Step | Hyp | Ref
| Expression |
1 | | prdsxms.i |
. . . 4
⊢ (𝜑 → 𝐼 ∈ Fin) |
2 | | topnfn 17145 |
. . . . 5
⊢ TopOpen
Fn V |
3 | | prdsxms.r |
. . . . . . 7
⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp) |
4 | 3 | ffnd 6610 |
. . . . . 6
⊢ (𝜑 → 𝑅 Fn 𝐼) |
5 | | dffn2 6611 |
. . . . . 6
⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V) |
6 | 4, 5 | sylib 217 |
. . . . 5
⊢ (𝜑 → 𝑅:𝐼⟶V) |
7 | | fnfco 6648 |
. . . . 5
⊢ ((TopOpen
Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen
∘ 𝑅) Fn 𝐼) |
8 | 2, 6, 7 | sylancr 587 |
. . . 4
⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼) |
9 | | prdsxms.c |
. . . . 5
⊢ 𝐶 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))} |
10 | 9 | ptval 22730 |
. . . 4
⊢ ((𝐼 ∈ Fin ∧ (TopOpen
∘ 𝑅) Fn 𝐼) →
(∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶)) |
11 | 1, 8, 10 | syl2anc 584 |
. . 3
⊢ (𝜑 →
(∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶)) |
12 | | eldifsn 4721 |
. . . . . . . 8
⊢ (𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) ↔
(𝑥 ∈ ran
(ball‘𝐷) ∧ 𝑥 ≠ ∅)) |
13 | | prdsxms.y |
. . . . . . . . . . . 12
⊢ 𝑌 = (𝑆Xs𝑅) |
14 | | prdsxms.s |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑆 ∈ 𝑊) |
15 | | prdsxms.d |
. . . . . . . . . . . 12
⊢ 𝐷 = (dist‘𝑌) |
16 | | prdsxms.b |
. . . . . . . . . . . 12
⊢ 𝐵 = (Base‘𝑌) |
17 | 13, 14, 1, 15, 16, 3 | prdsxmslem1 23693 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
18 | | blrn 23571 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ (∞Met‘𝐵) → (𝑥 ∈ ran (ball‘𝐷) ↔ ∃𝑝 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟))) |
19 | 17, 18 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑥 ∈ ran (ball‘𝐷) ↔ ∃𝑝 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟))) |
20 | 17 | adantr 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝐷 ∈ (∞Met‘𝐵)) |
21 | | simprl 768 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝑝 ∈ 𝐵) |
22 | | simprr 770 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝑟 ∈
ℝ*) |
23 | | xbln0 23576 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟)) |
24 | 20, 21, 22, 23 | syl3anc 1370 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟)) |
25 | 1 | 3ad2ant1 1132 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝐼 ∈ Fin) |
26 | 25 | mptexd 7109 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∈ V) |
27 | | ovex 7317 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) ∈ V |
28 | 27 | rgenw 3077 |
. . . . . . . . . . . . . . . . . 18
⊢
∀𝑛 ∈
𝐼 ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) ∈ V |
29 | | eqid 2739 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) |
30 | 29 | fnmpt 6582 |
. . . . . . . . . . . . . . . . . 18
⊢
(∀𝑛 ∈
𝐼 ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) ∈ V → (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼) |
31 | 28, 30 | mp1i 13 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼) |
32 | 3 | 3ad2ant1 1132 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑅:𝐼⟶∞MetSp) |
33 | 32 | ffvelrnda 6970 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ ∞MetSp) |
34 | | prdsxms.v |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝑉 = (Base‘(𝑅‘𝑘)) |
35 | | prdsxms.e |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐸 = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉)) |
36 | 34, 35 | xmsxmet 23618 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅‘𝑘) ∈ ∞MetSp → 𝐸 ∈ (∞Met‘𝑉)) |
37 | 33, 36 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) |
38 | | eqid 2739 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) |
39 | | eqid 2739 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) |
40 | 14 | 3ad2ant1 1132 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑆 ∈ 𝑊) |
41 | 33 | ralrimiva 3104 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → ∀𝑘 ∈ 𝐼 (𝑅‘𝑘) ∈ ∞MetSp) |
42 | | simp2l 1198 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑝 ∈ 𝐵) |
43 | 32 | feqmptd 6846 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑅 = (𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) |
44 | 43 | oveq2d 7300 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) |
45 | 13, 44 | eqtrid 2791 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑌 = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) |
46 | 45 | fveq2d 6787 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
