Step | Hyp | Ref
| Expression |
1 | | hoidmvle.s |
. 2
⊢ (𝜑 → X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
2 | | hoidmvle.d |
. . . 4
⊢ (𝜑 → 𝐷:ℕ⟶(ℝ
↑𝑚 𝑋)) |
3 | | ovex 6577 |
. . . . . . 7
⊢ (ℝ
↑𝑚 𝑋) ∈ V |
4 | | nnex 10903 |
. . . . . . 7
⊢ ℕ
∈ V |
5 | 3, 4 | pm3.2i 470 |
. . . . . 6
⊢ ((ℝ
↑𝑚 𝑋) ∈ V ∧ ℕ ∈
V) |
6 | 5 | a1i 11 |
. . . . 5
⊢ (𝜑 → ((ℝ
↑𝑚 𝑋) ∈ V ∧ ℕ ∈
V)) |
7 | | elmapg 7757 |
. . . . 5
⊢
(((ℝ ↑𝑚 𝑋) ∈ V ∧ ℕ ∈ V) →
(𝐷 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚 ℕ) ↔
𝐷:ℕ⟶(ℝ
↑𝑚 𝑋))) |
8 | 6, 7 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐷 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ) ↔ 𝐷:ℕ⟶(ℝ
↑𝑚 𝑋))) |
9 | 2, 8 | mpbird 246 |
. . 3
⊢ (𝜑 → 𝐷 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)) |
10 | | hoidmvle.c |
. . . . 5
⊢ (𝜑 → 𝐶:ℕ⟶(ℝ
↑𝑚 𝑋)) |
11 | | elmapg 7757 |
. . . . . 6
⊢
(((ℝ ↑𝑚 𝑋) ∈ V ∧ ℕ ∈ V) →
(𝐶 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚 ℕ) ↔
𝐶:ℕ⟶(ℝ
↑𝑚 𝑋))) |
12 | 6, 11 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐶 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ) ↔ 𝐶:ℕ⟶(ℝ
↑𝑚 𝑋))) |
13 | 10, 12 | mpbird 246 |
. . . 4
⊢ (𝜑 → 𝐶 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)) |
14 | | hoidmvle.b |
. . . . . 6
⊢ (𝜑 → 𝐵:𝑋⟶ℝ) |
15 | | reex 9906 |
. . . . . . . . 9
⊢ ℝ
∈ V |
16 | 15 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → ℝ ∈
V) |
17 | | hoidmvle.x |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ Fin) |
18 | 16, 17 | jca 553 |
. . . . . . 7
⊢ (𝜑 → (ℝ ∈ V ∧
𝑋 ∈
Fin)) |
19 | | elmapg 7757 |
. . . . . . 7
⊢ ((ℝ
∈ V ∧ 𝑋 ∈
Fin) → (𝐵 ∈
(ℝ ↑𝑚 𝑋) ↔ 𝐵:𝑋⟶ℝ)) |
20 | 18, 19 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐵 ∈ (ℝ ↑𝑚
𝑋) ↔ 𝐵:𝑋⟶ℝ)) |
21 | 14, 20 | mpbird 246 |
. . . . 5
⊢ (𝜑 → 𝐵 ∈ (ℝ ↑𝑚
𝑋)) |
22 | | hoidmvle.a |
. . . . . . 7
⊢ (𝜑 → 𝐴:𝑋⟶ℝ) |
23 | | elmapg 7757 |
. . . . . . . 8
⊢ ((ℝ
∈ V ∧ 𝑋 ∈
Fin) → (𝐴 ∈
(ℝ ↑𝑚 𝑋) ↔ 𝐴:𝑋⟶ℝ)) |
24 | 18, 23 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝐴 ∈ (ℝ ↑𝑚
𝑋) ↔ 𝐴:𝑋⟶ℝ)) |
25 | 22, 24 | mpbird 246 |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ (ℝ ↑𝑚
𝑋)) |
26 | | oveq2 6557 |
. . . . . . . . . 10
⊢ (𝑥 = ∅ → (ℝ
↑𝑚 𝑥) = (ℝ ↑𝑚
∅)) |
27 | 26 | eleq2d 2673 |
. . . . . . . . 9
⊢ (𝑥 = ∅ → (𝑎 ∈ (ℝ
↑𝑚 𝑥) ↔ 𝑎 ∈ (ℝ ↑𝑚
∅))) |
28 | 26 | eleq2d 2673 |
. . . . . . . . . . 11
⊢ (𝑥 = ∅ → (𝑏 ∈ (ℝ
↑𝑚 𝑥) ↔ 𝑏 ∈ (ℝ ↑𝑚
∅))) |
29 | 26 | oveq1d 6564 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = ∅ → ((ℝ
↑𝑚 𝑥) ↑𝑚 ℕ) =
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)) |
30 | 29 | eleq2d 2673 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ∅ → (𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚 ℕ) ↔
𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚
ℕ))) |
31 | 29 | eleq2d 2673 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = ∅ → (𝑑 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚 ℕ) ↔
𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚
ℕ))) |
32 | | ixpeq1 7805 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = ∅ → X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
33 | | ixpeq1 7805 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = ∅ → X𝑘 ∈
𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
34 | 33 | iuneq2d 4483 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = ∅ → ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
35 | 32, 34 | sseq12d 3597 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = ∅ → (X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
36 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = ∅ → (𝐿‘𝑥) = (𝐿‘∅)) |
37 | 36 | oveqd 6566 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = ∅ → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘∅)𝑏)) |
38 | 36 | oveqd 6566 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = ∅ → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))) |
39 | 38 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = ∅ → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))) |
40 | 39 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = ∅ →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))) |
41 | 37, 40 | breq12d 4596 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = ∅ → ((𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) ↔ (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))) |
42 | 35, 41 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = ∅ → ((X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))) |
43 | 31, 42 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = ∅ → ((𝑑 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚 ℕ) →
(X𝑘
∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑑 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ) → (X𝑘 ∈
∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))))) |
44 | 43 | ralbidv2 2967 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ∅ → (∀𝑑 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))) |
45 | 30, 44 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑥 = ∅ → ((𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚 ℕ) →
∀𝑑 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑐 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ) → ∀𝑑 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))))) |
46 | 45 | ralbidv2 2967 |
. . . . . . . . . . 11
⊢ (𝑥 = ∅ → (∀𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))) |
47 | 28, 46 | imbi12d 333 |
. . . . . . . . . 10
⊢ (𝑥 = ∅ → ((𝑏 ∈ (ℝ
↑𝑚 𝑥) → ∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑏 ∈ (ℝ ↑𝑚
∅) → ∀𝑐
∈ ((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))))) |
48 | 47 | ralbidv2 2967 |
. . . . . . . . 9
⊢ (𝑥 = ∅ → (∀𝑏 ∈ (ℝ
↑𝑚 𝑥)∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑𝑚
∅)∀𝑐 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))) |
49 | 27, 48 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑥 = ∅ → ((𝑎 ∈ (ℝ
↑𝑚 𝑥) → ∀𝑏 ∈ (ℝ ↑𝑚
𝑥)∀𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑎 ∈ (ℝ ↑𝑚
∅) → ∀𝑏
∈ (ℝ ↑𝑚 ∅)∀𝑐 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))))) |
50 | 49 | ralbidv2 2967 |
. . . . . . 7
⊢ (𝑥 = ∅ → (∀𝑎 ∈ (ℝ
↑𝑚 𝑥)∀𝑏 ∈ (ℝ ↑𝑚
𝑥)∀𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑎 ∈ (ℝ ↑𝑚
∅)∀𝑏 ∈
(ℝ ↑𝑚 ∅)∀𝑐 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))))) |
51 | | oveq2 6557 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (ℝ ↑𝑚
𝑥) = (ℝ
↑𝑚 𝑦)) |
52 | 51 | eleq2d 2673 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑𝑚
𝑥) ↔ 𝑎 ∈ (ℝ
↑𝑚 𝑦))) |
53 | 51 | eleq2d 2673 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (𝑏 ∈ (ℝ ↑𝑚
𝑥) ↔ 𝑏 ∈ (ℝ
↑𝑚 𝑦))) |
54 | 51 | oveq1d 6564 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) = ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)) |
55 | 54 | eleq2d 2673 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) ↔ 𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ))) |
56 | 54 | eleq2d 2673 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → (𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) ↔ 𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ))) |
57 | | ixpeq1 7805 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
58 | | ixpeq1 7805 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
59 | 58 | iuneq2d 4483 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
60 | 57, 59 | sseq12d 3597 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
61 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → (𝐿‘𝑥) = (𝐿‘𝑦)) |
62 | 61 | oveqd 6566 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘𝑦)𝑏)) |
63 | 61 | oveqd 6566 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))) |
64 | 63 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) |
65 | 64 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) |
66 | 62, 65 | breq12d 4596 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → ((𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) ↔ (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))) |
67 | 60, 66 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑦 → ((X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
68 | 56, 67 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → ((𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ) → (X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))) |
69 | 68 | ralbidv2 2967 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
70 | 55, 69 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ((𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) → ∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ) → ∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))) |
71 | 70 | ralbidv2 2967 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
72 | 53, 71 | imbi12d 333 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → ((𝑏 ∈ (ℝ ↑𝑚
𝑥) → ∀𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑏 ∈ (ℝ ↑𝑚
𝑦) → ∀𝑐 ∈ ((ℝ
↑𝑚 𝑦) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑘 ∈
𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))) |
73 | 72 | ralbidv2 2967 |
. . . . . . . . 9
⊢ (𝑥 = 𝑦 → (∀𝑏 ∈ (ℝ ↑𝑚
𝑥)∀𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑𝑚
𝑦)∀𝑐 ∈ ((ℝ
↑𝑚 𝑦) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑘 ∈
𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
74 | 52, 73 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑥 = 𝑦 → ((𝑎 ∈ (ℝ ↑𝑚
𝑥) → ∀𝑏 ∈ (ℝ
↑𝑚 𝑥)∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑎 ∈ (ℝ ↑𝑚
