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
17 | 16 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ) |
18 | 17 | ffvelrnda 6855 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
19 | 18 | recnd 10740 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℂ) |
20 | | nnrecre 11751 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ) |
21 | 20 | ad2antlr 727 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ) |
22 | 21 | recnd 10740 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℂ) |
23 | 15, 19, 22 | subsub4d 11099 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)) = ((𝐵‘𝑘) − ((𝐴‘𝑘) + (1 / 𝑛)))) |
24 | 11, 23 | eqtr4d 2776 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) |
25 | 24 | prodeq2dv 15362 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) |
26 | 25 | mpteq2dva 5122 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))) |
27 | 2, 26 | eqtrd 2773 |
. . 3
⊢ (𝜑 → 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))) |
28 | | nfv 1920 |
. . . 4
⊢
Ⅎ𝑘𝜑 |
29 | | rpssre 12472 |
. . . . . 6
⊢
ℝ+ ⊆ ℝ |
30 | | vonioolem1.t |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘)) |
31 | 16 | ffvelrnda 6855 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
32 | | difrp 12503 |
. . . . . . . 8
⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+)) |
33 | 31, 13, 32 | syl2anc 587 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+)) |
34 | 30, 33 | mpbid 235 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+) |
35 | 29, 34 | sseldi 3873 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ) |
36 | 35 | recnd 10740 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℂ) |
37 | | eqid 2738 |
. . . 4
⊢ (𝑛 ∈ ℕ ↦
∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) |
38 | 28, 5, 36, 37 | fprodsubrecnncnv 42975 |
. . 3
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
39 | 27, 38 | eqbrtrd 5049 |
. 2
⊢ (𝜑 → 𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
40 | | vonioolem1.z |
. . 3
⊢ 𝑍 =
(ℤ≥‘𝑁) |
41 | | nnex 11715 |
. . . . . 6
⊢ ℕ
∈ V |
42 | 41 | mptex 6990 |
. . . . 5
⊢ (𝑛 ∈ ℕ ↦
∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) ∈ V |
43 | 42 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) ∈ V) |
44 | 1, 43 | eqeltrid 2837 |
. . 3
⊢ (𝜑 → 𝑇 ∈ V) |
45 | | vonioolem1.s |
. . . 4
⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) |
46 | 41 | mptex 6990 |
. . . . 5
⊢ (𝑛 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑛))) ∈ V |
47 | 46 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ∈ V) |
48 | 45, 47 | eqeltrid 2837 |
. . 3
⊢ (𝜑 → 𝑆 ∈ V) |
49 | | vonioolem1.n |
. . . 4
⊢ 𝑁 = ((⌊‘(1 / 𝐸)) + 1) |
50 | | 1rp 12469 |
. . . . . . . . . 10
⊢ 1 ∈
ℝ+ |
51 | 50 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈
ℝ+) |
52 | | eqid 2738 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) |
53 | 28, 52, 34 | rnmptssd 42257 |
. . . . . . . . . 10
⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆
ℝ+) |
54 | | vonioolem1.e |
. . . . . . . . . . 11
⊢ 𝐸 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) |
55 | | ltso 10792 |
. . . . . . . . . . . . 13
⊢ < Or
ℝ |
56 | 55 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → < Or
ℝ) |
57 | 52 | rnmptfi 42229 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∈ Fin → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin) |
58 | 5, 57 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin) |
59 | | vonioolem1.u |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑋 ≠ ∅) |
60 | 28, 34, 52, 59 | rnmptn0 6070 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ≠ ∅) |
61 | 29 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ℝ+
⊆ ℝ) |
62 | 53, 61 | sstrd 3885 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ) |
63 | | fiinfcl 9031 |
. . . . . . . . . . . 12
⊢ (( <
Or ℝ ∧ (ran (𝑘
∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ≠ ∅ ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ)) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
64 | 56, 58, 60, 62, 63 | syl13anc 1373 |
. . . . . . . . . . 11
⊢ (𝜑 → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
65 | 54, 64 | eqeltrid 2837 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐸 ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
66 | 53, 65 | sseldd 3876 |
. . . . . . . . 9
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