47 | 16, 46 | eqtrid 2791 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
48 | 42, 47 | eleqtrd 2842 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
49 | 38, 39, 40, 25, 41, 34, 48 | prdsbascl 17203 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → ∀𝑘 ∈ 𝐼 (𝑝‘𝑘) ∈ 𝑉) |
50 | 49 | r19.21bi 3135 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → (𝑝‘𝑘) ∈ 𝑉) |
51 | | simp2r 1199 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑟 ∈
ℝ*) |
52 | 51 | adantr 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → 𝑟 ∈ ℝ*) |
53 | | eqid 2739 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(MetOpen‘𝐸) =
(MetOpen‘𝐸) |
54 | 53 | blopn 23665 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑝‘𝑘) ∈ 𝑉 ∧ 𝑟 ∈ ℝ*) → ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸)) |
55 | 37, 50, 52, 54 | syl3anc 1370 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸)) |
56 | | 2fveq3 6788 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝑘 → (dist‘(𝑅‘𝑛)) = (dist‘(𝑅‘𝑘))) |
57 | | 2fveq3 6788 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 = 𝑘 → (Base‘(𝑅‘𝑛)) = (Base‘(𝑅‘𝑘))) |
58 | 57, 34 | eqtr4di 2797 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = 𝑘 → (Base‘(𝑅‘𝑛)) = 𝑉) |
59 | 58 | sqxpeqd 5622 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝑘 → ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛))) = (𝑉 × 𝑉)) |
60 | 56, 59 | reseq12d 5895 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑘 → ((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))) = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉))) |
61 | 60, 35 | eqtr4di 2797 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑘 → ((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))) = 𝐸) |
62 | 61 | fveq2d 6787 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑘 → (ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛))))) = (ball‘𝐸)) |
63 | | fveq2 6783 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑘 → (𝑝‘𝑛) = (𝑝‘𝑘)) |
64 | | eqidd 2740 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑘 → 𝑟 = 𝑟) |
65 | 62, 63, 64 | oveq123d 7305 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑘 → ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) = ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
66 | | ovex 7317 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ V |
67 | 65, 29, 66 | fvmpt 6884 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ 𝐼 → ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) = ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
68 | 67 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) = ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
69 | | fvco3 6876 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑅:𝐼⟶∞MetSp ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘))) |
70 | | prdsxms.k |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐾 = (TopOpen‘(𝑅‘𝑘)) |
71 | 69, 70 | eqtr4di 2797 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅:𝐼⟶∞MetSp ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾) |
72 | 32, 71 | sylan 580 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾) |
73 | 70, 34, 35 | xmstopn 23613 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅‘𝑘) ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐸)) |
74 | 33, 73 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → 𝐾 = (MetOpen‘𝐸)) |
75 | 72, 74 | eqtrd 2779 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (MetOpen‘𝐸)) |
76 | 55, 68, 75 | 3eltr4d 2855 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) |
77 | 76 | ralrimiva 3104 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) |
78 | 32 | feqmptd 6846 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑅 = (𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))) |
79 | 78 | oveq2d 7300 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) |
80 | 13, 79 | eqtrid 2791 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑌 = (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) |
81 | 80 | fveq2d 6787 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (dist‘𝑌) = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))) |
82 | 15, 81 | eqtrid 2791 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝐷 = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))) |
83 | 82 | fveq2d 6787 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (ball‘𝐷) =
(ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))) |
84 | 83 | oveqd 7301 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑝(ball‘𝐷)𝑟) = (𝑝(ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))𝑟)) |
85 | | fveq2 6783 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑘 → (𝑅‘𝑛) = (𝑅‘𝑘)) |
86 | 85 | cbvmptv 5188 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)) = (𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)) |
87 | 86 | oveq2i 7295 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) |
88 | | eqid 2739 |
. . . . . . . . . . . . . . . . . . 19
⊢
(Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) |
89 | | eqid 2739 |
. . . . . . . . . . . . . . . . . . 19
⊢
(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) |
90 | 80 | fveq2d 6787 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))) |
91 | 16, 90 | eqtrid 2791 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))) |
92 | 42, 91 | eleqtrd 2842 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))) |
93 | | simp3 1137 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 0 < 𝑟) |
94 | 87, 88, 34, 35, 89, 40, 25, 33, 37, 92, 51, 93 | prdsbl 23656 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑝(ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
95 | 84, 94 | eqtrd 2779 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
96 | | fneq1 6533 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → (𝑔 Fn 𝐼 ↔ (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼)) |
97 | | fveq1 6782 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → (𝑔‘𝑘) = ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘)) |
98 | 97 | eleq1d 2824 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → ((𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ↔ ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))) |
99 | 98 | ralbidv 3113 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → (∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))) |
100 | 96, 99 | anbi12d 631 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ↔ ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)))) |
101 | 97, 67 | sylan9eq 2799 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) = ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
102 | 101 | ixpeq2dva 8709 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → X𝑘 ∈ 𝐼 (𝑔‘𝑘) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
103 | 102 | eqeq2d 2750 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → ((𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟))) |
104 | 100, 103 | anbi12d 631 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → (((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)) ↔ (((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)))) |
105 | 104 | spcegv 3537 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∈ V → ((((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
106 | 105 | 3impib 1115 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∈ V ∧ ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘))) |
107 | 26, 31, 77, 95, 106 | syl121anc 1374 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘))) |
108 | 107 | 3expia 1120 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → (0 <
𝑟 → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
109 | 24, 108 | sylbid 239 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
110 | 109 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
111 | | simpr 485 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → 𝑥 = (𝑝(ball‘𝐷)𝑟)) |
112 | 111 | neeq1d 3004 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ ↔ (𝑝(ball‘𝐷)𝑟) ≠ ∅)) |
113 | | df-3an 1088 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘))) |
114 | | ral0 4444 |
. . . . . . . . . . . . . . . . . . 19
⊢
∀𝑘 ∈
∅ (𝑔‘𝑘) = ∪
((TopOpen ∘ 𝑅)‘𝑘) |
115 | | difeq2 4052 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 = 𝐼 → (𝐼 ∖ 𝑧) = (𝐼 ∖ 𝐼)) |
116 | | difid 4305 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐼 ∖ 𝐼) = ∅ |
117 | 115, 116 | eqtrdi 2795 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝐼 → (𝐼 ∖ 𝑧) = ∅) |
118 | 117 | raleqdv 3349 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝐼 → (∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ ∅ (𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘))) |
