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 6528 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V) |
7 | 6 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) ∈ V) |
8 | 4, 7 | fvmpt2d 6332 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛)))) |
9 | | ovexd 6720 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ V) |
10 | 8, 9 | fvmpt2d 6332 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛))) |
11 | 10 | oveq2d 6706 |
. . . . . . 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 6399 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
14 | 13 | adantlr 751 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
15 | 14 | recnd 10106 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℂ) |
16 | | vonioolem1.a |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
17 | 16 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴:𝑋⟶ℝ) |
18 | 17 | ffvelrnda 6399 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
19 | 18 | recnd 10106 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℂ) |
20 | | nnrecre 11095 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ → (1 /
𝑛) ∈
ℝ) |
21 | 20 | ad2antlr 763 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ) |
22 | 21 | recnd 10106 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℂ) |
23 | 15, 19, 22 | subsub4d 10461 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)) = ((𝐵‘𝑘) − ((𝐴‘𝑘) + (1 / 𝑛)))) |
24 | 11, 23 | eqtr4d 2688 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) |
25 | 24 | prodeq2dv 14697 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) = ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) |
26 | 25 | mpteq2dva 4777 |
. . . 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 2685 |
. . 3
⊢ (𝜑 → 𝑇 = (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛)))) |
28 | | nfv 1883 |
. . . 4
⊢
Ⅎ𝑘𝜑 |
29 | | rpssre 11881 |
. . . . . 6
⊢
ℝ+ ⊆ ℝ |
30 | | vonioolem1.t |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) < (𝐵‘𝑘)) |
31 | 16 | ffvelrnda 6399 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
32 | | difrp 11906 |
. . . . . . . 8
⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+)) |
33 | 31, 13, 32 | syl2anc 694 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+)) |
34 | 30, 33 | mpbid 222 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+) |
35 | 29, 34 | sseldi 3634 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ) |
36 | 35 | recnd 10106 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℂ) |
37 | | eqid 2651 |
. . . 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 40440 |
. . 3
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (((𝐵‘𝑘) − (𝐴‘𝑘)) − (1 / 𝑛))) ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
39 | 27, 38 | eqbrtrd 4707 |
. 2
⊢ (𝜑 → 𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
40 | | vonioolem1.z |
. . 3
⊢ 𝑍 =
(ℤ≥‘𝑁) |
41 | | nnex 11064 |
. . . . . 6
⊢ ℕ
∈ V |
42 | 41 | mptex 6527 |
. . . . 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 | syl5eqel 2734 |
. . 3
⊢ (𝜑 → 𝑇 ∈ V) |
45 | | vonioolem1.s |
. . . 4
⊢ 𝑆 = (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) |
46 | 41 | mptex 6527 |
. . . . 5
⊢ (𝑛 ∈ ℕ ↦
((voln‘𝑋)‘(𝐷‘𝑛))) ∈ V |
47 | 46 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ((voln‘𝑋)‘(𝐷‘𝑛))) ∈ V) |
48 | 45, 47 | syl5eqel 2734 |
. . 3
⊢ (𝜑 → 𝑆 ∈ V) |
49 | | vonioolem1.n |
. . . 4
⊢ 𝑁 = ((⌊‘(1 / 𝐸)) + 1) |
50 | | 1rp 11874 |
. . . . . . . . . 10
⊢ 1 ∈
ℝ+ |
51 | 50 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → 1 ∈
ℝ+) |
52 | | eqid 2651 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) = (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) |
53 | 28, 52, 34 | rnmptssd 39699 |
. . . . . . . . . 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 10156 |
. . . . . . . . . . . . 13
⊢ < Or
ℝ |
56 | 55 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → < Or
ℝ) |
57 | 52 | rnmptfi 39665 |
. . . . . . . . . . . . 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 39727 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ≠ ∅) |
61 | 29 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ℝ+
⊆ ℝ) |
62 | 53, 61 | sstrd 3646 |
. . . . . . . . . . . 12
⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ) |
63 | | fiinfcl 8448 |
. . . . . . . . . . . 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 1368 |
. . . . . . . . . . 11
⊢ (𝜑 → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
65 | 54, 64 | syl5eqel 2734 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐸 ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