𝑦) → ∀𝑏 ∈ (ℝ
↑𝑚 𝑦)∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))))) |
75 | 74 | ralbidv2 2967 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (∀𝑎 ∈ (ℝ ↑𝑚
𝑥)∀𝑏 ∈ (ℝ
↑𝑚 𝑥)∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑎 ∈ (ℝ ↑𝑚
𝑦)∀𝑏 ∈ (ℝ
↑𝑚 𝑦)∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
76 | | oveq2 6557 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (ℝ ↑𝑚
𝑥) = (ℝ
↑𝑚 (𝑦 ∪ {𝑧}))) |
77 | 76 | eleq2d 2673 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑎 ∈ (ℝ ↑𝑚
𝑥) ↔ 𝑎 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})))) |
78 | 76 | eleq2d 2673 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑏 ∈ (ℝ ↑𝑚
𝑥) ↔ 𝑏 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})))) |
79 | 76 | oveq1d 6564 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((ℝ
↑𝑚 𝑥) ↑𝑚 ℕ) =
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) |
80 | 79 | eleq2d 2673 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) ↔ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ))) |
81 | 79 | eleq2d 2673 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) ↔ 𝑑 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ))) |
82 | | ixpeq1 7805 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
83 | | ixpeq1 7805 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
84 | 83 | iuneq2d 4483 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
(𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
85 | 82, 84 | sseq12d 3597 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
86 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐿‘𝑥) = (𝐿‘(𝑦 ∪ {𝑧}))) |
87 | 86 | oveqd 6566 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏)) |
88 | 86 | oveqd 6566 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))) |
89 | 88 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))) |
90 | 89 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = (𝑦 ∪ {𝑧}) →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))) |
91 | 87, 90 | breq12d 4596 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) ↔ (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))) |
92 | 85, 91 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))))) |
93 | 81, 92 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑑 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ) → (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))))) |
94 | 93 | ralbidv2 2967 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)(X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))))) |
95 | 80, 94 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) → ∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ) → ∀𝑑
∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))))) |
96 | 95 | ralbidv2 2967 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))))) |
97 | 78, 96 | imbi12d 333 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑏 ∈ (ℝ ↑𝑚
𝑥) → ∀𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧})) → ∀𝑐 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))))) |
98 | 97 | ralbidv2 2967 |
. . . . . . . . 9
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑏 ∈ (ℝ ↑𝑚
𝑥)∀𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))∀𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))))) |
99 | 77, 98 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑎 ∈ (ℝ ↑𝑚
𝑥) → ∀𝑏 ∈ (ℝ
↑𝑚 𝑥)∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧})) → ∀𝑏 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧}))∀𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))))) |
100 | 99 | ralbidv2 2967 |
. . . . . . 7
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑎 ∈ (ℝ ↑𝑚
𝑥)∀𝑏 ∈ (ℝ
↑𝑚 𝑥)∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))∀𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))∀𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))))) |
101 | | oveq2 6557 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑋 → (ℝ ↑𝑚
𝑥) = (ℝ
↑𝑚 𝑋)) |
102 | 101 | eleq2d 2673 |
. . . . . . . . 9
⊢ (𝑥 = 𝑋 → (𝑎 ∈ (ℝ ↑𝑚
𝑥) ↔ 𝑎 ∈ (ℝ
↑𝑚 𝑋))) |
103 | 101 | eleq2d 2673 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑋 → (𝑏 ∈ (ℝ ↑𝑚
𝑥) ↔ 𝑏 ∈ (ℝ
↑𝑚 𝑋))) |
104 | 101 | oveq1d 6564 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑋 → ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) = ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)) |
105 | 104 | eleq2d 2673 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑋 → (𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) ↔ 𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ))) |
106 | 104 | eleq2d 2673 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑋 → (𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) ↔ 𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ))) |
107 | | ixpeq1 7805 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
108 | | ixpeq1 7805 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑋 → X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
109 | 108 | iuneq2d 4483 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑋 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
110 | 107, 109 | sseq12d 3597 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑋 → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
111 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑋 → (𝐿‘𝑥) = (𝐿‘𝑋)) |
112 | 111 | oveqd 6566 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑋 → (𝑎(𝐿‘𝑥)𝑏) = (𝑎(𝐿‘𝑋)𝑏)) |
113 | 111 | oveqd 6566 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑋 → ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))) |
114 | 113 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑋 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) |
115 | 114 | fveq2d 6107 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑋 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) |
116 | 112, 115 | breq12d 4596 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑋 → ((𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))) ↔ (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
117 | 110, 116 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑋 → ((X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
118 | 106, 117 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑋 → ((𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) → (X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ) → (X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))) |
119 | 118 | ralbidv2 2967 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑋 → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
120 | 105, 119 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑋 → ((𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ) → ∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ) → ∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))) |
121 | 120 | ralbidv2 2967 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑋 → (∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
122 | 103, 121 | imbi12d 333 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑋 → ((𝑏 ∈ (ℝ ↑𝑚
𝑥) → ∀𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑏 ∈ (ℝ ↑𝑚
𝑋) → ∀𝑐 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ)(X𝑘 ∈
𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))) |
123 | 122 | ralbidv2 2967 |
. . . . . . . . 9
⊢ (𝑥 = 𝑋 → (∀𝑏 ∈ (ℝ ↑𝑚
𝑥)∀𝑐 ∈ ((ℝ
↑𝑚 𝑥) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑥) ↑𝑚 ℕ)(X𝑘 ∈
𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑𝑚
𝑋)∀𝑐 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ)(X𝑘 ∈
𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
124 | 102, 123 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑥 = 𝑋 → ((𝑎 ∈ (ℝ ↑𝑚
𝑥) → ∀𝑏 ∈ (ℝ
↑𝑚 𝑥)∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗)))))) ↔ (𝑎 ∈ (ℝ ↑𝑚
𝑋) → ∀𝑏 ∈ (ℝ
↑𝑚 𝑋)∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))))) |
125 | 124 | ralbidv2 2967 |
. . . . . . 7
⊢ (𝑥 = 𝑋 → (∀𝑎 ∈ (ℝ ↑𝑚
𝑥)∀𝑏 ∈ (ℝ
↑𝑚 𝑥)∀𝑐 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑥)
↑𝑚 ℕ)(X𝑘 ∈ 𝑥 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑥 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑥)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑥)(𝑑‘𝑗))))) ↔ ∀𝑎 ∈ (ℝ ↑𝑚
𝑋)∀𝑏 ∈ (ℝ
↑𝑚 𝑋)∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
126 | | hoidmvle.l |
. . . . . . . . . . . . . . . 16
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚
𝑥), 𝑏 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
127 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 = 𝑒 → (𝑎‘𝑘) = (𝑒‘𝑘)) |
128 | 127 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎 = 𝑒 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑒‘𝑘)[,)(𝑏‘𝑘))) |
129 | 128 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑎 = 𝑒 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘)))) |
130 | 129 | prodeq2ad 38659 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑎 = 𝑒 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘)))) |
131 | 130 | ifeq2d 4055 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑒 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))))) |
132 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑏 = 𝑓 → (𝑏‘𝑘) = (𝑓‘𝑘)) |
133 | 132 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑏 = 𝑓 → ((𝑒‘𝑘)[,)(𝑏‘𝑘)) = ((𝑒‘𝑘)[,)(𝑓‘𝑘))) |
