Step |
Hyp |
Ref |
Expression |
1 |
|
vonioolem1.x |
⊢ ( 𝜑 → 𝑋 ∈ Fin ) |
2 |
|
vonioolem1.a |
⊢ ( 𝜑 → 𝐴 : 𝑋 ⟶ ℝ ) |
3 |
|
vonioolem1.b |
⊢ ( 𝜑 → 𝐵 : 𝑋 ⟶ ℝ ) |
4 |
|
vonioolem1.u |
⊢ ( 𝜑 → 𝑋 ≠ ∅ ) |
5 |
|
vonioolem1.t |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) ) |
6 |
|
vonioolem1.c |
⊢ 𝐶 = ( 𝑛 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) ) |
7 |
|
vonioolem1.d |
⊢ 𝐷 = ( 𝑛 ∈ ℕ ↦ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
8 |
|
vonioolem1.s |
⊢ 𝑆 = ( 𝑛 ∈ ℕ ↦ ( ( voln ‘ 𝑋 ) ‘ ( 𝐷 ‘ 𝑛 ) ) ) |
9 |
|
vonioolem1.r |
⊢ 𝑇 = ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ) |
10 |
|
vonioolem1.e |
⊢ 𝐸 = inf ( ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) , ℝ , < ) |
11 |
|
vonioolem1.n |
⊢ 𝑁 = ( ( ⌊ ‘ ( 1 / 𝐸 ) ) + 1 ) |
12 |
|
vonioolem1.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑁 ) |
13 |
9
|
a1i |
⊢ ( 𝜑 → 𝑇 = ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ) ) |
14 |
6
|
a1i |
⊢ ( 𝜑 → 𝐶 = ( 𝑛 ∈ ℕ ↦ ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) ) ) |
15 |
1
|
mptexd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) ∈ V ) |
16 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) ∈ V ) |
17 |
14 16
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐶 ‘ 𝑛 ) = ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) ) |
18 |
|
ovexd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ∈ V ) |
19 |
17 18
|
fvmpt2d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) |
20 |
19
|
oveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) = ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) ) |
21 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) |
22 |
21
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) |
23 |
22
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑘 ) ∈ ℂ ) |
24 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐴 : 𝑋 ⟶ ℝ ) |
25 |
24
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℝ ) |
26 |
25
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℂ ) |
27 |
|
nnrecre |
⊢ ( 𝑛 ∈ ℕ → ( 1 / 𝑛 ) ∈ ℝ ) |
28 |
27
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1 / 𝑛 ) ∈ ℝ ) |
29 |
28
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( 1 / 𝑛 ) ∈ ℂ ) |
30 |
23 26 29
|
subsub4d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) − ( 1 / 𝑛 ) ) = ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) ) |
31 |
20 30
|
eqtr4d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) = ( ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) − ( 1 / 𝑛 ) ) ) |
32 |
31
|
prodeq2dv |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) = ∏ 𝑘 ∈ 𝑋 ( ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) − ( 1 / 𝑛 ) ) ) |
33 |
32
|
mpteq2dva |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ) = ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) − ( 1 / 𝑛 ) ) ) ) |
34 |
13 33
|
eqtrd |
⊢ ( 𝜑 → 𝑇 = ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) − ( 1 / 𝑛 ) ) ) ) |
35 |
|
nfv |
⊢ Ⅎ 𝑘 𝜑 |
36 |
|
rpssre |
⊢ ℝ+ ⊆ ℝ |
37 |
2
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℝ ) |
38 |
|
difrp |
⊢ ( ( ( 𝐴 ‘ 𝑘 ) ∈ ℝ ∧ ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) → ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) ↔ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ ) ) |
39 |
37 21 38
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐴 ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) ↔ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ ) ) |
40 |
5 39
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ+ ) |
41 |
36 40
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ ) |
42 |
41
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℂ ) |
