Step | Hyp | Ref
| Expression |
1 | | nfv 1921 |
. . . . 5
⊢
Ⅎ𝑠𝜑 |
2 | | nfre1 3237 |
. . . . 5
⊢
Ⅎ𝑠∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠) |
3 | 1, 2 | nfan 1906 |
. . . 4
⊢
Ⅎ𝑠(𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) |
4 | | hoidmvlelem5.l |
. . . 4
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑m 𝑥), 𝑏 ∈ (ℝ ↑m 𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
5 | | hoidmvlelem5.w |
. . . . . 6
⊢ 𝑊 = (𝑌 ∪ {𝑍}) |
6 | | hoidmvlelem5.f |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ Fin) |
7 | | hoidmvlelem5.y |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
8 | | ssfi 8938 |
. . . . . . . 8
⊢ ((𝑋 ∈ Fin ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ Fin) |
9 | 6, 7, 8 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → 𝑌 ∈ Fin) |
10 | | snfi 8817 |
. . . . . . . 8
⊢ {𝑍} ∈ Fin |
11 | 10 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → {𝑍} ∈ Fin) |
12 | | unfi 8937 |
. . . . . . 7
⊢ ((𝑌 ∈ Fin ∧ {𝑍} ∈ Fin) → (𝑌 ∪ {𝑍}) ∈ Fin) |
13 | 9, 11, 12 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (𝑌 ∪ {𝑍}) ∈ Fin) |
14 | 5, 13 | eqeltrid 2845 |
. . . . 5
⊢ (𝜑 → 𝑊 ∈ Fin) |
15 | 14 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → 𝑊 ∈ Fin) |
16 | | hoidmvlelem5.a |
. . . . 5
⊢ (𝜑 → 𝐴:𝑊⟶ℝ) |
17 | 16 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → 𝐴:𝑊⟶ℝ) |
18 | | hoidmvlelem5.b |
. . . . 5
⊢ (𝜑 → 𝐵:𝑊⟶ℝ) |
19 | 18 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → 𝐵:𝑊⟶ℝ) |
20 | | simpr 485 |
. . . 4
⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) |
21 | 3, 4, 15, 17, 19, 20 | hoidmvval0 44096 |
. . 3
⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → (𝐴(𝐿‘𝑊)𝐵) = 0) |
22 | | nnex 11979 |
. . . . . 6
⊢ ℕ
∈ V |
23 | 22 | a1i 11 |
. . . . 5
⊢ (𝜑 → ℕ ∈
V) |
24 | | icossicc 13167 |
. . . . . . 7
⊢
(0[,)+∞) ⊆ (0[,]+∞) |
25 | 14 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑊 ∈ Fin) |
26 | | hoidmvlelem5.c |
. . . . . . . . . 10
⊢ (𝜑 → 𝐶:ℕ⟶(ℝ ↑m
𝑊)) |
27 | 26 | ffvelrnda 6958 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑m 𝑊)) |
28 | | elmapi 8620 |
. . . . . . . . 9
⊢ ((𝐶‘𝑗) ∈ (ℝ ↑m 𝑊) → (𝐶‘𝑗):𝑊⟶ℝ) |
29 | 27, 28 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ) |
30 | | hoidmvlelem5.d |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷:ℕ⟶(ℝ ↑m
𝑊)) |
31 | 30 | ffvelrnda 6958 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑m 𝑊)) |
32 | | elmapi 8620 |
. . . . . . . . 9
⊢ ((𝐷‘𝑗) ∈ (ℝ ↑m 𝑊) → (𝐷‘𝑗):𝑊⟶ℝ) |
33 | 31, 32 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ) |
34 | 4, 25, 29, 33 | hoidmvcl 44091 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,)+∞)) |
35 | 24, 34 | sselid 3924 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) ∈ (0[,]+∞)) |
36 | 35 | fmpttd 6986 |
. . . . 5
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))):ℕ⟶(0[,]+∞)) |
37 | 23, 36 | sge0ge0 43893 |
. . . 4
⊢ (𝜑 → 0 ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
38 | 37 | adantr 481 |
. . 3
⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → 0 ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
39 | 21, 38 | eqbrtrd 5101 |
. 2
⊢ ((𝜑 ∧ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → (𝐴(𝐿‘𝑊)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
40 | | icossxr 13163 |
. . . . . . 7
⊢
(0[,)+∞) ⊆ ℝ* |
41 | 4, 14, 16, 18 | hoidmvcl 44091 |
. . . . . . 7
⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ∈ (0[,)+∞)) |
42 | 40, 41 | sselid 3924 |
. . . . . 6
⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ∈
ℝ*) |
43 | 42 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) ∈
ℝ*) |
44 | 23, 36 | sge0xrcl 43894 |
. . . . . 6
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈
ℝ*) |
45 | 44 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈
ℝ*) |
46 | | rge0ssre 13187 |
. . . . . . . . 9
⊢
(0[,)+∞) ⊆ ℝ |
47 | 46, 41 | sselid 3924 |
. . . . . . . 8
⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ∈ ℝ) |
48 | | ltpnf 12855 |
. . . . . . . 8
⊢ ((𝐴(𝐿‘𝑊)𝐵) ∈ ℝ → (𝐴(𝐿‘𝑊)𝐵) < +∞) |
49 | 47, 48 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) < +∞) |
50 | 49 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) < +∞) |
51 | | id 22 |
. . . . . . . 8
⊢
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞ →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) |
52 | 51 | eqcomd 2746 |