134 | 133 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑏 = 𝑓 → (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘)))) |
135 | 134 | prodeq2ad 38659 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 = 𝑓 → ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘)))) |
136 | 135 | ifeq2d 4055 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑓 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘))))) |
137 | 131, 136 | cbvmpt2v 6633 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 ∈ (ℝ
↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) = (𝑒 ∈ (ℝ ↑𝑚
𝑥), 𝑓 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘))))) |
138 | 137 | mpteq2i 4669 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ
↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) = (𝑥 ∈ Fin ↦ (𝑒 ∈ (ℝ ↑𝑚
𝑥), 𝑓 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘)))))) |
139 | 126, 138 | eqtri 2632 |
. . . . . . . . . . . . . . 15
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑒 ∈ (ℝ ↑𝑚
𝑥), 𝑓 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑒‘𝑘)[,)(𝑓‘𝑘)))))) |
140 | | elmapi 7765 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 ∈ (ℝ
↑𝑚 ∅) → 𝑎:∅⟶ℝ) |
141 | 140 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ∈ (ℝ
↑𝑚 ∅) ∧ 𝑏 ∈ (ℝ ↑𝑚
∅)) → 𝑎:∅⟶ℝ) |
142 | | elmapi 7765 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 ∈ (ℝ
↑𝑚 ∅) → 𝑏:∅⟶ℝ) |
143 | 142 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ∈ (ℝ
↑𝑚 ∅) ∧ 𝑏 ∈ (ℝ ↑𝑚
∅)) → 𝑏:∅⟶ℝ) |
144 | 139, 141,
143 | hoidmv0val 39473 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ∈ (ℝ
↑𝑚 ∅) ∧ 𝑏 ∈ (ℝ ↑𝑚
∅)) → (𝑎(𝐿‘∅)𝑏) = 0) |
145 | 144 | ad5ant23 1296 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑎 ∈ (ℝ
↑𝑚 ∅)) ∧ 𝑏 ∈ (ℝ ↑𝑚
∅)) ∧ 𝑐 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)) → (𝑎(𝐿‘∅)𝑏) = 0) |
146 | | nfv 1830 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗(𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚
ℕ)) |
147 | 4 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ)) →
ℕ ∈ V) |
148 | | icossicc 12131 |
. . . . . . . . . . . . . . . 16
⊢
(0[,)+∞) ⊆ (0[,]+∞) |
149 | | 0fin 8073 |
. . . . . . . . . . . . . . . . . 18
⊢ ∅
∈ Fin |
150 | 149 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ)) ∧
𝑗 ∈ ℕ) →
∅ ∈ Fin) |
151 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℝ
↑𝑚 ∅) ∈ V |
152 | 151 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
(ℝ ↑𝑚 ∅) ∈ V) |
153 | 4 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
ℕ ∈ V) |
154 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚
ℕ)) |
155 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
𝑗 ∈
ℕ) |
156 | 152, 153,
154, 155 | fvmap 38382 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
(𝑐‘𝑗) ∈ (ℝ ↑𝑚
∅)) |
157 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑐‘𝑗) ∈ (ℝ ↑𝑚
∅) → (𝑐‘𝑗):∅⟶ℝ) |
158 | 156, 157 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
(𝑐‘𝑗):∅⟶ℝ) |
159 | 158 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ)) ∧
𝑗 ∈ ℕ) →
(𝑐‘𝑗):∅⟶ℝ) |
160 | 151 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
(ℝ ↑𝑚 ∅) ∈ V) |
161 | 4 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
ℕ ∈ V) |
162 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚
ℕ)) |
163 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
𝑗 ∈
ℕ) |
164 | 160, 161,
162, 163 | fvmap 38382 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
(𝑑‘𝑗) ∈ (ℝ ↑𝑚
∅)) |
165 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑑‘𝑗) ∈ (ℝ ↑𝑚
∅) → (𝑑‘𝑗):∅⟶ℝ) |
166 | 164, 165 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑗 ∈ ℕ) →
(𝑑‘𝑗):∅⟶ℝ) |
167 | 166 | adantll 746 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ)) ∧
𝑗 ∈ ℕ) →
(𝑑‘𝑗):∅⟶ℝ) |
168 | 126, 150,
159, 167 | hoidmvcl 39472 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ)) ∧
𝑗 ∈ ℕ) →
((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)) ∈ (0[,)+∞)) |
169 | 148, 168 | sseldi 3566 |
. . . . . . . . . . . . . . 15
⊢ (((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ)) ∧
𝑗 ∈ ℕ) →
((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)) ∈ (0[,]+∞)) |
170 | 146, 147,
169 | sge0ge0mpt 39331 |
. . . . . . . . . . . . . 14
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ) ∧
𝑑 ∈ ((ℝ
↑𝑚 ∅) ↑𝑚 ℕ)) →
0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))) |
171 | 170 | adantll 746 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑎 ∈ (ℝ
↑𝑚 ∅)) ∧ 𝑏 ∈ (ℝ ↑𝑚
∅)) ∧ 𝑐 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)) → 0 ≤ (Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))) |
172 | 145, 171 | eqbrtrd 4605 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑎 ∈ (ℝ
↑𝑚 ∅)) ∧ 𝑏 ∈ (ℝ ↑𝑚
∅)) ∧ 𝑐 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗))))) |
173 | 172 | a1d 25 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑎 ∈ (ℝ
↑𝑚 ∅)) ∧ 𝑏 ∈ (ℝ ↑𝑚
∅)) ∧ 𝑐 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 ∅) ↑𝑚
ℕ)) → (X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))) |
174 | 173 | ralrimiva 2949 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑎 ∈ (ℝ ↑𝑚
∅)) ∧ 𝑏 ∈
(ℝ ↑𝑚 ∅)) ∧ 𝑐 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)) → ∀𝑑 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))) |
175 | 174 | ralrimiva 2949 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ (ℝ ↑𝑚
∅)) ∧ 𝑏 ∈
(ℝ ↑𝑚 ∅)) → ∀𝑐 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))) |
176 | 175 | ralrimiva 2949 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ (ℝ ↑𝑚
∅)) → ∀𝑏
∈ (ℝ ↑𝑚 ∅)∀𝑐 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))) |
177 | 176 | ralrimiva 2949 |
. . . . . . 7
⊢ (𝜑 → ∀𝑎 ∈ (ℝ ↑𝑚
∅)∀𝑏 ∈
(ℝ ↑𝑚 ∅)∀𝑐 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
∅) ↑𝑚 ℕ)(X𝑘 ∈ ∅ ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ ∅ (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘∅)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘∅)(𝑑‘𝑗)))))) |
178 | | simpl 472 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀𝑎 ∈ (ℝ ↑𝑚
𝑦)∀𝑏 ∈ (ℝ
↑𝑚 𝑦)∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))) → (𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦)))) |
179 | 128 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑒 → X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘))) |
180 | 179 | sseq1d 3595 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑒 → (X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
181 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑎 = 𝑒 → (𝑎(𝐿‘𝑦)𝑏) = (𝑒(𝐿‘𝑦)𝑏)) |
182 | 181 | breq1d 4593 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 = 𝑒 → ((𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) ↔ (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))) |
183 | 180, 182 | imbi12d 333 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 = 𝑒 → ((X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
184 | 183 | ralbidv 2969 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 = 𝑒 → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
185 | 184 | ralbidv 2969 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝑒 → (∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
186 | 185 | ralbidv 2969 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑒 → (∀𝑏 ∈ (ℝ ↑𝑚
𝑦)∀𝑐 ∈ ((ℝ
↑𝑚 𝑦) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑘 ∈
𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑𝑚
𝑦)∀𝑐 ∈ ((ℝ
↑𝑚 𝑦) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑘 ∈
𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
187 | 133 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑏 = 𝑓 → X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘))) |
188 | 187 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 = 𝑓 → (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
189 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑏 = 𝑓 → (𝑒(𝐿‘𝑦)𝑏) = (𝑒(𝐿‘𝑦)𝑓)) |
190 | 189 | breq1d 4593 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑏 = 𝑓 → ((𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) ↔ (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))) |
191 | 188, 190 | imbi12d 333 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑏 = 𝑓 → ((X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
192 | 191 | ralbidv 2969 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑏 = 𝑓 → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
193 | 192 | ralbidv 2969 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑓 → (∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
194 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑐 = 𝑔 → (𝑐‘𝑗) = (𝑔‘𝑗)) |
195 | 194 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑐 = 𝑔 → ((𝑐‘𝑗)‘𝑘) = ((𝑔‘𝑗)‘𝑘)) |
196 | 195 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑐 = 𝑔 → (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
197 | 196 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = 𝑔 → X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
198 | 197 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑐 = 𝑔 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈
𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
199 | 198 | iuneq2dv 4478 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑐 = 𝑔 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
200 | 199 | sseq2d 3596 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 = 𝑔 → (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