66 | 53, 65 | sseldd 3637 |
. . . . . . . . 9
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
67 | 51, 66 | rpdivcld 11927 |
. . . . . . . 8
⊢ (𝜑 → (1 / 𝐸) ∈
ℝ+) |
68 | 67 | rpred 11910 |
. . . . . . 7
⊢ (𝜑 → (1 / 𝐸) ∈ ℝ) |
69 | 67 | rpge0d 11914 |
. . . . . . 7
⊢ (𝜑 → 0 ≤ (1 / 𝐸)) |
70 | | flge0nn0 12661 |
. . . . . . 7
⊢ (((1 /
𝐸) ∈ ℝ ∧ 0
≤ (1 / 𝐸)) →
(⌊‘(1 / 𝐸))
∈ ℕ0) |
71 | 68, 69, 70 | syl2anc 694 |
. . . . . 6
⊢ (𝜑 → (⌊‘(1 / 𝐸)) ∈
ℕ0) |
72 | | nn0p1nn 11370 |
. . . . . 6
⊢
((⌊‘(1 / 𝐸)) ∈ ℕ0 →
((⌊‘(1 / 𝐸)) +
1) ∈ ℕ) |
73 | 71, 72 | syl 17 |
. . . . 5
⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈
ℕ) |
74 | 73 | nnzd 11519 |
. . . 4
⊢ (𝜑 → ((⌊‘(1 / 𝐸)) + 1) ∈
ℤ) |
75 | 49, 74 | syl5eqel 2734 |
. . 3
⊢ (𝜑 → 𝑁 ∈ ℤ) |
76 | 49 | recnnltrp 39906 |
. . . . . . . . . . . . 13
⊢ (𝐸 ∈ ℝ+
→ (𝑁 ∈ ℕ
∧ (1 / 𝑁) < 𝐸)) |
77 | 66, 76 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 ∈ ℕ ∧ (1 / 𝑁) < 𝐸)) |
78 | 77 | simpld 474 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑁 ∈ ℕ) |
79 | | uznnssnn 11773 |
. . . . . . . . . . 11
⊢ (𝑁 ∈ ℕ →
(ℤ≥‘𝑁) ⊆ ℕ) |
80 | 78, 79 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 →
(ℤ≥‘𝑁) ⊆ ℕ) |
81 | 40, 80 | syl5eqss 3682 |
. . . . . . . . 9
⊢ (𝜑 → 𝑍 ⊆ ℕ) |
82 | 81 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑍 ⊆ ℕ) |
83 | | simpr 476 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) |
84 | 82, 83 | sseldd 3637 |
. . . . . . 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 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑋 ∈ Fin) |
88 | | eqid 2651 |
. . . . . . . . . 10
⊢ dom
(voln‘𝑋) = dom
(voln‘𝑋) |
89 | 18, 21 | readdcld 10107 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) ∈ ℝ) |
90 | | eqid 2651 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) = (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))) |
91 | 89, 90 | fmptd 6425 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ) |
92 | 8 | feq1d 6068 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐶‘𝑛):𝑋⟶ℝ ↔ (𝑘 ∈ 𝑋 ↦ ((𝐴‘𝑘) + (1 / 𝑛))):𝑋⟶ℝ)) |
93 | 91, 92 | mpbird 247 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐶‘𝑛):𝑋⟶ℝ) |
94 | 12 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐵:𝑋⟶ℝ) |
95 | 87, 88, 93, 94 | hoimbl 41166 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ dom (voln‘𝑋)) |
96 | 95 | elexd 3245 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) ∈ V) |
97 | 86, 96 | fvmpt2d 6332 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) |
98 | 84, 97 | syldan 486 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐷‘𝑛) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) |
99 | 98 | fveq2d 6233 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))) |
100 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ∈ Fin) |
101 | 59 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑋 ≠ ∅) |
102 | 84, 93 | syldan 486 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐶‘𝑛):𝑋⟶ℝ) |
103 | 12 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐵:𝑋⟶ℝ) |
104 | | eqid 2651 |
. . . . . 6
⊢ X𝑘 ∈
𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)) |
105 | 100, 101,
102, 103, 104 | vonn0hoi 41205 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘X𝑘 ∈ 𝑋 (((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘)))) |
106 | 102 | ffvelrnda 6399 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) ∈ ℝ) |
107 | 84, 14 | syldanl 735 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐵‘𝑘) ∈ ℝ) |
108 | | volico 40518 |
. . . . . . . 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)) |
109 | 106, 107,
108 | syl2anc 694 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = if(((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)), 0)) |
110 | 84, 10 | syldanl 735 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) = ((𝐴‘𝑘) + (1 / 𝑛))) |
111 | 84, 21 | syldanl 735 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ) |
112 | 78 | nnrecred 11104 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1 / 𝑁) ∈ ℝ) |
113 | 112 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ) |
114 | 35 | adantlr 751 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ℝ) |
115 | 40 | eleq2i 2722 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑁)) |
116 | 115 | biimpi 206 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑁)) |
117 | | eluzle 11738 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈
(ℤ≥‘𝑁) → 𝑁 ≤ 𝑛) |
118 | 116, 117 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ 𝑍 → 𝑁 ≤ 𝑛) |
119 | 118 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ≤ 𝑛) |