. . . . . . 7
⊢
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞ → +∞ =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
53 | 52 | adantl 482 |
. . . . . 6
⊢ ((𝜑 ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → +∞ =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
54 | 50, 53 | breqtrd 5105 |
. . . . 5
⊢ ((𝜑 ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) <
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
55 | 43, 45, 54 | xrltled 12883 |
. . . 4
⊢ ((𝜑 ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
56 | 55 | adantlr 712 |
. . 3
⊢ (((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
57 | | simpll 764 |
. . . 4
⊢ (((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → 𝜑) |
58 | | simpr 485 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) |
59 | 16 | ffvelrnda 6958 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑊) → (𝐴‘𝑠) ∈ ℝ) |
60 | 18 | ffvelrnda 6958 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑊) → (𝐵‘𝑠) ∈ ℝ) |
61 | 59, 60 | ltnled 11122 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑊) → ((𝐴‘𝑠) < (𝐵‘𝑠) ↔ ¬ (𝐵‘𝑠) ≤ (𝐴‘𝑠))) |
62 | 61 | ralbidva 3122 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ↔ ∀𝑠 ∈ 𝑊 ¬ (𝐵‘𝑠) ≤ (𝐴‘𝑠))) |
63 | | ralnex 3166 |
. . . . . . . . 9
⊢
(∀𝑠 ∈
𝑊 ¬ (𝐵‘𝑠) ≤ (𝐴‘𝑠) ↔ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) |
64 | 63 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑠 ∈ 𝑊 ¬ (𝐵‘𝑠) ≤ (𝐴‘𝑠) ↔ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠))) |
65 | 62, 64 | bitrd 278 |
. . . . . . 7
⊢ (𝜑 → (∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ↔ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠))) |
66 | 65 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → (∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ↔ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠))) |
67 | 58, 66 | mpbird 256 |
. . . . 5
⊢ ((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) |
68 | 67 | adantr 481 |
. . . 4
⊢ (((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) |
69 | | simpr 485 |
. . . . . 6
⊢ ((𝜑 ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) |
70 | 22 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → ℕ ∈
V) |
71 | 36 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))):ℕ⟶(0[,]+∞)) |
72 | 70, 71 | sge0repnf 43895 |
. . . . . 6
⊢ ((𝜑 ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) →
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ ↔ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞)) |
73 | 69, 72 | mpbird 256 |
. . . . 5
⊢ ((𝜑 ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) |
74 | 73 | adantlr 712 |
. . . 4
⊢ (((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) |
75 | | simpll 764 |
. . . . . . 7
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠))) |
76 | | fveq2 6771 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖)) |
77 | | fveq2 6771 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖)) |
78 | 76, 77 | oveq12d 7289 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)) = ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖))) |
79 | 78 | cbvmptv 5192 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖))) |
80 | 79 | fveq2i 6774 |
. . . . . . . . . 10
⊢
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) =
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) |
81 | 80 | eleq1i 2831 |
. . . . . . . . 9
⊢
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ ↔
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) |
82 | 81 | biimpi 215 |
. . . . . . . 8
⊢
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ →
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) |
83 | 82 | ad2antlr 724 |
. . . . . . 7
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) →
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) |
84 | | simpr 485 |
. . . . . . 7
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈
ℝ+) |
85 | 6 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑋 ∈ Fin) |
86 | 7 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑌 ⊆ 𝑋) |
87 | | hoidmvlelem5.n |
. . . . . . . . 9
⊢ (𝜑 → 𝑌 ≠ ∅) |
88 | 87 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑌 ≠ ∅) |
89 | | hoidmvlelem5.z |
. . . . . . . . 9
⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌)) |
90 | 89 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑍 ∈ (𝑋 ∖ 𝑌)) |