201 | 194 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = 𝑔 → ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)) = ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))) |
202 | 201 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑐 = 𝑔 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) |
203 | 202 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑐 = 𝑔 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) |
204 | 203 | breq2d 4595 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 = 𝑔 → ((𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) ↔ (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))) |
205 | 200, 204 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 = 𝑔 → ((X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
206 | 205 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = 𝑔 → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))))) |
207 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑑 = ℎ → (𝑑‘𝑗) = (ℎ‘𝑗)) |
208 | 207 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑑 = ℎ → ((𝑑‘𝑗)‘𝑘) = ((ℎ‘𝑗)‘𝑘)) |
209 | 208 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑑 = ℎ → (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))) |
210 | 209 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑑 = ℎ → X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))) |
211 | 210 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑑 = ℎ ∧ 𝑗 ∈ ℕ) → X𝑘 ∈
𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))) |
212 | 211 | iuneq2dv 4478 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑑 = ℎ → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))) |
213 | 212 | sseq2d 3596 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 = ℎ → (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))) |
214 | 207 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑑 = ℎ → ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)) = ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))) |
215 | 214 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑑 = ℎ → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))) |
216 | 215 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑑 = ℎ →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))))) |
217 | 216 | breq2d 4595 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑑 = ℎ → ((𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))) ↔ (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))))) |
218 | 213, 217 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑑 = ℎ → ((X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ (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))))))) |
219 | 218 | cbvralv 3147 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀𝑑 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑘 ∈
𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀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)))))) |
220 | 219 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = 𝑔 → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀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))))))) |
221 | 206, 220 | bitrd 267 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = 𝑔 → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀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))))))) |
222 | 221 | cbvralv 3147 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑐 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑘 ∈
𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀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)))))) |
223 | 222 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑏 = 𝑓 → (∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀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))))))) |
224 | 193, 223 | bitrd 267 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = 𝑓 → (∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀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))))))) |
225 | 224 | cbvralv 3147 |
. . . . . . . . . . . . . 14
⊢
(∀𝑏 ∈
(ℝ ↑𝑚 𝑦)∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀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)))))) |
226 | 225 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑒 → (∀𝑏 ∈ (ℝ ↑𝑚
𝑦)∀𝑐 ∈ ((ℝ
↑𝑚 𝑦) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑘 ∈
𝑦 ((𝑒‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀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))))))) |
227 | 186, 226 | bitrd 267 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑒 → (∀𝑏 ∈ (ℝ ↑𝑚
𝑦)∀𝑐 ∈ ((ℝ
↑𝑚 𝑦) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑘 ∈
𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀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))))))) |
228 | 227 | cbvralv 3147 |
. . . . . . . . . . 11
⊢
(∀𝑎 ∈
(ℝ ↑𝑚 𝑦)∀𝑏 ∈ (ℝ ↑𝑚
𝑦)∀𝑐 ∈ ((ℝ
↑𝑚 𝑦) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑘 ∈
𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) ↔ ∀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)))))) |
229 | 228 | biimpi 205 |
. . . . . . . . . 10
⊢
(∀𝑎 ∈
(ℝ ↑𝑚 𝑦)∀𝑏 ∈ (ℝ ↑𝑚
𝑦)∀𝑐 ∈ ((ℝ
↑𝑚 𝑦) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑘 ∈
𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) → ∀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)))))) |
230 | 229 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀𝑎 ∈ (ℝ ↑𝑚
𝑦)∀𝑏 ∈ (ℝ
↑𝑚 𝑦)∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))) → ∀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)))))) |
231 | | simplll 794 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝜑) |
232 | | eldifi 3694 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑧 ∈ (𝑋 ∖ 𝑦) → 𝑧 ∈ 𝑋) |
233 | 232 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦)) → 𝑧 ∈ 𝑋) |
234 | 233 | adantrl 748 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) → 𝑧 ∈ 𝑋) |
235 | 234 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝑧 ∈ 𝑋) |
236 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑎 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) |
237 | | uneq1 3722 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = (∅ ∪ {𝑧})) |
238 | | 0un 38240 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (∅
∪ {𝑧}) = {𝑧} |
239 | 238 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑦 = ∅ → (∅ ∪
{𝑧}) = {𝑧}) |
240 | 237, 239 | eqtr2d 2645 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑦 = ∅ → {𝑧} = (𝑦 ∪ {𝑧})) |
241 | 240 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = {𝑧}) |
242 | 241 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑦 = ∅ → (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) = (ℝ ↑𝑚
{𝑧})) |
243 | 242 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑎 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) = (ℝ ↑𝑚
{𝑧})) |
244 | 236, 243 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑎 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) |
245 | 244 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) |
246 | 231, 235,
245 | jca31 555 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → ((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧}))) |
247 | 246 | adantllr 751 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → ((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧}))) |
248 | 247 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑦 = ∅) → ((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧}))) |
249 | 248 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑦 =
∅) → ((𝜑 ∧
𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧}))) |
250 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑏 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) |
251 | 242 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑏 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) = (ℝ ↑𝑚
{𝑧})) |
252 | 250, 251 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑏 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑦 = ∅) → 𝑏 ∈ (ℝ ↑𝑚
{𝑧})) |
253 | 252 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑏 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑦 =
∅) → 𝑏 ∈
(ℝ ↑𝑚 {𝑧})) |
254 | 253 | adantlll 750 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑦 =
∅) → 𝑏 ∈
(ℝ ↑𝑚 {𝑧})) |
255 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧
𝑦 = ∅) → 𝑐 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) |
256 | 242 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = ∅ → ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) =
((ℝ ↑𝑚 {𝑧}) ↑𝑚
ℕ)) |
257 | 256 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧
𝑦 = ∅) →
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) =
((ℝ ↑𝑚 {𝑧}) ↑𝑚
ℕ)) |
258 | 255, 257 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑐 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧
𝑦 = ∅) → 𝑐 ∈ ((ℝ
↑𝑚 {𝑧}) ↑𝑚
ℕ)) |
259 | 258 | adantll 746 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑦 =
∅) → 𝑐 ∈
((ℝ ↑𝑚 {𝑧}) ↑𝑚
ℕ)) |
260 | 249, 254,
259 | jca31 555 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑦 =
∅) → ((((𝜑 ∧
𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ))) |
261 | 260 | adantlr 747 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ 𝑦 = ∅) →
((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ))) |
262 | 261 | adantlr 747 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ))) |