43 |
|
eqid |
⊢ ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) − ( 1 / 𝑛 ) ) ) = ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) − ( 1 / 𝑛 ) ) ) |
44 |
35 1 42 43
|
fprodsubrecnncnv |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) − ( 1 / 𝑛 ) ) ) ⇝ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
45 |
34 44
|
eqbrtrd |
⊢ ( 𝜑 → 𝑇 ⇝ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
46 |
|
nnex |
⊢ ℕ ∈ V |
47 |
46
|
mptex |
⊢ ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ) ∈ V |
48 |
47
|
a1i |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ) ∈ V ) |
49 |
9 48
|
eqeltrid |
⊢ ( 𝜑 → 𝑇 ∈ V ) |
50 |
46
|
mptex |
⊢ ( 𝑛 ∈ ℕ ↦ ( ( voln ‘ 𝑋 ) ‘ ( 𝐷 ‘ 𝑛 ) ) ) ∈ V |
51 |
50
|
a1i |
⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ( ( voln ‘ 𝑋 ) ‘ ( 𝐷 ‘ 𝑛 ) ) ) ∈ V ) |
52 |
8 51
|
eqeltrid |
⊢ ( 𝜑 → 𝑆 ∈ V ) |
53 |
|
1rp |
⊢ 1 ∈ ℝ+ |
54 |
53
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℝ+ ) |
55 |
|
eqid |
⊢ ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) = ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
56 |
35 55 40
|
rnmptssd |
⊢ ( 𝜑 → ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ⊆ ℝ+ ) |
57 |
|
ltso |
⊢ < Or ℝ |
58 |
57
|
a1i |
⊢ ( 𝜑 → < Or ℝ ) |
59 |
55
|
rnmptfi |
⊢ ( 𝑋 ∈ Fin → ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ∈ Fin ) |
60 |
1 59
|
syl |
⊢ ( 𝜑 → ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ∈ Fin ) |
61 |
35 40 55 4
|
rnmptn0 |
⊢ ( 𝜑 → ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ≠ ∅ ) |
62 |
36
|
a1i |
⊢ ( 𝜑 → ℝ+ ⊆ ℝ ) |
63 |
56 62
|
sstrd |
⊢ ( 𝜑 → ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ⊆ ℝ ) |
64 |
|
fiinfcl |
⊢ ( ( < 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 ) ) ) ) |
65 |
58 60 61 63 64
|
syl13anc |
⊢ ( 𝜑 → inf ( ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) , ℝ , < ) ∈ ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ) |
66 |
10 65
|
eqeltrid |
⊢ ( 𝜑 → 𝐸 ∈ ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ) |
67 |
56 66
|
sseldd |
⊢ ( 𝜑 → 𝐸 ∈ ℝ+ ) |
68 |
54 67
|
rpdivcld |
⊢ ( 𝜑 → ( 1 / 𝐸 ) ∈ ℝ+ ) |
69 |
68
|
rpred |
⊢ ( 𝜑 → ( 1 / 𝐸 ) ∈ ℝ ) |
70 |
68
|
rpge0d |
⊢ ( 𝜑 → 0 ≤ ( 1 / 𝐸 ) ) |
71 |
|
flge0nn0 |
⊢ ( ( ( 1 / 𝐸 ) ∈ ℝ ∧ 0 ≤ ( 1 / 𝐸 ) ) → ( ⌊ ‘ ( 1 / 𝐸 ) ) ∈ ℕ0 ) |
72 |
69 70 71
|
syl2anc |
⊢ ( 𝜑 → ( ⌊ ‘ ( 1 / 𝐸 ) ) ∈ ℕ0 ) |
73 |
|
nn0p1nn |
⊢ ( ( ⌊ ‘ ( 1 / 𝐸 ) ) ∈ ℕ0 → ( ( ⌊ ‘ ( 1 / 𝐸 ) ) + 1 ) ∈ ℕ ) |
74 |
72 73
|
syl |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 1 / 𝐸 ) ) + 1 ) ∈ ℕ ) |
75 |
74
|
nnzd |
⊢ ( 𝜑 → ( ( ⌊ ‘ ( 1 / 𝐸 ) ) + 1 ) ∈ ℤ ) |
76 |
11 75
|
eqeltrid |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
77 |
11
|
recnnltrp |
⊢ ( 𝐸 ∈ ℝ+ → ( 𝑁 ∈ ℕ ∧ ( 1 / 𝑁 ) < 𝐸 ) ) |
78 |
67 77
|
syl |
⊢ ( 𝜑 → ( 𝑁 ∈ ℕ ∧ ( 1 / 𝑁 ) < 𝐸 ) ) |
79 |
78
|
simpld |
⊢ ( 𝜑 → 𝑁 ∈ ℕ ) |
80 |
|
uznnssnn |
⊢ ( 𝑁 ∈ ℕ → ( ℤ≥ ‘ 𝑁 ) ⊆ ℕ ) |
81 |
79 80
|
syl |
⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ ℕ ) |
82 |
12 81
|
eqsstrid |
⊢ ( 𝜑 → 𝑍 ⊆ ℕ ) |
83 |
82
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑍 ⊆ ℕ ) |
84 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ 𝑍 ) |
85 |
83 84
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ ℕ ) |
86 |
7
|
a1i |
⊢ ( 𝜑 → 𝐷 = ( 𝑛 ∈ ℕ ↦ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
87 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝑋 ∈ Fin ) |
88 |
|
eqid |
⊢ dom ( voln ‘ 𝑋 ) = dom ( voln ‘ 𝑋 ) |
89 |
25 28
|
readdcld |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ∈ ℝ ) |
90 |
89
|
fmpttd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) : 𝑋 ⟶ ℝ ) |
91 |
17
|