119 | 118 | rspcev 3562 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐼 ∈ Fin ∧ ∀𝑘 ∈ ∅ (𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
120 | 1, 114, 119 | sylancl 586 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
121 | 120 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) →
∃𝑧 ∈ Fin
∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
122 | 121 | biantrud 532 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)))) |
123 | 113, 122 | bitr4id 290 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ↔ (𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)))) |
124 | | eqeq1 2743 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘))) |
125 | 123, 124 | bi2anan9 636 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
126 | 125 | exbidv 1925 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)) ↔ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
127 | 110, 112,
126 | 3imtr4d 294 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
128 | 127 | ex 413 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))) |
129 | 128 | rexlimdvva 3224 |
. . . . . . . . . 10
⊢ (𝜑 → (∃𝑝 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))) |
130 | 19, 129 | sylbid 239 |
. . . . . . . . 9
⊢ (𝜑 → (𝑥 ∈ ran (ball‘𝐷) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))) |
131 | 130 | impd 411 |
. . . . . . . 8
⊢ (𝜑 → ((𝑥 ∈ ran (ball‘𝐷) ∧ 𝑥 ≠ ∅) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
132 | 12, 131 | syl5bi 241 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
133 | 132 | alrimiv 1931 |
. . . . . 6
⊢ (𝜑 → ∀𝑥(𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
134 | | ssab 3996 |
. . . . . 6
⊢ ((ran
(ball‘𝐷) ∖
{∅}) ⊆ {𝑥
∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))} ↔ ∀𝑥(𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
135 | 133, 134 | sylibr 233 |
. . . . 5
⊢ (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆
{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))}) |
136 | 135, 9 | sseqtrrdi 3973 |
. . . 4
⊢ (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆
𝐶) |
137 | | ssv 3946 |
. . . . . . . . . 10
⊢
∞MetSp ⊆ V |
138 | | fnssres 6564 |
. . . . . . . . . 10
⊢ ((TopOpen
Fn V ∧ ∞MetSp ⊆ V) → (TopOpen ↾ ∞MetSp) Fn
∞MetSp) |
139 | 2, 137, 138 | mp2an 689 |
. . . . . . . . 9
⊢ (TopOpen
↾ ∞MetSp) Fn ∞MetSp |
140 | | fvres 6802 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∞MetSp →
((TopOpen ↾ ∞MetSp)‘𝑥) = (TopOpen‘𝑥)) |
141 | | xmstps 23615 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ∞MetSp →
𝑥 ∈
TopSp) |
142 | | eqid 2739 |
. . . . . . . . . . . . 13
⊢
(TopOpen‘𝑥) =
(TopOpen‘𝑥) |
143 | 142 | tpstop 22095 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ TopSp →
(TopOpen‘𝑥) ∈
Top) |
144 | 141, 143 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∞MetSp →
(TopOpen‘𝑥) ∈
Top) |
145 | 140, 144 | eqeltrd 2840 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ∞MetSp →
((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top) |
146 | 145 | rgen 3075 |
. . . . . . . . 9
⊢
∀𝑥 ∈
∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top |
147 | | ffnfv 7001 |
. . . . . . . . 9
⊢ ((TopOpen
↾ ∞MetSp):∞MetSp⟶Top ↔ ((TopOpen ↾
∞MetSp) Fn ∞MetSp ∧ ∀𝑥 ∈ ∞MetSp ((TopOpen ↾
∞MetSp)‘𝑥)
∈ Top)) |
148 | 139, 146,
147 | mpbir2an 708 |
. . . . . . . 8
⊢ (TopOpen
↾ ∞MetSp):∞MetSp⟶Top |
149 | | fco2 6636 |
. . . . . . . 8
⊢
(((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen
∘ 𝑅):𝐼⟶Top) |
150 | 148, 3, 149 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top) |
151 | | eqid 2739 |
. . . . . . . 8
⊢ X𝑛 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) = X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) |
152 | 9, 151 | ptbasfi 22741 |
. . . . . . 7
⊢ ((𝐼 ∈ Fin ∧ (TopOpen
∘ 𝑅):𝐼⟶Top) → 𝐶 = (fi‘({X𝑛 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢))))) |
153 | 1, 150, 152 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → 𝐶 = (fi‘({X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢))))) |
154 | | eqid 2739 |
. . . . . . . . 9
⊢
(MetOpen‘𝐷) =
(MetOpen‘𝐷) |
155 | 154 | mopntop 23602 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Top) |
156 | 17, 155 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (MetOpen‘𝐷) ∈ Top) |
157 | 13, 16, 14, 1, 4 | prdsbas2 17189 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 = X𝑘 ∈ 𝐼 (Base‘(𝑅‘𝑘))) |
158 | 3, 71 | sylan 580 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾) |
159 | 3 | ffvelrnda 6970 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ ∞MetSp) |
160 | | xmstps 23615 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅‘𝑘) ∈ ∞MetSp → (𝑅‘𝑘) ∈ TopSp) |
161 | 159, 160 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ TopSp) |
162 | 34, 70 | istps 22092 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅‘𝑘) ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑉)) |
163 | 161, 162 | sylib 217 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐾 ∈ (TopOn‘𝑉)) |
164 | 158, 163 | eqeltrd 2840 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉)) |
165 | | toponuni 22072 |
. . . . . . . . . . . . . . 15
⊢
(((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉) → 𝑉 = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
166 | 164, 165 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑉 = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
167 | 34, 166 | eqtr3id 2793 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (Base‘(𝑅‘𝑘)) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
168 | 167 | ixpeq2dva 8709 |
. . . . . . . . . . . 12
⊢ (𝜑 → X𝑘 ∈
𝐼 (Base‘(𝑅‘𝑘)) = X𝑘 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
169 | 157, 168 | eqtrd 2779 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 = X𝑘 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
170 | | fveq2 6783 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑛 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛)) |
171 | 170 | unieqd 4854 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑛 → ∪
((TopOpen ∘ 𝑅)‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑛)) |
172 | 171 | cbvixpv 8712 |
. . . . . . . . . . 11
⊢ X𝑘 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑘) = X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) |
173 | 169, 172 | eqtrdi 2795 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 = X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛)) |
174 | 154 | mopntopon 23601 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ (TopOn‘𝐵)) |
175 | 17, 174 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (MetOpen‘𝐷) ∈ (TopOn‘𝐵)) |
176 | | toponmax 22084 |
. . . . . . . . . . 11
⊢
((MetOpen‘𝐷)
∈ (TopOn‘𝐵)
→ 𝐵 ∈
(MetOpen‘𝐷)) |
177 | 175, 176 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ (MetOpen‘𝐷)) |
178 | 173, 177 | eqeltrrd 2841 |
. . . . . . . . 9
⊢ (𝜑 → X𝑛 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ∈ (MetOpen‘𝐷)) |
179 | 178 | snssd 4743 |
. . . . . . . 8
⊢ (𝜑 → {X𝑛 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ⊆ (MetOpen‘𝐷)) |
180 | 173 | mpteq1d 5170 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘))) |
181 | 180 | ad2antrr 723 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘))) |
182 | 181 | cnveqd 5787 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘))) |
183 | 182 | imaeq1d 5971 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) |
184 | | fveq1 6782 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = 𝑝 → (𝑤‘𝑘) = (𝑝‘𝑘)) |
185 | 184 | eleq1d 2824 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 𝑝 → ((𝑤‘𝑘) ∈ 𝑢 ↔ (𝑝‘𝑘) ∈ 𝑢)) |
186 | | eqid 2739 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) |
187 | 186 | mptpreima 6146 |
. . . . . . . . . . . . . . . . . . 19
⊢ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝐵 ∣ (𝑤‘𝑘) ∈ 𝑢} |
188 | 185, 187 | elrab2 3628 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢)) |
189 | 159, 36 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) |
190 | 189 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝐸 ∈ (∞Met‘𝑉)) |
191 | | simprl 768 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑢 ∈ 𝐾) |
192 | 159, 73 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐾 = (MetOpen‘𝐸)) |
193 | 192 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝐾 = (MetOpen‘𝐸)) |
194 | 191, 193 | eleqtrd 2842 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑢 ∈ (MetOpen‘𝐸)) |
195 | | simprrr 779 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → (𝑝‘𝑘) ∈ 𝑢) |
196 | 53 | mopni2 23658 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑢 ∈ (MetOpen‘𝐸) ∧ (𝑝‘𝑘) ∈ 𝑢) → ∃𝑟 ∈ ℝ+ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢) |
197 | 190, 194,
195, 196 | syl3anc 1370 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → ∃𝑟 ∈ ℝ+ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢) |
198 | 17 | ad3antrrr 727 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝐷 ∈ (∞Met‘𝐵)) |
199 | | simprrl 778 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑝 ∈ 𝐵) |
200 | 199 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ 𝐵) |
201 | | rpxr 12748 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑟 ∈ ℝ+
→ 𝑟 ∈
ℝ*) |
202 | 201 | ad2antrl 725 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ*) |
203 | 154 | blopn 23665 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷)) |
204 | 198, 200,
202, 203 | syl3anc 1370 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷)) |
205 | | simprl 768 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ+) |
206 | | blcntr 23575 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟)) |
207 | 198, 200,
205, 206 | syl3anc 1370 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟)) |
208 | | blssm 23580 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵) |
209 | 198, 200,
202, 208 | syl3anc 1370 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵) |
210 | | simplrr 775 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢) |
211 | | simplll 772 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝜑) |
212 | | rpgt0 12751 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑟 ∈ ℝ+
→ 0 < 𝑟) |
213 | 212 | ad2antrl 725 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 0 < 𝑟) |
214 | 211, 200,
202, 213, 95 | syl121anc 1374 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
215 | 214 | eleq2d 2825 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑤 ∈ (𝑝(ball‘𝐷)𝑟) ↔ 𝑤 ∈ X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟))) |
216 | 215 | biimpa 477 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑤 ∈ X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
217 | | vex 3437 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 𝑤 ∈ V |
218 | 217 | elixp 8701 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑤 ∈ X𝑘 ∈
𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟) ↔ (𝑤 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟))) |
219 | 218 | simprbi 497 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑤 ∈ X𝑘 ∈
𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟) → ∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
220 | 216, 219 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → ∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
221 | | simp-4r 781 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑘 ∈ 𝐼) |
222 | | rsp 3132 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(∀𝑘 ∈
𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟) → (𝑘 ∈ 𝐼 → (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟))) |
223 | 220, 221,
222 | sylc 65 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
224 | 210, 223 | sseldd 3923 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤‘𝑘) ∈ 𝑢) |
225 | 209, 224 | ssrabdv 4008 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ {𝑤 ∈ 𝐵 ∣ (𝑤‘𝑘) ∈ 𝑢}) |
226 | 225, 187 | sseqtrrdi 3973 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)) |
227 | | eleq2 2828 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑝 ∈ 𝑦 ↔ 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))) |
228 | | sseq1 3947 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑝(ball‘𝐷)𝑟) ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))) |
229 | 227, 228 | anbi12d 631 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑦 = (𝑝(ball‘𝐷)𝑟) → ((𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)) ↔ (𝑝 ∈ (𝑝(ball‘𝐷)𝑟) ∧ (𝑝(ball‘𝐷)𝑟) ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))) |
230 | 229 | rspcev 3562 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷) ∧ (𝑝 ∈ (𝑝(ball‘𝐷)𝑟) ∧ (𝑝(ball‘𝐷)𝑟) ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))) |
231 | 204, 207,
226, 230 | syl12anc 834 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))) |
232 | 197, 231 | rexlimddv 3221 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))) |
233 | 232 | expr 457 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ((𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))) |
234 | 188, 233 | syl5bi 241 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))) |
235 | 234 | ralrimiv 3103 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ∀𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))) |