263 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑑 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧
𝑦 = ∅) → 𝑑 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) |
264 | 256 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑑 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧
𝑦 = ∅) →
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) =
((ℝ ↑𝑚 {𝑧}) ↑𝑚
ℕ)) |
265 | 263, 264 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑑 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧
𝑦 = ∅) → 𝑑 ∈ ((ℝ
↑𝑚 {𝑧}) ↑𝑚
ℕ)) |
266 | 265 | adantlr 747 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑑 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧
X𝑘
∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) |
267 | 266 | adantlll 750 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) |
268 | | simpl 472 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ∧ 𝑦 = ∅) → X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
269 | 240 | ixpeq1d 7806 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = ∅ → X𝑘 ∈
{𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
270 | 269 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ∧ 𝑦 = ∅) → X𝑘 ∈
{𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
271 | 240 | ixpeq1d 7806 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = ∅ → X𝑘 ∈
{𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) |
272 | 271 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑦 = ∅ ∧ 𝑖 ∈ ℕ) → X𝑘 ∈
{𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) |
273 | 272 | iuneq2dv 4478 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪
𝑖 ∈ ℕ X𝑘 ∈
(𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) |
274 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑖 = 𝑗 → (𝑐‘𝑖) = (𝑐‘𝑗)) |
275 | 274 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖)‘𝑘) = ((𝑐‘𝑗)‘𝑘)) |
276 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑖 = 𝑗 → (𝑑‘𝑖) = (𝑑‘𝑗)) |
277 | 276 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑖 = 𝑗 → ((𝑑‘𝑖)‘𝑘) = ((𝑑‘𝑗)‘𝑘)) |
278 | 275, 277 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑖 = 𝑗 → (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
279 | 278 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑖 = 𝑗 → X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
280 | 279 | cbviunv 4495 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ∪ 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
(𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) |
281 | 280 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
(𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
282 | 273, 281 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = ∅ → ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
(𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
283 | 282 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ∧ 𝑦 = ∅) → ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
(𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
284 | 270, 283 | sseq12d 3597 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ∧ 𝑦 = ∅) → (X𝑘 ∈
{𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) ↔ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
285 | 268, 284 | mpbird 246 |
. . . . . . . . . . . . . . . . . 18
⊢ ((X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ∧ 𝑦 = ∅) → X𝑘 ∈
{𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) |
286 | 285 | adantll 746 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → X𝑘 ∈
{𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) |
287 | 262, 267,
286 | jca31 555 |
. . . . . . . . . . . . . . . 16
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → ((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))) |
288 | | simpr 476 |
. . . . . . . . . . . . . . . 16
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → 𝑦 = ∅) |
289 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑎 = 𝑢 → (𝑎‘𝑘) = (𝑢‘𝑘)) |
290 | 289 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑎 = 𝑢 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑢‘𝑘)[,)(𝑏‘𝑘))) |
291 | 290 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑎 = 𝑢 → (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘)))) |
292 | 291 | prodeq2ad 38659 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑎 = 𝑢 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘)))) |
293 | 292 | ifeq2d 4055 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑎 = 𝑢 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))))) |
294 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 = 𝑙 → (𝑢‘𝑘) = (𝑢‘𝑙)) |
295 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 = 𝑙 → (𝑏‘𝑘) = (𝑏‘𝑙)) |
296 | 294, 295 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 = 𝑙 → ((𝑢‘𝑘)[,)(𝑏‘𝑘)) = ((𝑢‘𝑙)[,)(𝑏‘𝑙))) |
297 | 296 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 = 𝑙 → (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙)))) |
298 | 297 | cbvprodv 14485 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
∏𝑘 ∈
𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))) |
299 | 298 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑏 = 𝑣 → ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙)))) |
300 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑏 = 𝑣 → (𝑏‘𝑙) = (𝑣‘𝑙)) |
301 | 300 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑏 = 𝑣 → ((𝑢‘𝑙)[,)(𝑏‘𝑙)) = ((𝑢‘𝑙)[,)(𝑣‘𝑙))) |
302 | 301 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑏 = 𝑣 → (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))) = (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙)))) |
303 | 302 | prodeq2ad 38659 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑏 = 𝑣 → ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑏‘𝑙))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙)))) |
304 | 299, 303 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 = 𝑣 → ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙)))) |
305 | 304 | ifeq2d 4055 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑏 = 𝑣 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑢‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑥 = ∅, 0, ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙))))) |
306 | 293, 305 | cbvmpt2v 6633 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎 ∈ (ℝ
↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) = (𝑢 ∈ (ℝ ↑𝑚
𝑥), 𝑣 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙))))) |
307 | 306 | mpteq2i 4669 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ
↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) = (𝑥 ∈ Fin ↦ (𝑢 ∈ (ℝ ↑𝑚
𝑥), 𝑣 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙)))))) |
308 | 126, 307 | eqtri 2632 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑢 ∈ (ℝ ↑𝑚
𝑥), 𝑣 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑙 ∈ 𝑥 (vol‘((𝑢‘𝑙)[,)(𝑣‘𝑙)))))) |
309 | | simp-6r 807 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → 𝑧 ∈ 𝑋) |
310 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑧} = {𝑧} |
311 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑎 ∈ (ℝ
↑𝑚 {𝑧}) → 𝑎:{𝑧}⟶ℝ) |
312 | 311 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) → 𝑎:{𝑧}⟶ℝ) |
313 | 312 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → 𝑎:{𝑧}⟶ℝ) |
314 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑏 ∈ (ℝ
↑𝑚 {𝑧}) → 𝑏:{𝑧}⟶ℝ) |
315 | 314 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) → 𝑏:{𝑧}⟶ℝ) |
316 | 315 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → 𝑏:{𝑧}⟶ℝ) |
317 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 ∈ ((ℝ
↑𝑚 {𝑧}) ↑𝑚 ℕ) →
𝑐:ℕ⟶(ℝ
↑𝑚 {𝑧})) |
318 | 317 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) → 𝑐:ℕ⟶(ℝ
↑𝑚 {𝑧})) |
319 | 318 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → 𝑐:ℕ⟶(ℝ
↑𝑚 {𝑧})) |
320 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑑 ∈ ((ℝ
↑𝑚 {𝑧}) ↑𝑚 ℕ) →
𝑑:ℕ⟶(ℝ
↑𝑚 {𝑧})) |
321 | 320 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → 𝑑:ℕ⟶(ℝ
↑𝑚 {𝑧})) |
322 | | id 22 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (X𝑘 ∈
{𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) → X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) |
323 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 = 𝑙 → (𝑎‘𝑘) = (𝑎‘𝑙)) |
324 | 323, 295 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 = 𝑙 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))) |
325 | | eqcom 2617 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 = 𝑙 ↔ 𝑙 = 𝑘) |
326 | 325 | imbi1i 338 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑘 = 𝑙 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙)))) |
327 | | eqcom 2617 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙)) ↔ ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
328 | 327 | imbi2i 325 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑙 = 𝑘 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘)))) |
329 | 326, 328 | bitri 263 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑘 = 𝑙 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝑎‘𝑙)[,)(𝑏‘𝑙))) ↔ (𝑙 = 𝑘 → ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘)))) |
330 | 324, 329 | mpbi 219 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑙 = 𝑘 → ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = ((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
331 | 330 | cbvixpv 7812 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ X𝑙 ∈
{𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) |
332 | 331 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (X𝑘 ∈
{𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) → X𝑙 ∈ {𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) = X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘))) |
333 | 278 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑖 = 𝑗 → X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
334 | 333 | cbviunv 4495 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
{𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) |
335 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 = 𝑙 → ((𝑐‘𝑗)‘𝑘) = ((𝑐‘𝑗)‘𝑙)) |
336 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 = 𝑙 → ((𝑑‘𝑗)‘𝑘) = ((𝑑‘𝑗)‘𝑙)) |
337 | 335, 336 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 = 𝑙 → (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))) |
338 | 337 | cbvixpv 7812 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ X𝑘 ∈
{𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) |
339 | 338 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 ∈ ℕ → X𝑘 ∈
{𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))) |
340 | 339 | iuneq2i 4475 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑙 ∈
{𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) |
341 | 334, 340 | eqtr2i 2633 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = ∪
𝑖 ∈ ℕ X𝑘 ∈
{𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) |
342 | 341 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (X𝑘 ∈
{𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) → ∪
𝑗 ∈ ℕ X𝑙 ∈
{𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = ∪
𝑖 ∈ ℕ X𝑘 ∈
{𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) |
343 | 332, 342 | sseq12d 3597 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (X𝑘 ∈
{𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) → (X𝑙 ∈ {𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) ↔ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)))) |
344 | 322, 343 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (X𝑘 ∈
{𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘)) → X𝑙 ∈ {𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))) |
345 | 344 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → X𝑙 ∈ {𝑧} ((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑗 ∈ ℕ X𝑙 ∈ {𝑧} (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))) |
346 | 308, 309,
310, 313, 316, 319, 321, 345 | hoidmv1le 39484 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) → (𝑎(𝐿‘{𝑧})𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗))))) |
347 | 346 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) ∧ 𝑦 = ∅) → (𝑎(𝐿‘{𝑧})𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗))))) |
348 | 237, 239 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = ∅ → (𝑦 ∪ {𝑧}) = {𝑧}) |
349 | 348 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = ∅ → (𝐿‘(𝑦 ∪ {𝑧})) = (𝐿‘{𝑧})) |
350 | 349 | oveqd 6566 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = ∅ → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) = (𝑎(𝐿‘{𝑧})𝑏)) |
351 | 350 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) ∧ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) = (𝑎(𝐿‘{𝑧})𝑏)) |
352 | 349 | oveqd 6566 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑦 = ∅ → ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)) = ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗))) |
353 | 352 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 = ∅ → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗)))) |
354 | 353 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = ∅ →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗))))) |
355 | 354 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) ∧ 𝑦 = ∅) →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗))))) |
356 | 351, 355 | breq12d 4596 |
. . . . . . . . . . . . . . . . 17
⊢
((((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) ∧ 𝑦 = ∅) → ((𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))) ↔ (𝑎(𝐿‘{𝑧})𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘{𝑧})(𝑑‘𝑗)))))) |
357 | 347, 356 | mpbird 246 |
. . . . . . . . . . . . . . . 16
⊢
((((((((𝜑 ∧ 𝑧 ∈ 𝑋) ∧ 𝑎 ∈ (ℝ ↑𝑚
{𝑧})) ∧ 𝑏 ∈ (ℝ
↑𝑚 {𝑧})) ∧ 𝑐 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ 𝑑 ∈ ((ℝ ↑𝑚
{𝑧})
↑𝑚 ℕ)) ∧ X𝑘 ∈ {𝑧} ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑖 ∈ ℕ X𝑘 ∈ {𝑧} (((𝑐‘𝑖)‘𝑘)[,)((𝑑‘𝑖)‘𝑘))) ∧ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))) |
358 | 287, 288,
357 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))) |
359 | 17 | ad8antr 772 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑋 ∈ Fin) |
360 | | simplrl 796 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) → 𝑦 ⊆ 𝑋) |
361 | 360 | ad5ant12 1292 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) → 𝑦 ⊆
𝑋) |
362 | 361 | ad5ant12 1292 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑦 ⊆ 𝑋) |
363 | | simplrr 797 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) → 𝑧 ∈ (𝑋 ∖ 𝑦)) |
364 | 363 | ad5ant12 1292 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) → 𝑧 ∈
(𝑋 ∖ 𝑦)) |
365 | 364 | ad5ant12 1292 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑧 ∈ (𝑋 ∖ 𝑦)) |
366 | | eqid 2610 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∪ {𝑧}) = (𝑦 ∪ {𝑧}) |
367 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑎 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ) |
368 | 367 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ) |
369 | 368 | ad4ant23 1289 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ) |
370 | 369 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑎:(𝑦 ∪ {𝑧})⟶ℝ) |
371 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑏 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ) |
372 | 371 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑎 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ) |
373 | 372 | ad4ant23 1289 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ) |
374 | 373 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑏:(𝑦 ∪ {𝑧})⟶ℝ) |
375 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)
→ 𝑐:ℕ⟶(ℝ
↑𝑚 (𝑦 ∪ {𝑧}))) |
376 | 375 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) → 𝑐:ℕ⟶(ℝ
↑𝑚 (𝑦 ∪ {𝑧}))) |
377 | 376 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑐:ℕ⟶(ℝ
↑𝑚 (𝑦 ∪ {𝑧}))) |
378 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑑 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)
→ 𝑑:ℕ⟶(ℝ
↑𝑚 (𝑦 ∪ {𝑧}))) |
379 | 378 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑑 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ) ∧
X𝑘
∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑑:ℕ⟶(ℝ
↑𝑚 (𝑦 ∪ {𝑧}))) |
380 | 379 | adantlll 750 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑑:ℕ⟶(ℝ
↑𝑚 (𝑦 ∪ {𝑧}))) |
381 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑘 = 𝑙 → (𝑒‘𝑘) = (𝑒‘𝑙)) |
382 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑘 = 𝑙 → (𝑓‘𝑘) = (𝑓‘𝑙)) |
383 | 381, 382 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 = 𝑙 → ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = ((𝑒‘𝑙)[,)(𝑓‘𝑙))) |
384 | 383 | cbvixpv 7812 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ X𝑘 ∈
𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) |
385 | 384 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (ℎ = 𝑜 → X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙))) |
386 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑗 = 𝑖 → (𝑔‘𝑗) = (𝑔‘𝑖)) |
387 | 386 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 = 𝑖 → ((𝑔‘𝑗)‘𝑘) = ((𝑔‘𝑖)‘𝑘)) |
388 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑗 = 𝑖 → (ℎ‘𝑗) = (ℎ‘𝑖)) |
389 | 388 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑗 = 𝑖 → ((ℎ‘𝑗)‘𝑘) = ((ℎ‘𝑖)‘𝑘)) |
390 | 387, 389 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 = 𝑖 → (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘))) |
391 | 390 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 = 𝑖 → X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘))) |
392 | 391 | cbviunv 4495 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪
𝑖 ∈ ℕ X𝑘 ∈
𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) |
393 | 392 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (ℎ = 𝑜 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪
𝑖 ∈ ℕ X𝑘 ∈
𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘))) |
394 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑘 = 𝑙 → ((𝑔‘𝑖)‘𝑘) = ((𝑔‘𝑖)‘𝑙)) |
395 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑘 = 𝑙 → ((ℎ‘𝑖)‘𝑘) = ((ℎ‘𝑖)‘𝑙)) |
396 | 394, 395 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑘 = 𝑙 → (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙))) |
397 | 396 | cbvixpv 7812 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ X𝑘 ∈
𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)) |
398 | 397 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (ℎ = 𝑜 → X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙))) |
399 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (ℎ = 𝑜 → (ℎ‘𝑖) = (𝑜‘𝑖)) |
400 | 399 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (ℎ = 𝑜 → ((ℎ‘𝑖)‘𝑙) = ((𝑜‘𝑖)‘𝑙)) |
401 | 400 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (ℎ = 𝑜 → (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)) = (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙))) |
402 | 401 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (ℎ = 𝑜 → X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((ℎ‘𝑖)‘𝑙)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙))) |
403 | 398, 402 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (ℎ = 𝑜 → X𝑘 ∈ 𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙))) |
404 | 403 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((ℎ = 𝑜 ∧ 𝑖 ∈ ℕ) → X𝑘 ∈
𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙))) |
405 | 404 | iuneq2dv 4478 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (ℎ = 𝑜 → ∪
𝑖 ∈ ℕ X𝑘 ∈
𝑦 (((𝑔‘𝑖)‘𝑘)[,)((ℎ‘𝑖)‘𝑘)) = ∪
𝑖 ∈ ℕ X𝑙 ∈
𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙))) |
406 | 393, 405 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (ℎ = 𝑜 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪
𝑖 ∈ ℕ X𝑙 ∈
𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙))) |
407 | 385, 406 | sseq12d 3597 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (ℎ = 𝑜 → (X𝑘 ∈ 𝑦 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)))) |
408 | 386, 388 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑗 = 𝑖 → ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)) = ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖))) |
409 | 408 | cbvmptv 4678 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖))) |
410 | 409 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (ℎ = 𝑜 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖)))) |
411 | 399 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (ℎ = 𝑜 → ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖)) = ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))) |
412 | 411 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (ℎ = 𝑜 → (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(ℎ‘𝑖))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))) |
413 | 410, 412 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (ℎ = 𝑜 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗))) = (𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))) |
414 | 413 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (ℎ = 𝑜 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))) =
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))) |
415 | 414 | breq2d 4595 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (ℎ = 𝑜 → ((𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑦)(ℎ‘𝑗)))) ↔ (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))) |
416 | 407, 415 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ℎ = 𝑜 → ((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))))) ↔ (X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖))))))) |
417 | 416 | cbvralv 3147 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∀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))))) ↔ ∀𝑜 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))) |
418 | 417 | ralbii 2963 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∀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))))) ↔ ∀𝑔 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑜 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))) |
419 | 418 | ralbii 2963 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(∀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))))) ↔ ∀𝑓 ∈ (ℝ ↑𝑚
𝑦)∀𝑔 ∈ ((ℝ
↑𝑚 𝑦) ↑𝑚
ℕ)∀𝑜 ∈
((ℝ ↑𝑚 𝑦) ↑𝑚 ℕ)(X𝑙 ∈
𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))) |
420 | 419 | ralbii 2963 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∀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))))) ↔ ∀𝑒 ∈ (ℝ ↑𝑚
𝑦)∀𝑓 ∈ (ℝ
↑𝑚 𝑦)∀𝑔 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑜 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))) |
421 | 420 | biimpi 205 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀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))))) → ∀𝑒 ∈ (ℝ ↑𝑚
𝑦)∀𝑓 ∈ (ℝ
↑𝑚 𝑦)∀𝑔 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑜 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))) |
422 | 421 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) → ∀𝑒 ∈ (ℝ ↑𝑚
𝑦)∀𝑓 ∈ (ℝ
↑𝑚 𝑦)∀𝑔 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑜 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))) |
423 | 422 | ad6antr 768 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → ∀𝑒 ∈ (ℝ ↑𝑚
𝑦)∀𝑓 ∈ (ℝ
↑𝑚 𝑦)∀𝑔 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑜 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑙 ∈ 𝑦 ((𝑒‘𝑙)[,)(𝑓‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ 𝑦 (((𝑔‘𝑖)‘𝑙)[,)((𝑜‘𝑖)‘𝑙)) → (𝑒(𝐿‘𝑦)𝑓) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑔‘𝑖)(𝐿‘𝑦)(𝑜‘𝑖)))))) |
424 | 324 | cbvixpv 7812 |
. . . . . . . . . . . . . . . . . . . 20
⊢ X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑙)[,)(𝑏‘𝑙)) |
425 | 337 | cbvixpv 7812 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ X𝑘 ∈
(𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) |
426 | 425 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑖 → X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙))) |
427 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = 𝑖 → (𝑐‘𝑗) = (𝑐‘𝑖)) |
428 | 427 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = 𝑖 → ((𝑐‘𝑗)‘𝑙) = ((𝑐‘𝑖)‘𝑙)) |
429 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑗 = 𝑖 → (𝑑‘𝑗) = (𝑑‘𝑖)) |
430 | 429 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 = 𝑖 → ((𝑑‘𝑗)‘𝑙) = ((𝑑‘𝑖)‘𝑙)) |
431 | 428, 430 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 = 𝑖 → (((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = (((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙))) |
432 | 431 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑖 → X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑙)[,)((𝑑‘𝑗)‘𝑙)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙))) |
433 | 426, 432 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑖 → X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙))) |
434 | 433 | cbviunv 4495 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪
𝑖 ∈ ℕ X𝑙 ∈
(𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙)) |
435 | 424, 434 | sseq12i 3594 |
. . . . . . . . . . . . . . . . . . 19
⊢ (X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙))) |
436 | 435 | biimpi 205 |
. . . . . . . . . . . . . . . . . 18
⊢ (X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → X𝑙 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙))) |
437 | 436 | ad2antlr 759 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → X𝑙 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑙)[,)(𝑏‘𝑙)) ⊆ ∪ 𝑖 ∈ ℕ X𝑙 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑖)‘𝑙)[,)((𝑑‘𝑖)‘𝑙))) |
438 | | neqne 2790 |
. . . . . . . . . . . . . . . . . 18
⊢ (¬
𝑦 = ∅ → 𝑦 ≠ ∅) |
439 | 438 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → 𝑦 ≠ ∅) |
440 | 308, 359,
362, 365, 366, 370, 374, 377, 380, 423, 437, 439 | hoidmvlelem5 39489 |
. . . . . . . . . . . . . . . 16
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖))))) |
441 | 274, 276 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝑗 → ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖)) = ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))) |
442 | 441 | cbvmptv 4678 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 ∈ ℕ ↦ ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖))) = (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))) |
443 | 442 | fveq2i 6106 |
. . . . . . . . . . . . . . . . 17
⊢
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))) |
444 | 443 | breq2i 4591 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝑐‘𝑖)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑖)))) ↔ (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))) |
445 | 440, 444 | sylib 207 |
. . . . . . . . . . . . . . 15
⊢
(((((((((𝜑 ∧
(𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) ∧ ¬ 𝑦 = ∅) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))) |
446 | 358, 445 | pm2.61dan 828 |
. . . . . . . . . . . . . 14
⊢
((((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
∧ X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))) |
447 | 446 | ex 449 |
. . . . . . . . . . . . 13
⊢
(((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) ∧ 𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ))
→ (X𝑘 ∈ (𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))) |
448 | 447 | ralrimiva 2949 |
. . . . . . . . . . . 12
⊢
((((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)) → ∀𝑑
∈ ((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))) |
449 | 448 | ralrimiva 2949 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) ∧ 𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) → ∀𝑐 ∈ ((ℝ
↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))) |
450 | 449 | ralrimiva 2949 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) ∧ 𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))) → ∀𝑏 ∈ (ℝ
↑𝑚 (𝑦 ∪ {𝑧}))∀𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))) |
451 | 450 | ralrimiva 2949 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀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)))))) → ∀𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))∀𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))∀𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))) |