67 | 51, 66 | rpdivcld 12524 |
. . . . . . . 8
⊢ (𝜑 → (1 / 𝐸) ∈
ℝ+) |
68 | 67 | rpred 12507 |
. . . . . . 7
⊢ (𝜑 → (1 / 𝐸) ∈ ℝ) |
69 | 67 | rpge0d 12511 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ (1 / 𝐸)) |
70 | | flge0nn0 13274 |
. . . . . . 7
⊢ (((1 /
𝐸) ∈ ℝ ∧ 0
≤ (1 / 𝐸)) →
(⌊‘(1 / 𝐸))
∈ ℕ0) |
71 | 68, 69, 70 | syl2anc 587 |
. . . . . 6
⊢ (𝜑 → (⌊‘(1 / 𝐸)) ∈
ℕ0) |
72 | | nn0p1nn 12008 |
. . . . . 6
⊢
((⌊‘(1 / 𝐸)) ∈ ℕ0 →
((⌊‘(1 / 𝐸)) +
1) ∈ ℕ) |
73 | 71, 72 | syl 17 |
. . . . 5
⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈
ℕ) |
74 | 73 | nnzd 12160 |
. . . 4
⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈
ℤ) |
75 | 49, 74 | eqeltrid 2837 |
. . 3
⊢ (𝜑 → 𝑁 ∈ ℤ) |
76 | 49 | recnnltrp 42438 |
. . . . . . . . . . . . 13
⊢ (𝐸 ∈ ℝ+
→ (𝑁 ∈ ℕ
∧ (1 / 𝑁) < 𝐸)) |
77 | 66, 76 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸)) |
78 | 77 | simpld 498 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℕ) |
79 | | uznnssnn 12370 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ →
(ℤ≥‘𝑁) ⊆ ℕ) |
80 | 78, 79 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 →
(ℤ≥‘𝑁) ⊆ ℕ) |
81 | 40, 80 | eqsstrid 3923 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍 ⊆ ℕ) |
82 | 81 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑍 ⊆ ℕ) |
83 | | simpr 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) |
84 | 82, 83 | sseldd 3876 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℕ) |
85 | | vonioolem1.d |
. . . . . . . . 9
⊢ 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) |
86 | 85 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 𝐷 = (𝑛 ∈ ℕ ↦ X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))) |
87 | 5 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin) |
88 | | eqid 2738 |
. . . . . . . . . 10
⊢ dom
(voln‘𝑋) = dom
(voln‘𝑋) |
89 | 18, 21 | readdcld 10741 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ) |
90 | 89 | fmpttd 6883 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ) |
91 | 8 | feq1d 6483 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)) |
92 | 90, 91 | mpbird 260 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ) |
93 | 12 | adantr 484 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ) |
94 | 87, 88, 92, 93 | hoimbl 43695 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋)) |
95 | 94 | elexd 3417 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V) |
96 | 86, 95 | fvmpt2d 6782 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) |
97 | 84, 96 | syldan 594 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) |
98 | 97 | fveq2d 6672 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))) |
99 | 5 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ∈ Fin) |
100 | 59 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ≠ ∅) |
101 | 84, 92 | syldan 594 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐶‘𝑛):𝑋⟶ℝ) |
102 | 12 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐵:𝑋⟶ℝ) |
103 | | eqid 2738 |
. . . . . 6
⊢ X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) |
104 | 99, 100, 101, 102, 103 | vonn0hoi 43734 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))) |
105 | 101 | ffvelrnda 6855 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ) |
106 | 84, 14 | syldanl 605 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
107 | | volico 43050 |
. . . . . . . 8
⊢ ((((𝐶‘𝑛)‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = if(((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)), 0)) |
108 | 105, 106,
107 | syl2anc 587 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = if(((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)), 0)) |
109 | 84, 10 | syldanl 605 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛))) |
110 | 84, 21 | syldanl 605 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ) |
111 | 78 | nnrecred 11760 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1 / 𝑁) ∈ ℝ) |
112 | 111 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ) |
113 | 35 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ) |
114 | 40 | eleq2i 2824 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑁)) |
115 | 114 | biimpi 219 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑁)) |
116 | | eluzle 12330 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈
(ℤ≥‘𝑁) → 𝑁 ≤ 𝑛) |