feq1d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( ( 𝐶 ‘ 𝑛 ) : 𝑋 ⟶ ℝ ↔ ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) : 𝑋 ⟶ ℝ ) ) |
92 |
90 91
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐶 ‘ 𝑛 ) : 𝑋 ⟶ ℝ ) |
93 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → 𝐵 : 𝑋 ⟶ ℝ ) |
94 |
87 88 92 93
|
hoimbl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∈ dom ( voln ‘ 𝑋 ) ) |
95 |
94
|
elexd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ∈ V ) |
96 |
86 95
|
fvmpt2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ ℕ ) → ( 𝐷 ‘ 𝑛 ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
97 |
85 96
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐷 ‘ 𝑛 ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) |
98 |
97
|
fveq2d |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( voln ‘ 𝑋 ) ‘ ( 𝐷 ‘ 𝑛 ) ) = ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
99 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑋 ∈ Fin ) |
100 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑋 ≠ ∅ ) |
101 |
85 92
|
syldan |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝐶 ‘ 𝑛 ) : 𝑋 ⟶ ℝ ) |
102 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝐵 : 𝑋 ⟶ ℝ ) |
103 |
|
eqid |
⊢ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) = X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) |
104 |
99 100 101 102 103
|
vonn0hoi |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( voln ‘ 𝑋 ) ‘ X 𝑘 ∈ 𝑋 ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) ) |
105 |
101
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ∈ ℝ ) |
106 |
85 22
|
syldanl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) |
107 |
|
volico |
⊢ ( ( ( ( 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 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = if ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) , ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) , 0 ) ) |
109 |
85 19
|
syldanl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) = ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) ) |
110 |
85 28
|
syldanl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( 1 / 𝑛 ) ∈ ℝ ) |
111 |
79
|
nnrecred |
⊢ ( 𝜑 → ( 1 / 𝑁 ) ∈ ℝ ) |
112 |
111
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( 1 / 𝑁 ) ∈ ℝ ) |
113 |
41
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ℝ ) |
114 |
12
|
eleq2i |
⊢ ( 𝑛 ∈ 𝑍 ↔ 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
115 |
114
|
biimpi |
⊢ ( 𝑛 ∈ 𝑍 → 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
116 |
|
eluzle |
⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑁 ) → 𝑁 ≤ 𝑛 ) |
117 |
115 116
|
syl |
⊢ ( 𝑛 ∈ 𝑍 → 𝑁 ≤ 𝑛 ) |
118 |
117
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑁 ≤ 𝑛 ) |
119 |
79
|
nnrpd |
⊢ ( 𝜑 → 𝑁 ∈ ℝ+ ) |
120 |
119
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑁 ∈ ℝ+ ) |
121 |
|
nnrp |
⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℝ+ ) |
122 |
85 121
|
syl |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → 𝑛 ∈ ℝ+ ) |
123 |
120 122
|
lerecd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑁 ≤ 𝑛 ↔ ( 1 / 𝑛 ) ≤ ( 1 / 𝑁 ) ) ) |
124 |
118 123
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 1 / 𝑛 ) ≤ ( 1 / 𝑁 ) ) |
125 |
124
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( 1 / 𝑛 ) ≤ ( 1 / 𝑁 ) ) |
126 |
111
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 1 / 𝑁 ) ∈ ℝ ) |
127 |
36 67
|
sseldi |
⊢ ( 𝜑 → 𝐸 ∈ ℝ ) |
128 |
127
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝐸 ∈ ℝ ) |
129 |
78
|
simprd |
⊢ ( 𝜑 → ( 1 / 𝑁 ) < 𝐸 ) |
130 |
129
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 1 / 𝑁 ) < 𝐸 ) |
131 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ⊆ ℝ ) |
132 |
60
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ∈ Fin ) |
133 |
|
id |
⊢ ( 𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋 ) |
134 |
|
ovexd |
⊢ ( 𝑘 ∈ 𝑋 → ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ V ) |
135 |
55
|
elrnmpt1 |
⊢ ( ( 𝑘 ∈ 𝑋 ∧ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ V ) → ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ) |
136 |
133 134 135
|
syl2anc |
⊢ ( 𝑘 ∈ 𝑋 → ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ) |
137 |
136
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ∈ ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ) |
138 |
|
infrefilb |
⊢ ( ( 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 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → inf ( ran ( 𝑘 ∈ 𝑋 ↦ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) , ℝ , < ) ≤ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
140 |
10 139
|
eqbrtrid |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → 𝐸 ≤ ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
141 |
126 128 41 130 140
|
ltletrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑋 ) → ( 1 / 𝑁 ) < ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
142 |
141
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( 1 / 𝑁 ) < ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
143 |
110 112 113 125 142
|
lelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( 1 / 𝑛 ) < ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |
144 |
85 25
|
syldanl |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( 𝐴 ‘ 𝑘 ) ∈ ℝ ) |
145 |
144 110 106
|
ltaddsub2d |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) < ( 𝐵 ‘ 𝑘 ) ↔ ( 1 / 𝑛 ) < ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ) |
146 |
143 145
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐴 ‘ 𝑘 ) + ( 1 / 𝑛 ) ) < ( 𝐵 ‘ 𝑘 ) ) |
147 |
109 146
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) < ( 𝐵 ‘ 𝑘 ) ) |
148 |
147
|
iftrued |
⊢ ( ( ( 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 |
⊢ ( ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) ∧ 𝑘 ∈ 𝑋 ) → ( vol ‘ ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ) |
150 |
149
|
prodeq2dv |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∏ 𝑘 ∈ 𝑋 ( vol ‘ ( ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) [,) ( 𝐵 ‘ 𝑘 ) ) ) = ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ) |
151 |
98 104 150
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( voln ‘ 𝑋 ) ‘ ( 𝐷 ‘ 𝑛 ) ) = ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ) |
152 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( ( voln ‘ 𝑋 ) ‘ ( 𝐷 ‘ 𝑛 ) ) ∈ V ) |
153 |
8
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ ℕ ∧ ( ( voln ‘ 𝑋 ) ‘ ( 𝐷 ‘ 𝑛 ) ) ∈ V ) → ( 𝑆 ‘ 𝑛 ) = ( ( voln ‘ 𝑋 ) ‘ ( 𝐷 ‘ 𝑛 ) ) ) |
154 |
85 152 153
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑆 ‘ 𝑛 ) = ( ( voln ‘ 𝑋 ) ‘ ( 𝐷 ‘ 𝑛 ) ) ) |
155 |
|
prodex |
⊢ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ∈ V |
156 |
155
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ∈ V ) |
157 |
9
|
fvmpt2 |
⊢ ( ( 𝑛 ∈ ℕ ∧ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ∈ V ) → ( 𝑇 ‘ 𝑛 ) = ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ) |
158 |
85 156 157
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑇 ‘ 𝑛 ) = ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( ( 𝐶 ‘ 𝑛 ) ‘ 𝑘 ) ) ) |
159 |
151 154 158
|
3eqtr4rd |
⊢ ( ( 𝜑 ∧ 𝑛 ∈ 𝑍 ) → ( 𝑇 ‘ 𝑛 ) = ( 𝑆 ‘ 𝑛 ) ) |
160 |
12 49 52 76 159
|
climeq |
⊢ ( 𝜑 → ( 𝑇 ⇝ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ↔ 𝑆 ⇝ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) ) |
161 |
45 160
|
mpbid |
⊢ ( 𝜑 → 𝑆 ⇝ ∏ 𝑘 ∈ 𝑋 ( ( 𝐵 ‘ 𝑘 ) − ( 𝐴 ‘ 𝑘 ) ) ) |