452 | 178, 230,
451 | syl2anc 691 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) ∧ ∀𝑎 ∈ (ℝ ↑𝑚
𝑦)∀𝑏 ∈ (ℝ
↑𝑚 𝑦)∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗)))))) → ∀𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))∀𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))∀𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗)))))) |
453 | 452 | ex 449 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ⊆ 𝑋 ∧ 𝑧 ∈ (𝑋 ∖ 𝑦))) → (∀𝑎 ∈ (ℝ ↑𝑚
𝑦)∀𝑏 ∈ (ℝ
↑𝑚 𝑦)∀𝑐 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑦)
↑𝑚 ℕ)(X𝑘 ∈ 𝑦 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑦 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑦)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑦)(𝑑‘𝑗))))) → ∀𝑎 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))∀𝑏 ∈ (ℝ ↑𝑚
(𝑦 ∪ {𝑧}))∀𝑐 ∈ ((ℝ ↑𝑚
(𝑦 ∪ {𝑧})) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 (𝑦 ∪ {𝑧})) ↑𝑚 ℕ)(X𝑘 ∈
(𝑦 ∪ {𝑧})((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ (𝑦 ∪ {𝑧})(((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘(𝑦 ∪ {𝑧}))𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘(𝑦 ∪ {𝑧}))(𝑑‘𝑗))))))) |
454 | 50, 75, 100, 125, 177, 453, 17 | findcard2d 8087 |
. . . . . 6
⊢ (𝜑 → ∀𝑎 ∈ (ℝ ↑𝑚
𝑋)∀𝑏 ∈ (ℝ
↑𝑚 𝑋)∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
455 | | fveq1 6102 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = 𝐴 → (𝑎‘𝑘) = (𝐴‘𝑘)) |
456 | 455 | oveq1d 6564 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝐴 → ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = ((𝐴‘𝑘)[,)(𝑏‘𝑘))) |
457 | 456 | ixpeq2dv 7810 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝐴 → X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘))) |
458 | 457 | sseq1d 3595 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝐴 → (X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
459 | | oveq1 6556 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝐴 → (𝑎(𝐿‘𝑋)𝑏) = (𝐴(𝐿‘𝑋)𝑏)) |
460 | 459 | breq1d 4593 |
. . . . . . . . . . 11
⊢ (𝑎 = 𝐴 → ((𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) ↔ (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
461 | 458, 460 | imbi12d 333 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐴 → ((X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
462 | 461 | ralbidv 2969 |
. . . . . . . . 9
⊢ (𝑎 = 𝐴 → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
463 | 462 | ralbidv 2969 |
. . . . . . . 8
⊢ (𝑎 = 𝐴 → (∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
464 | 463 | ralbidv 2969 |
. . . . . . 7
⊢ (𝑎 = 𝐴 → (∀𝑏 ∈ (ℝ ↑𝑚
𝑋)∀𝑐 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ)(X𝑘 ∈
𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ ∀𝑏 ∈ (ℝ ↑𝑚
𝑋)∀𝑐 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ)(X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
465 | 464 | rspcva 3280 |
. . . . . 6
⊢ ((𝐴 ∈ (ℝ
↑𝑚 𝑋) ∧ ∀𝑎 ∈ (ℝ ↑𝑚
𝑋)∀𝑏 ∈ (ℝ
↑𝑚 𝑋)∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝑎‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝑎(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) → ∀𝑏 ∈ (ℝ ↑𝑚
𝑋)∀𝑐 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ)(X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
466 | 25, 454, 465 | syl2anc 691 |
. . . . 5
⊢ (𝜑 → ∀𝑏 ∈ (ℝ ↑𝑚
𝑋)∀𝑐 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ)(X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
467 | | fveq1 6102 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝐵 → (𝑏‘𝑘) = (𝐵‘𝑘)) |
468 | 467 | oveq2d 6565 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝐵 → ((𝐴‘𝑘)[,)(𝑏‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
469 | 468 | ixpeq2dv 7810 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐵 → X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) = X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
470 | 469 | sseq1d 3595 |
. . . . . . . . 9
⊢ (𝑏 = 𝐵 → (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
471 | | oveq2 6557 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐵 → (𝐴(𝐿‘𝑋)𝑏) = (𝐴(𝐿‘𝑋)𝐵)) |
472 | 471 | breq1d 4593 |
. . . . . . . . 9
⊢ (𝑏 = 𝐵 → ((𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) ↔ (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
473 | 470, 472 | imbi12d 333 |
. . . . . . . 8
⊢ (𝑏 = 𝐵 → ((X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
474 | 473 | ralbidv 2969 |
. . . . . . 7
⊢ (𝑏 = 𝐵 → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
475 | 474 | ralbidv 2969 |
. . . . . 6
⊢ (𝑏 = 𝐵 → (∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ ∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
476 | 475 | rspcva 3280 |
. . . . 5
⊢ ((𝐵 ∈ (ℝ
↑𝑚 𝑋) ∧ ∀𝑏 ∈ (ℝ ↑𝑚
𝑋)∀𝑐 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ)(X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝑏‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝑏) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) → ∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
477 | 21, 466, 476 | syl2anc 691 |
. . . 4
⊢ (𝜑 → ∀𝑐 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
478 | | fveq1 6102 |
. . . . . . . . . . . . 13
⊢ (𝑐 = 𝐶 → (𝑐‘𝑗) = (𝐶‘𝑗)) |
479 | 478 | fveq1d 6105 |
. . . . . . . . . . . 12
⊢ (𝑐 = 𝐶 → ((𝑐‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑘)) |
480 | 479 | oveq1d 6564 |
. . . . . . . . . . 11
⊢ (𝑐 = 𝐶 → (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
481 | 480 | ixpeq2dv 7810 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐶 → X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
482 | 481 | adantr 480 |
. . . . . . . . 9
⊢ ((𝑐 = 𝐶 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
483 | 482 | iuneq2dv 4478 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘))) |
484 | 483 | sseq2d 3596 |
. . . . . . 7
⊢ (𝑐 = 𝐶 → (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)))) |
485 | 478 | oveq1d 6564 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐶 → ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)) = ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))) |
486 | 485 | mpteq2dv 4673 |
. . . . . . . . 9
⊢ (𝑐 = 𝐶 → (𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) |
487 | 486 | fveq2d 6107 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) |
488 | 487 | breq2d 4595 |
. . . . . . 7
⊢ (𝑐 = 𝐶 → ((𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) ↔ (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
489 | 484, 488 | imbi12d 333 |
. . . . . 6
⊢ (𝑐 = 𝐶 → ((X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
490 | 489 | ralbidv 2969 |
. . . . 5
⊢ (𝑐 = 𝐶 → (∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ ∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))))) |
491 | 490 | rspcva 3280 |
. . . 4
⊢ ((𝐶 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚 ℕ) ∧
∀𝑐 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚
ℕ)∀𝑑 ∈
((ℝ ↑𝑚 𝑋) ↑𝑚 ℕ)(X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝑐‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑐‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) → ∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
492 | 13, 477, 491 | syl2anc 691 |
. . 3
⊢ (𝜑 → ∀𝑑 ∈ ((ℝ ↑𝑚
𝑋)
↑𝑚 ℕ)(X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) |
493 | | fveq1 6102 |
. . . . . . . . . . 11
⊢ (𝑑 = 𝐷 → (𝑑‘𝑗) = (𝐷‘𝑗)) |
494 | 493 | fveq1d 6105 |
. . . . . . . . . 10
⊢ (𝑑 = 𝐷 → ((𝑑‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑘)) |
495 | 494 | oveq2d 6565 |
. . . . . . . . 9
⊢ (𝑑 = 𝐷 → (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
496 | 495 | ixpeq2dv 7810 |
. . . . . . . 8
⊢ (𝑑 = 𝐷 → X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
497 | 496 | adantr 480 |
. . . . . . 7
⊢ ((𝑑 = 𝐷 ∧ 𝑗 ∈ ℕ) → X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
498 | 497 | iuneq2dv 4478 |
. . . . . 6
⊢ (𝑑 = 𝐷 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
499 | 498 | sseq2d 3596 |
. . . . 5
⊢ (𝑑 = 𝐷 → (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
500 | 493 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝑑 = 𝐷 → ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)) = ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))) |
501 | 500 | mpteq2dv 4673 |
. . . . . . 7
⊢ (𝑑 = 𝐷 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) |
502 | 501 | fveq2d 6107 |
. . . . . 6
⊢ (𝑑 = 𝐷 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))) |
503 | 502 | breq2d 4595 |
. . . . 5
⊢ (𝑑 = 𝐷 → ((𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))) ↔ (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))) |
504 | 499, 503 | imbi12d 333 |
. . . 4
⊢ (𝑑 = 𝐷 → ((X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗))))) ↔ (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))))) |
505 | 504 | rspcva 3280 |
. . 3
⊢ ((𝐷 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚 ℕ) ∧
∀𝑑 ∈ ((ℝ
↑𝑚 𝑋) ↑𝑚 ℕ)(X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝑑‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝑑‘𝑗)))))) → (X𝑘 ∈ 𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))) |
506 | 9, 492, 505 | syl2anc 691 |
. 2
⊢ (𝜑 → (X𝑘 ∈
𝑋 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))))) |
507 | 1, 506 | mpd 15 |
1
⊢ (𝜑 → (𝐴(𝐿‘𝑋)𝐵) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))) |