Proof of Theorem ovn0lem
Step | Hyp | Ref
| Expression |
1 | | iccssxr 12256 |
. . 3
⊢
(0[,]+∞) ⊆ ℝ* |
2 | | ovn0lem.infm |
. . 3
⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈
(0[,]+∞)) |
3 | 1, 2 | sseldi 3601 |
. 2
⊢ (𝜑 → inf(𝑀, ℝ*, < ) ∈
ℝ*) |
4 | | 0xr 10086 |
. . 3
⊢ 0 ∈
ℝ* |
5 | 4 | a1i 11 |
. 2
⊢ (𝜑 → 0 ∈
ℝ*) |
6 | | ovn0lem.m |
. . . . 5
⊢ 𝑀 = {𝑧 ∈ ℝ* ∣
∃𝑖 ∈ (((ℝ
× ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))} |
7 | | ssrab2 3687 |
. . . . 5
⊢ {𝑧 ∈ ℝ*
∣ ∃𝑖 ∈
(((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))} ⊆
ℝ* |
8 | 6, 7 | eqsstri 3635 |
. . . 4
⊢ 𝑀 ⊆
ℝ* |
9 | 8 | a1i 11 |
. . 3
⊢ (𝜑 → 𝑀 ⊆
ℝ*) |
10 | | 1re 10039 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℝ |
11 | | 0re 10040 |
. . . . . . . . . . . . . 14
⊢ 0 ∈
ℝ |
12 | 10, 11 | pm3.2i 471 |
. . . . . . . . . . . . 13
⊢ (1 ∈
ℝ ∧ 0 ∈ ℝ) |
13 | | opelxp 5146 |
. . . . . . . . . . . . 13
⊢ (〈1,
0〉 ∈ (ℝ × ℝ) ↔ (1 ∈ ℝ ∧ 0
∈ ℝ)) |
14 | 12, 13 | mpbir 221 |
. . . . . . . . . . . 12
⊢ 〈1,
0〉 ∈ (ℝ × ℝ) |
15 | 14 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑙 ∈ 𝑋) → 〈1, 0〉 ∈ (ℝ
× ℝ)) |
16 | | eqid 2622 |
. . . . . . . . . . 11
⊢ (𝑙 ∈ 𝑋 ↦ 〈1, 0〉) = (𝑙 ∈ 𝑋 ↦ 〈1, 0〉) |
17 | 15, 16 | fmptd 6385 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑙 ∈ 𝑋 ↦ 〈1, 0〉):𝑋⟶(ℝ ×
ℝ)) |
18 | | reex 10027 |
. . . . . . . . . . . . 13
⊢ ℝ
∈ V |
19 | 18, 18 | xpex 6962 |
. . . . . . . . . . . 12
⊢ (ℝ
× ℝ) ∈ V |
20 | 19 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → (ℝ × ℝ)
∈ V) |
21 | | ovn0lem.x |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈ Fin) |
22 | | elmapg 7870 |
. . . . . . . . . . 11
⊢
(((ℝ × ℝ) ∈ V ∧ 𝑋 ∈ Fin) → ((𝑙 ∈ 𝑋 ↦ 〈1, 0〉) ∈ ((ℝ
× ℝ) ↑𝑚 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ 〈1, 0〉):𝑋⟶(ℝ ×
ℝ))) |
23 | 20, 21, 22 | syl2anc 693 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑙 ∈ 𝑋 ↦ 〈1, 0〉) ∈ ((ℝ
× ℝ) ↑𝑚 𝑋) ↔ (𝑙 ∈ 𝑋 ↦ 〈1, 0〉):𝑋⟶(ℝ ×
ℝ))) |
24 | 17, 23 | mpbird 247 |
. . . . . . . . 9
⊢ (𝜑 → (𝑙 ∈ 𝑋 ↦ 〈1, 0〉) ∈ ((ℝ
× ℝ) ↑𝑚 𝑋)) |
25 | 24 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ 〈1, 0〉) ∈ ((ℝ
× ℝ) ↑𝑚 𝑋)) |
26 | | ovn0lem.i |
. . . . . . . 8
⊢ 𝐼 = (𝑗 ∈ ℕ ↦ (𝑙 ∈ 𝑋 ↦ 〈1, 0〉)) |
27 | 25, 26 | fmptd 6385 |
. . . . . . 7
⊢ (𝜑 → 𝐼:ℕ⟶((ℝ × ℝ)
↑𝑚 𝑋)) |
28 | | ovexd 6680 |
. . . . . . . 8
⊢ (𝜑 → ((ℝ × ℝ)
↑𝑚 𝑋) ∈ V) |
29 | | nnex 11026 |
. . . . . . . . 9
⊢ ℕ
∈ V |
30 | 29 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ℕ ∈
V) |
31 | | elmapg 7870 |
. . . . . . . 8
⊢
((((ℝ × ℝ) ↑𝑚 𝑋) ∈ V ∧ ℕ ∈
V) → (𝐼 ∈
(((ℝ × ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ↔
𝐼:ℕ⟶((ℝ
× ℝ) ↑𝑚 𝑋))) |
32 | 28, 30, 31 | syl2anc 693 |
. . . . . . 7
⊢ (𝜑 → (𝐼 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚 ℕ) ↔
𝐼:ℕ⟶((ℝ
× ℝ) ↑𝑚 𝑋))) |
33 | 27, 32 | mpbird 247 |
. . . . . 6
⊢ (𝜑 → 𝐼 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚
ℕ)) |
34 | | ovn0lem.n0 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑋 ≠ ∅) |
35 | | n0 3931 |
. . . . . . . . . . . 12
⊢ (𝑋 ≠ ∅ ↔
∃𝑙 𝑙 ∈ 𝑋) |
36 | 34, 35 | sylib 208 |
. . . . . . . . . . 11
⊢ (𝜑 → ∃𝑙 𝑙 ∈ 𝑋) |
37 | 36 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∃𝑙 𝑙 ∈ 𝑋) |
38 | | nfv 1843 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) |
39 | | nfcv 2764 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘(vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) |
40 | 21 | ad2antrr 762 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → 𝑋 ∈ Fin) |
41 | 27 | ffvelrnda 6359 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) ∈ ((ℝ × ℝ)
↑𝑚 𝑋)) |
42 | | elmapi 7879 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐼‘𝑗) ∈ ((ℝ × ℝ)
↑𝑚 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ ×
ℝ)) |
43 | 41, 42 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗):𝑋⟶(ℝ ×
ℝ)) |
44 | 43 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗):𝑋⟶(ℝ ×
ℝ)) |
45 | | simpr 477 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) |
46 | 44, 45 | fvovco 39381 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘)))) |
47 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) |
48 | 25 | elexd 3214 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 ↦ 〈1, 0〉) ∈
V) |
49 | 26 | fvmpt2 6291 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑗 ∈ ℕ ∧ (𝑙 ∈ 𝑋 ↦ 〈1, 0〉) ∈ V) →
(𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ 〈1, 0〉)) |
50 | 47, 48, 49 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ 〈1, 0〉)) |
51 | 50 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑗) = (𝑙 ∈ 𝑋 ↦ 〈1, 0〉)) |
52 | | eqidd 2623 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑙 = 𝑘) → 〈1, 0〉 = 〈1,
0〉) |
53 | 14 | elexi 3213 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ 〈1,
0〉 ∈ V |
54 | 53 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 〈1, 0〉 ∈
V) |
55 | 51, 52, 45, 54 | fvmptd 6288 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐼‘𝑗)‘𝑘) = 〈1, 0〉) |
56 | 55 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = (1st ‘〈1,
0〉)) |
57 | 10 | elexi 3213 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 1 ∈
V |
58 | 4 | elexi 3213 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 0 ∈
V |
59 | 57, 58 | op1st 7176 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1st ‘〈1, 0〉) = 1 |
60 | 59 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘〈1,
0〉) = 1) |
61 | 56, 60 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1st ‘((𝐼‘𝑗)‘𝑘)) = 1) |
62 | 55 | fveq2d 6195 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = (2nd ‘〈1,
0〉)) |
63 | 57, 58 | op2nd 7177 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(2nd ‘〈1, 0〉) = 0 |
64 | 63 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘〈1,
0〉) = 0) |
65 | 62, 64 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (2nd ‘((𝐼‘𝑗)‘𝑘)) = 0) |
66 | 61, 65 | oveq12d 6668 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((1st ‘((𝐼‘𝑗)‘𝑘))[,)(2nd ‘((𝐼‘𝑗)‘𝑘))) = (1[,)0)) |
67 | | 0le1 10551 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 0 ≤
1 |
68 | 10 | rexri 10097 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 1 ∈
ℝ* |
69 | | ico0 12221 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((1
∈ ℝ* ∧ 0 ∈ ℝ*) → ((1[,)0)
= ∅ ↔ 0 ≤ 1)) |