117 | 115, 116 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ 𝑍 → 𝑁 ≤ 𝑛) |
118 | 117 | adantl 485 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ≤ 𝑛) |
119 | 78 | nnrpd 12505 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈
ℝ+) |
120 | 119 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ∈
ℝ+) |
121 | | nnrp 12476 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
122 | 84, 121 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℝ+) |
123 | 120, 122 | lerecd 12526 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁 ≤ 𝑛 ↔ (1 / 𝑛) ≤ (1 / 𝑁))) |
124 | 118, 123 | mpbid 235 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (1 / 𝑛) ≤ (1 / 𝑁)) |
125 | 124 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ≤ (1 / 𝑁)) |
126 | 111 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ) |
127 | 29, 66 | sseldi 3873 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐸 ∈ ℝ) |
128 | 127 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ∈ ℝ) |
129 | 77 | simprd 499 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1 / 𝑁) < 𝐸) |
130 | 129 | adantr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < 𝐸) |
131 | 62 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ) |
132 | 58 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin) |
133 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋) |
134 | | ovexd 7199 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V) |
135 | 52 | elrnmpt1 5795 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
136 | 133, 134,
135 | syl2anc 587 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
137 | 136 | adantl 485 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
138 | | infrefilb 11697 |
. . . . . . . . . . . . . . 15
⊢ ((ran
(𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin ∧ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ≤ ((𝐵‘𝑘) − (𝐴‘𝑘))) |
139 | 131, 132,
137, 138 | syl3anc 1372 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ≤ ((𝐵‘𝑘) − (𝐴‘𝑘))) |
140 | 54, 139 | eqbrtrid 5062 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑘) − (𝐴‘𝑘))) |
141 | 126, 128,
35, 130, 140 | ltletrd 10871 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < ((𝐵‘𝑘) − (𝐴‘𝑘))) |
142 | 141 | adantlr 715 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < ((𝐵‘𝑘) − (𝐴‘𝑘))) |
143 | 110, 112,
113, 125, 142 | lelttrd 10869 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘))) |
144 | 84, 18 | syldanl 605 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
145 | 144, 110,
106 | ltaddsub2d 11312 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (((𝐴‘𝑘) + (1 / 𝑛)) < (𝐵‘𝑘) ↔ (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
146 | 143, 145 | mpbird 260 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) < (𝐵‘𝑘)) |
147 | 109, 146 | eqbrtrd 5049 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘)) |
148 | 147 | iftrued 4419 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → if(((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)), 0) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
149 | 108, 148 | eqtrd 2773 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
150 | 149 | prodeq2dv 15362 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
151 | 98, 104, 150 | 3eqtrd 2777 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
152 | | fvexd 6683 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V) |
153 | 45 | fvmpt2 6780 |
. . . . 5
⊢ ((𝑛 ∈ ℕ ∧
((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛))) |
154 | 84, 152, 153 | syl2anc 587 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛))) |
155 | | prodex 15346 |
. . . . . 6
⊢
∏𝑘 ∈
𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) ∈ V |
156 | 155 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) ∈ V) |
157 | 1 | fvmpt2 6780 |
. . . . 5
⊢ ((𝑛 ∈ ℕ ∧
∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) ∈ V) → (𝑇‘𝑛) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
158 | 84, 156, 157 | syl2anc 587 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
159 | 151, 154,
158 | 3eqtr4rd 2784 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = (𝑆‘𝑛)) |
160 | 40, 44, 48, 75, 159 | climeq 15007 |
. 2
⊢ (𝜑 → (𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)) ↔ 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
161 | 39, 160 | mpbid 235 |
1
⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |