Step | Hyp | Ref
| Expression |
1 | | eqid 2610 |
. 2
⊢
(ℤ≥‘𝐴) = (ℤ≥‘𝐴) |
2 | | caubl.2 |
. . . 4
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
3 | 2 | 3ad2ant1 1075 |
. . 3
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋)) |
4 | | caublcls.6 |
. . . 4
⊢ 𝐽 = (MetOpen‘𝐷) |
5 | 4 | mopntopon 22054 |
. . 3
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
6 | 3, 5 | syl 17 |
. 2
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝑋)) |
7 | | simp3 1056 |
. . 3
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℕ) |
8 | 7 | nnzd 11357 |
. 2
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝐴 ∈ ℤ) |
9 | | simp2 1055 |
. 2
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (1st
∘ 𝐹)(⇝𝑡‘𝐽)𝑃) |
10 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑟 = 𝐴 → (𝐹‘𝑟) = (𝐹‘𝐴)) |
11 | 10 | fveq2d 6107 |
. . . . . . . 8
⊢ (𝑟 = 𝐴 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝐴))) |
12 | 11 | sseq1d 3595 |
. . . . . . 7
⊢ (𝑟 = 𝐴 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) |
13 | 12 | imbi2d 329 |
. . . . . 6
⊢ (𝑟 = 𝐴 → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) ↔ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))) |
14 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑟 = 𝑘 → (𝐹‘𝑟) = (𝐹‘𝑘)) |
15 | 14 | fveq2d 6107 |
. . . . . . . 8
⊢ (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘𝑘))) |
16 | 15 | sseq1d 3595 |
. . . . . . 7
⊢ (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) |
17 | 16 | imbi2d 329 |
. . . . . 6
⊢ (𝑟 = 𝑘 → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) ↔ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))) |
18 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑟 = (𝑘 + 1) → (𝐹‘𝑟) = (𝐹‘(𝑘 + 1))) |
19 | 18 | fveq2d 6107 |
. . . . . . . 8
⊢ (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹‘𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1)))) |
20 | 19 | sseq1d 3595 |
. . . . . . 7
⊢ (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) |
21 | 20 | imbi2d 329 |
. . . . . 6
⊢ (𝑟 = (𝑘 + 1) → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑟)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) ↔ ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))) |
22 | | ssid 3587 |
. . . . . . 7
⊢
((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) |
23 | 22 | 2a1i 12 |
. . . . . 6
⊢ (𝐴 ∈ ℤ → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) |
24 | | caubl.4 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛))) |
25 | | eluznn 11634 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ℕ ∧ 𝑘 ∈
(ℤ≥‘𝐴)) → 𝑘 ∈ ℕ) |
26 | | oveq1 6556 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → (𝑛 + 1) = (𝑘 + 1)) |
27 | 26 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1))) |
28 | 27 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1)))) |
29 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘)) |
30 | 29 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘𝑛)) = ((ball‘𝐷)‘(𝐹‘𝑘))) |
31 | 28, 30 | sseq12d 3597 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)))) |
32 | 31 | rspccva 3281 |
. . . . . . . . . . 11
⊢
((∀𝑛 ∈
ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))) |
33 | 24, 25, 32 | syl2an 493 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ≥‘𝐴))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))) |
34 | 33 | anassrs 678 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘))) |
35 | | sstr2 3575 |
. . . . . . . . 9
⊢
(((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝑘)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) |
36 | 34, 35 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) |
37 | 36 | expcom 450 |
. . . . . . 7
⊢ (𝑘 ∈
(ℤ≥‘𝐴) → ((𝜑 ∧ 𝐴 ∈ ℕ) → (((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))) |
38 | 37 | a2d 29 |
. . . . . 6
⊢ (𝑘 ∈
(ℤ≥‘𝐴) → (((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))))) |
39 | 13, 17, 21, 17, 23, 38 | uzind4 11622 |
. . . . 5
⊢ (𝑘 ∈
(ℤ≥‘𝐴) → ((𝜑 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴)))) |
40 | 39 | impcom 445 |
. . . 4
⊢ (((𝜑 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) |
41 | 40 | 3adantl2 1211 |
. . 3
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) ⊆ ((ball‘𝐷)‘(𝐹‘𝐴))) |
42 | 3 | adantr 480 |
. . . . 5
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝐷 ∈ (∞Met‘𝑋)) |
43 | | simpl1 1057 |
. . . . . . . 8
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝜑) |
44 | | caubl.3 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:ℕ⟶(𝑋 ×
ℝ+)) |
45 | 43, 44 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝐹:ℕ⟶(𝑋 ×
ℝ+)) |
46 | 25 | 3ad2antl3 1218 |
. . . . . . 7
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → 𝑘 ∈ ℕ) |
47 | 45, 46 | ffvelrnd 6268 |
. . . . . 6
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝑘) ∈ (𝑋 ×
ℝ+)) |
48 | | xp1st 7089 |
. . . . . 6
⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) →
(1st ‘(𝐹‘𝑘)) ∈ 𝑋) |
49 | 47, 48 | syl 17 |
. . . . 5
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (1st
‘(𝐹‘𝑘)) ∈ 𝑋) |
50 | | xp2nd 7090 |
. . . . . 6
⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) →
(2nd ‘(𝐹‘𝑘)) ∈
ℝ+) |
51 | 47, 50 | syl 17 |
. . . . 5
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (2nd
‘(𝐹‘𝑘)) ∈
ℝ+) |
52 | | blcntr 22028 |
. . . . 5
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st
‘(𝐹‘𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹‘𝑘)) ∈ ℝ+) →
(1st ‘(𝐹‘𝑘)) ∈ ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘)))) |
53 | 42, 49, 51, 52 | syl3anc 1318 |
. . . 4
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (1st
‘(𝐹‘𝑘)) ∈ ((1st
‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘)))) |
54 | | fvco3 6185 |
. . . . 5
⊢ ((𝐹:ℕ⟶(𝑋 × ℝ+)
∧ 𝑘 ∈ ℕ)
→ ((1st ∘ 𝐹)‘𝑘) = (1st ‘(𝐹‘𝑘))) |
55 | 45, 46, 54 | syl2anc 691 |
. . . 4
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st
∘ 𝐹)‘𝑘) = (1st
‘(𝐹‘𝑘))) |
56 | | 1st2nd2 7096 |
. . . . . . 7
⊢ ((𝐹‘𝑘) ∈ (𝑋 × ℝ+) → (𝐹‘𝑘) = 〈(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))〉) |
57 | 47, 56 | syl 17 |
. . . . . 6
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → (𝐹‘𝑘) = 〈(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))〉) |
58 | 57 | fveq2d 6107 |
. . . . 5
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) = ((ball‘𝐷)‘〈(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))〉)) |
59 | | df-ov 6552 |
. . . . 5
⊢
((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘))) = ((ball‘𝐷)‘〈(1st ‘(𝐹‘𝑘)), (2nd ‘(𝐹‘𝑘))〉) |
60 | 58, 59 | syl6eqr 2662 |
. . . 4
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((ball‘𝐷)‘(𝐹‘𝑘)) = ((1st ‘(𝐹‘𝑘))(ball‘𝐷)(2nd ‘(𝐹‘𝑘)))) |
61 | 53, 55, 60 | 3eltr4d 2703 |
. . 3
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st
∘ 𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹‘𝑘))) |
62 | 41, 61 | sseldd 3569 |
. 2
⊢ (((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ≥‘𝐴)) → ((1st
∘ 𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹‘𝐴))) |
63 | 44 | ffvelrnda 6267 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) ∈ (𝑋 ×
ℝ+)) |
64 | 63 | 3adant2 1073 |
. . . . . 6
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) ∈ (𝑋 ×
ℝ+)) |
65 | | 1st2nd2 7096 |
. . . . . 6
⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) → (𝐹‘𝐴) = 〈(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))〉) |
66 | 64, 65 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (𝐹‘𝐴) = 〈(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))〉) |
67 | 66 | fveq2d 6107 |
. . . 4
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) = ((ball‘𝐷)‘〈(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))〉)) |
68 | | df-ov 6552 |
. . . 4
⊢
((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) = ((ball‘𝐷)‘〈(1st ‘(𝐹‘𝐴)), (2nd ‘(𝐹‘𝐴))〉) |
69 | 67, 68 | syl6eqr 2662 |
. . 3
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) = ((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴)))) |
70 | | xp1st 7089 |
. . . . 5
⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) →
(1st ‘(𝐹‘𝐴)) ∈ 𝑋) |
71 | 64, 70 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (1st
‘(𝐹‘𝐴)) ∈ 𝑋) |
72 | | xp2nd 7090 |
. . . . . 6
⊢ ((𝐹‘𝐴) ∈ (𝑋 × ℝ+) →
(2nd ‘(𝐹‘𝐴)) ∈
ℝ+) |
73 | 64, 72 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (2nd
‘(𝐹‘𝐴)) ∈
ℝ+) |
74 | 73 | rpxrd 11749 |
. . . 4
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → (2nd
‘(𝐹‘𝐴)) ∈
ℝ*) |
75 | | blssm 22033 |
. . . 4
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st
‘(𝐹‘𝐴)) ∈ 𝑋 ∧ (2nd ‘(𝐹‘𝐴)) ∈ ℝ*) →
((1st ‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) ⊆ 𝑋) |
76 | 3, 71, 74, 75 | syl3anc 1318 |
. . 3
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((1st
‘(𝐹‘𝐴))(ball‘𝐷)(2nd ‘(𝐹‘𝐴))) ⊆ 𝑋) |
77 | 69, 76 | eqsstrd 3602 |
. 2
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘𝐴)) ⊆ 𝑋) |
78 | 1, 6, 8, 9, 62, 77 | lmcls 20916 |
1
⊢ ((𝜑 ∧ (1st ∘
𝐹)(⇝𝑡‘𝐽)𝑃 ∧ 𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹‘𝐴)))) |