70 | 68, 4, 69 | mp2an 708 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((1[,)0)
= ∅ ↔ 0 ≤ 1) |
71 | 67, 70 | mpbir 221 |
. . . . . . . . . . . . . . . . . . 19
⊢ (1[,)0) =
∅ |
72 | 71 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1[,)0) = ∅) |
73 | 46, 66, 72 | 3eqtrd 2660 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (([,) ∘ (𝐼‘𝑗))‘𝑘) = ∅) |
74 | 73 | fveq2d 6195 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘∅)) |
75 | | vol0 40175 |
. . . . . . . . . . . . . . . . 17
⊢
(vol‘∅) = 0 |
76 | 75 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘∅) =
0) |
77 | 74, 76 | eqtrd 2656 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0) |
78 | | 0cn 10032 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℂ |
79 | 78 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 0 ∈ ℂ) |
80 | 77, 79 | eqeltrd 2701 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ ℂ) |
81 | 80 | adantlr 751 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) ∈ ℂ) |
82 | | fveq2 6191 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑙 → (([,) ∘ (𝐼‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑙)) |
83 | 82 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑙 → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙))) |
84 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → 𝑙 ∈ 𝑋) |
85 | | eleq1 2689 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑙 → (𝑘 ∈ 𝑋 ↔ 𝑙 ∈ 𝑋)) |
86 | 85 | anbi2d 740 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑙 → (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ↔ ((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋))) |
87 | 83 | eqeq1d 2624 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑙 → ((vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0 ↔ (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0)) |
88 | 86, 87 | imbi12d 334 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑙 → ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0) ↔ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0))) |
89 | 88, 77 | chvarv 2263 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑙)) = 0) |
90 | 38, 39, 40, 81, 83, 84, 89 | fprod0 39828 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑙 ∈ 𝑋) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0) |
91 | 90 | ex 450 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑙 ∈ 𝑋 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)) |
92 | 91 | exlimdv 1861 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (∃𝑙 𝑙 ∈ 𝑋 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0)) |
93 | 37, 92 | mpd 15 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)) = 0) |
94 | 93 | mpteq2dva 4744 |
. . . . . . . 8
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ 0)) |
95 | 94 | fveq2d 6195 |
. . . . . . 7
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ 0))) |
96 | | nfv 1843 |
. . . . . . . 8
⊢
Ⅎ𝑗𝜑 |
97 | 96, 30 | sge0z 40592 |
. . . . . . 7
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ 0)) =
0) |
98 | | eqidd 2623 |
. . . . . . 7
⊢ (𝜑 → 0 = 0) |
99 | 95, 97, 98 | 3eqtrrd 2661 |
. . . . . 6