91 | 16 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐴:𝑊⟶ℝ) |
92 | 18 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐵:𝑊⟶ℝ) |
93 | | fveq2 6771 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑘 → (𝐴‘𝑠) = (𝐴‘𝑘)) |
94 | | fveq2 6771 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑘 → (𝐵‘𝑠) = (𝐵‘𝑘)) |
95 | 93, 94 | breq12d 5092 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑘 → ((𝐴‘𝑠) < (𝐵‘𝑠) ↔ (𝐴‘𝑘) < (𝐵‘𝑘))) |
96 | 95 | cbvralvw 3381 |
. . . . . . . . . . . 12
⊢
(∀𝑠 ∈
𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ↔ ∀𝑘 ∈ 𝑊 (𝐴‘𝑘) < (𝐵‘𝑘)) |
97 | 96 | biimpi 215 |
. . . . . . . . . . 11
⊢
(∀𝑠 ∈
𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) → ∀𝑘 ∈ 𝑊 (𝐴‘𝑘) < (𝐵‘𝑘)) |
98 | 97 | adantr 481 |
. . . . . . . . . 10
⊢
((∀𝑠 ∈
𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ∧ 𝑘 ∈ 𝑊) → ∀𝑘 ∈ 𝑊 (𝐴‘𝑘) < (𝐵‘𝑘)) |
99 | | simpr 485 |
. . . . . . . . . 10
⊢
((∀𝑠 ∈
𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊) |
100 | | rspa 3133 |
. . . . . . . . . 10
⊢
((∀𝑘 ∈
𝑊 (𝐴‘𝑘) < (𝐵‘𝑘) ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘)) |
101 | 98, 99, 100 | syl2anc 584 |
. . . . . . . . 9
⊢
((∀𝑠 ∈
𝑊 (𝐴‘𝑠) < (𝐵‘𝑠) ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘)) |
102 | 101 | ad5ant25 759 |
. . . . . . . 8
⊢
(((((𝜑 ∧
∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘)) |
103 | 26 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐶:ℕ⟶(ℝ
↑m 𝑊)) |
104 | 30 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝐷:ℕ⟶(ℝ
↑m 𝑊)) |
105 | 81 | biimpri 227 |
. . . . . . . . 9
⊢
((Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) |
106 | 105 | ad2antlr 724 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) |
107 | | fveq1 6770 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑐 → (𝑑‘𝑖) = (𝑐‘𝑖)) |
108 | 107 | breq1d 5089 |
. . . . . . . . . . . . . 14
⊢ (𝑑 = 𝑐 → ((𝑑‘𝑖) ≤ 𝑥 ↔ (𝑐‘𝑖) ≤ 𝑥)) |
109 | 108, 107 | ifbieq1d 4489 |
. . . . . . . . . . . . 13
⊢ (𝑑 = 𝑐 → if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥) = if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)) |
110 | 107, 109 | ifeq12d 4486 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝑐 → if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)) = if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))) |
111 | 110 | mpteq2dv 5181 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑐 → (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))) = (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))) |
112 | | eleq1w 2823 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → (𝑖 ∈ 𝑌 ↔ 𝑗 ∈ 𝑌)) |
113 | | fveq2 6771 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → (𝑐‘𝑖) = (𝑐‘𝑗)) |
114 | 113 | breq1d 5089 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖) ≤ 𝑥 ↔ (𝑐‘𝑗) ≤ 𝑥)) |
115 | 114, 113 | ifbieq1d 4489 |
. . . . . . . . . . . . . 14
⊢ (𝑖 = 𝑗 → if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥) = if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)) |
116 | 112, 113,
115 | ifbieq12d 4493 |
. . . . . . . . . . . . 13
⊢ (𝑖 = 𝑗 → if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)) = if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))) |
117 | 116 | cbvmptv 5192 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))) = (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))) |
118 | 117 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝑐 → (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))) = (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))) |
119 | 111, 118 | eqtrd 2780 |
. . . . . . . . . 10
⊢ (𝑑 = 𝑐 → (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))) = (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))) |
120 | 119 | cbvmptv 5192 |
. . . . . . . . 9
⊢ (𝑑 ∈ (ℝ
↑m 𝑊)
↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))) = (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))) |
121 | 120 | mpteq2i 5184 |
. . . . . . . 8
⊢ (𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ
↑m 𝑊)
↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑m 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))) |
122 | | eqid 2740 |
. . . . . . . 8
⊢ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) |
123 | | simpr 485 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → 𝑟 ∈
ℝ+) |