120 | 78 | nnrpd 11908 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝑁 ∈
ℝ+) |
121 | 120 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑁 ∈
ℝ+) |
122 | | nnrp 11880 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ → 𝑛 ∈
ℝ+) |
123 | 84, 122 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℝ+) |
124 | 121, 123 | lerecd 11929 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑁 ≤ 𝑛 ↔ (1 / 𝑛) ≤ (1 / 𝑁))) |
125 | 119, 124 | mpbid 222 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (1 / 𝑛) ≤ (1 / 𝑁)) |
126 | 125 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ≤ (1 / 𝑁)) |
127 | 112 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) ∈ ℝ) |
128 | 29, 66 | sseldi 3634 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐸 ∈ ℝ) |
129 | 128 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ∈ ℝ) |
130 | 77 | simprd 478 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1 / 𝑁) < 𝐸) |
131 | 130 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < 𝐸) |
132 | 62 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ⊆ ℝ) |
133 | 58 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))) ∈ Fin) |
134 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋) |
135 | | ovexd 6720 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V) |
136 | 52 | elrnmpt1 5406 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ V) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
137 | 134, 135,
136 | syl2anc 694 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ 𝑋 → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
138 | 137 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
139 | | infrefilb 39913 |
. . . . . . . . . . . . . . 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))) |
140 | 132, 133,
138, 139 | syl3anc 1366 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐵‘𝑘) − (𝐴‘𝑘))), ℝ, < ) ≤ ((𝐵‘𝑘) − (𝐴‘𝑘))) |
141 | 54, 140 | syl5eqbr 4720 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐸 ≤ ((𝐵‘𝑘) − (𝐴‘𝑘))) |
142 | 127, 129,
35, 131, 141 | ltletrd 10235 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < ((𝐵‘𝑘) − (𝐴‘𝑘))) |
143 | 142 | adantlr 751 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑁) < ((𝐵‘𝑘) − (𝐴‘𝑘))) |
144 | 111, 113,
114, 126, 143 | lelttrd 10233 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘))) |
145 | 84, 18 | syldanl 735 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (𝐴‘𝑘) ∈ ℝ) |
146 | 145, 111,
107 | ltaddsub2d 10666 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (((𝐴‘𝑘) + (1 / 𝑛)) < (𝐵‘𝑘) ↔ (1 / 𝑛) < ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
147 | 144, 146 | mpbird 247 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐴‘𝑘) + (1 / 𝑛)) < (𝐵‘𝑘)) |
148 | 110, 147 | eqbrtrd 4707 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘)) |
149 | 148 | iftrued 4127 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → if(((𝐶‘𝑛)‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)), 0) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
150 | 109, 149 | eqtrd 2685 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑘 ∈ 𝑋) → (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
151 | 150 | prodeq2dv 14697 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑛)‘𝑘)[,)(𝐵‘𝑘))) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
152 | 99, 105, 151 | 3eqtrd 2689 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
153 | | fvexd 6241 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V) |
154 | 45 | fvmpt2 6330 |
. . . . 5
⊢ ((𝑛 ∈ ℕ ∧
((voln‘𝑋)‘(𝐷‘𝑛)) ∈ V) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛))) |
155 | 84, 153, 154 | syl2anc 694 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑆‘𝑛) = ((voln‘𝑋)‘(𝐷‘𝑛))) |
156 | | prodex 14681 |
. . . . . 6
⊢
∏𝑘 ∈
𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) ∈ V |
157 | 156 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) ∈ V) |
158 | 1 | fvmpt2 6330 |
. . . . 5
⊢ ((𝑛 ∈ ℕ ∧
∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘)) ∈ V) → (𝑇‘𝑛) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
159 | 84, 157, 158 | syl2anc 694 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − ((𝐶‘𝑛)‘𝑘))) |
160 | 152, 155,
159 | 3eqtr4rd 2696 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑇‘𝑛) = (𝑆‘𝑛)) |
161 | 40, 44, 48, 75, 160 | climeq 14342 |
. 2
⊢ (𝜑 → (𝑇 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)) ↔ 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘)))) |
162 | 39, 161 | mpbid 222 |
1
⊢ (𝜑 → 𝑆 ⇝ ∏𝑘 ∈ 𝑋 ((𝐵‘𝑘) − (𝐴‘𝑘))) |