⊢ (𝜑 → 0 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))) |
100 | | fveq1 6190 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = 𝐼 → (𝑖‘𝑗) = (𝐼‘𝑗)) |
101 | 100 | coeq2d 5284 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝐼 → ([,) ∘ (𝑖‘𝑗)) = ([,) ∘ (𝐼‘𝑗))) |
102 | 101 | fveq1d 6193 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝐼 → (([,) ∘ (𝑖‘𝑗))‘𝑘) = (([,) ∘ (𝐼‘𝑗))‘𝑘)) |
103 | 102 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ (𝑖 = 𝐼 → (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))) |
104 | 103 | ralrimivw 2967 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝐼 → ∀𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))) |
105 | 104 | prodeq2d 14652 |
. . . . . . . . . 10
⊢ (𝑖 = 𝐼 → ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)) = ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))) |
106 | 105 | mpteq2dv 4745 |
. . . . . . . . 9
⊢ (𝑖 = 𝐼 → (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))) |
107 | 106 | fveq2d 6195 |
. . . . . . . 8
⊢ (𝑖 = 𝐼 →
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))) |
108 | 107 | eqeq2d 2632 |
. . . . . . 7
⊢ (𝑖 = 𝐼 → (0 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ 0 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘)))))) |
109 | 108 | rspcev 3309 |
. . . . . 6
⊢ ((𝐼 ∈ (((ℝ ×
ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ) ∧
0 = (Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝐼‘𝑗))‘𝑘))))) → ∃𝑖 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚 ℕ)0 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) |
110 | 33, 99, 109 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → ∃𝑖 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚 ℕ)0 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘))))) |
111 | 5, 110 | jca 554 |
. . . 4
⊢ (𝜑 → (0 ∈
ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚 ℕ)0 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) |
112 | | eqeq1 2626 |
. . . . . 6
⊢ (𝑧 = 0 → (𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ 0 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) |
113 | 112 | rexbidv 3052 |
. . . . 5
⊢ (𝑧 = 0 → (∃𝑖 ∈ (((ℝ ×
ℝ) ↑𝑚 𝑋) ↑𝑚 ℕ)𝑧 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))) ↔ ∃𝑖 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚 ℕ)0 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) |
114 | 113, 6 | elrab2 3366 |
. . . 4
⊢ (0 ∈
𝑀 ↔ (0 ∈
ℝ* ∧ ∃𝑖 ∈ (((ℝ × ℝ)
↑𝑚 𝑋) ↑𝑚 ℕ)0 =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(([,) ∘ (𝑖‘𝑗))‘𝑘)))))) |
115 | 111, 114 | sylibr 224 |
. . 3
⊢ (𝜑 → 0 ∈ 𝑀) |
116 | | infxrlb 12164 |
. . 3
⊢ ((𝑀 ⊆ ℝ*
∧ 0 ∈ 𝑀) →
inf(𝑀, ℝ*,
< ) ≤ 0) |
117 | 9, 115, 116 | syl2anc 693 |
. 2
⊢ (𝜑 → inf(𝑀, ℝ*, < ) ≤
0) |
118 | | pnfxr 10092 |
. . . 4
⊢ +∞
∈ ℝ* |
119 | 118 | a1i 11 |
. . 3
⊢ (𝜑 → +∞ ∈
ℝ*) |
120 | | iccgelb 12230 |
. . 3
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧
inf(𝑀, ℝ*,
< ) ∈ (0[,]+∞)) → 0 ≤ inf(𝑀, ℝ*, <
)) |
121 | 5, 119, 2, 120 | syl3anc 1326 |
. 2
⊢ (𝜑 → 0 ≤ inf(𝑀, ℝ*, <
)) |
122 | 3, 5, 117, 121 | xrletrid 11986 |
1
⊢ (𝜑 → inf(𝑀, ℝ*, < ) =
0) |