124 | | oveq1 7278 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑧 → (𝑤 − (𝐴‘𝑍)) = (𝑧 − (𝐴‘𝑍))) |
125 | 124 | oveq2d 7287 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑧 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑤 − (𝐴‘𝑍))) = (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑧 − (𝐴‘𝑍)))) |
126 | | breq2 5083 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑤 = 𝑥 → ((𝑑‘𝑖) ≤ 𝑤 ↔ (𝑑‘𝑖) ≤ 𝑥)) |
127 | | eqidd 2741 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑤 = 𝑥 → (𝑑‘𝑖) = (𝑑‘𝑖)) |
128 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑤 = 𝑥 → 𝑤 = 𝑥) |
129 | 126, 127,
128 | ifbieq12d 4493 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑤 = 𝑥 → if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤) = if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)) |
130 | 129 | ifeq2d 4485 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑤 = 𝑥 → if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)) = if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))) |
131 | 130 | mpteq2dv 5181 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = 𝑥 → (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤))) = (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))) |
132 | 131 | mpteq2dv 5181 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 𝑥 → (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))) = (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))))) |
133 | 132 | cbvmptv 5192 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ
↑m 𝑊)
↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤))))) = (𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥))))) |
134 | 133 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 = 𝑧 → (𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤))))) = (𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))) |
135 | | id 22 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑤 = 𝑧 → 𝑤 = 𝑧) |
136 | 134, 135 | fveq12d 6778 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑧 → ((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤) = ((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)) |
137 | 136 | fveq1d 6773 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑧 → (((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙)) = (((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙))) |
138 | 137 | oveq2d 7287 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑧 → ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙))) = ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙)))) |
139 | 138 | mpteq2dv 5181 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑧 → (𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙)))) = (𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙))))) |
140 | | fveq2 6771 |
. . . . . . . . . . . . . . . 16
⊢ (𝑙 = 𝑗 → (𝐶‘𝑙) = (𝐶‘𝑗)) |
141 | | 2fveq3 6776 |
. . . . . . . . . . . . . . . 16
⊢ (𝑙 = 𝑗 → (((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙)) = (((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗))) |
142 | 140, 141 | oveq12d 7289 |
. . . . . . . . . . . . . . 15
⊢ (𝑙 = 𝑗 → ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙))) = ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗)))) |
143 | 142 | cbvmptv 5192 |
. . . . . . . . . . . . . 14
⊢ (𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗)))) |
144 | 143 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑧 → (𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑙)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗))))) |
145 | 139, 144 | eqtrd 2780 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑧 → (𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙)))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗))))) |
146 | 145 | fveq2d 6775 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑧 →
(Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙))))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗)))))) |
147 | 146 | oveq2d 7287 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑧 → ((1 + 𝑟) ·
(Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙)))))) = ((1 + 𝑟) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗))))))) |
148 | 125, 147 | breq12d 5092 |
. . . . . . . . 9
⊢ (𝑤 = 𝑧 → ((((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑤 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) ·
(Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙)))))) ↔ (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗)))))))) |
149 | 148 | cbvrabv 3425 |
. . . . . . . 8
⊢ {𝑤 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑤 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) ·
(Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙))))))} = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(((𝑥 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑥, (𝑑‘𝑖), 𝑥)))))‘𝑧)‘(𝐷‘𝑗))))))} |
150 | | eqid 2740 |
. . . . . . . 8
⊢
sup({𝑤 ∈
((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑤 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) ·
(Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙))))))}, ℝ, < ) = sup({𝑤 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) · (𝑤 − (𝐴‘𝑍))) ≤ ((1 + 𝑟) ·
(Σ^‘(𝑙 ∈ ℕ ↦ ((𝐶‘𝑙)(𝐿‘𝑊)(((𝑤 ∈ ℝ ↦ (𝑑 ∈ (ℝ ↑m 𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑑‘𝑖), if((𝑑‘𝑖) ≤ 𝑤, (𝑑‘𝑖), 𝑤)))))‘𝑤)‘(𝐷‘𝑙))))))}, ℝ, < ) |
151 | | hoidmvlelem5.i |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑒 ∈ (ℝ ↑m 𝑌)∀𝑓 ∈ (ℝ ↑m 𝑌)∀𝑔 ∈ ((ℝ ↑m 𝑌) ↑m
ℕ)∀ℎ ∈
((ℝ ↑m 𝑌) ↑m ℕ)(X𝑘 ∈
𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
152 | 151 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) →
∀𝑒 ∈ (ℝ
↑m 𝑌)∀𝑓 ∈ (ℝ ↑m 𝑌)∀𝑔 ∈ ((ℝ ↑m 𝑌) ↑m
ℕ)∀ℎ ∈
((ℝ ↑m 𝑌) ↑m ℕ)(X𝑘 ∈
𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
153 | | hoidmvlelem5.s |
. . . . . . . . 9
⊢ (𝜑 → X𝑘 ∈
𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
154 | 153 | ad3antrrr 727 |
. . . . . . . 8
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → X𝑘 ∈
𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
155 | 4, 85, 86, 88, 90, 5, 91, 92, 102, 103, 104, 106, 121, 122, 123, 149, 150, 152, 154 | hoidmvlelem4 44107 |
. . . . . . 7
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝑟) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))) |
156 | 75, 83, 84, 155 | syl21anc 835 |
. . . . . 6
⊢ ((((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) ∧ 𝑟 ∈ ℝ+) → (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝑟) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))) |
157 | 156 | ralrimiva 3110 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → ∀𝑟 ∈ ℝ+
(𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝑟) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))))) |
158 | | nfv 1921 |
. . . . . 6
⊢
Ⅎ𝑟((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) |
159 | 42 | ad2antrr 723 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → (𝐴(𝐿‘𝑊)𝐵) ∈
ℝ*) |
160 | | 0xr 11023 |
. . . . . . . 8
⊢ 0 ∈
ℝ* |
161 | 160 | a1i 11 |
. . . . . . 7
⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → 0 ∈
ℝ*) |
162 | | pnfxr 11030 |
. . . . . . . 8
⊢ +∞
∈ ℝ* |
163 | 162 | a1i 11 |
. . . . . . 7
⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → +∞ ∈
ℝ*) |
164 | 44 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈
ℝ*) |
165 | 37 | ad2antrr 723 |
. . . . . . 7
⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → 0 ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
166 | | ltpnf 12855 |
. . . . . . . 8
⊢
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞) |
167 | 166 | adantl 482 |
. . . . . . 7
⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) < +∞) |
168 | 161, 163,
164, 165, 167 | elicod 13128 |
. . . . . 6
⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ (0[,)+∞)) |
169 | 158, 159,
168 | xralrple2 42864 |
. . . . 5
⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → ((𝐴(𝐿‘𝑊)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ↔ ∀𝑟 ∈ ℝ+ (𝐴(𝐿‘𝑊)𝐵) ≤ ((1 + 𝑟) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))))) |
170 | 157, 169 | mpbird 256 |
. . . 4
⊢ (((𝜑 ∧ ∀𝑠 ∈ 𝑊 (𝐴‘𝑠) < (𝐵‘𝑠)) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) → (𝐴(𝐿‘𝑊)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
171 | 57, 68, 74, 170 | syl21anc 835 |
. . 3
⊢ (((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) = +∞) → (𝐴(𝐿‘𝑊)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
172 | 56, 171 | pm2.61dan 810 |
. 2
⊢ ((𝜑 ∧ ¬ ∃𝑠 ∈ 𝑊 (𝐵‘𝑠) ≤ (𝐴‘𝑠)) → (𝐴(𝐿‘𝑊)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |
173 | 39, 172 | pm2.61dan 810 |
1
⊢ (𝜑 → (𝐴(𝐿‘𝑊)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))))) |