236 | 156 | ad2antrr 723 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (MetOpen‘𝐷) ∈ Top) |
237 | | eltop2 22134 |
. . . . . . . . . . . . . . . . 17
⊢
((MetOpen‘𝐷)
∈ Top → ((◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))) |
238 | 236, 237 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ((◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))) |
239 | 235, 238 | mpbird 256 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
240 | 183, 239 | eqeltrrd 2841 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
241 | 240 | ralrimiva 3104 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑢 ∈ 𝐾 (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
242 | 158 | raleqdv 3349 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑢 ∈ 𝐾 (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))) |
243 | 241, 242 | mpbird 256 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
244 | 243 | ralrimiva 3104 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑘 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
245 | | fveq2 6783 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑚)) |
246 | | fveq2 6783 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑚 → (𝑤‘𝑘) = (𝑤‘𝑚)) |
247 | 246 | mpteq2dv 5177 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑚 → (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚))) |
248 | 247 | cnveqd 5787 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑚 → ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚))) |
249 | 248 | imaeq1d 5971 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑚 → (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)) |
250 | 249 | eleq1d 2824 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → ((◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷))) |
251 | 245, 250 | raleqbidv 3337 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑚 → (∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷))) |
252 | 251 | cbvralvw 3384 |
. . . . . . . . . . 11
⊢
(∀𝑘 ∈
𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑚 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
253 | 244, 252 | sylib 217 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑚 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
254 | | eqid 2739 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)) = (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)) |
255 | 254 | fmpox 7916 |
. . . . . . . . . 10
⊢
(∀𝑚 ∈
𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)):∪ 𝑚 ∈ 𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷)) |
256 | 253, 255 | sylib 217 |
. . . . . . . . 9
⊢ (𝜑 → (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)):∪ 𝑚 ∈ 𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷)) |
257 | 256 | frnd 6617 |
. . . . . . . 8
⊢ (𝜑 → ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)) ⊆ (MetOpen‘𝐷)) |
258 | 179, 257 | unssd 4121 |
. . . . . . 7
⊢ (𝜑 → ({X𝑛 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢))) ⊆ (MetOpen‘𝐷)) |
259 | | fiss 9192 |
. . . . . . 7
⊢
(((MetOpen‘𝐷)
∈ Top ∧ ({X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢))) ⊆ (MetOpen‘𝐷)) → (fi‘({X𝑛 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)))) ⊆ (fi‘(MetOpen‘𝐷))) |
260 | 156, 258,
259 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → (fi‘({X𝑛 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)))) ⊆ (fi‘(MetOpen‘𝐷))) |
261 | 153, 260 | eqsstrd 3960 |
. . . . 5
⊢ (𝜑 → 𝐶 ⊆ (fi‘(MetOpen‘𝐷))) |
262 | | fitop 22058 |
. . . . . . 7
⊢
((MetOpen‘𝐷)
∈ Top → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷)) |
263 | 156, 262 | syl 17 |
. . . . . 6
⊢ (𝜑 →
(fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷)) |
264 | 154 | mopnval 23600 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) = (topGen‘ran
(ball‘𝐷))) |
265 | 17, 264 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (MetOpen‘𝐷) = (topGen‘ran
(ball‘𝐷))) |
266 | | tgdif0 22151 |
. . . . . . 7
⊢
(topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘ran
(ball‘𝐷)) |
267 | 265, 266 | eqtr4di 2797 |
. . . . . 6
⊢ (𝜑 → (MetOpen‘𝐷) = (topGen‘(ran
(ball‘𝐷) ∖
{∅}))) |
268 | 263, 267 | eqtrd 2779 |
. . . . 5
⊢ (𝜑 →
(fi‘(MetOpen‘𝐷)) = (topGen‘(ran (ball‘𝐷) ∖
{∅}))) |
269 | 261, 268 | sseqtrd 3962 |
. . . 4
⊢ (𝜑 → 𝐶 ⊆ (topGen‘(ran
(ball‘𝐷) ∖
{∅}))) |
270 | | 2basgen 22149 |
. . . 4
⊢ (((ran
(ball‘𝐷) ∖
{∅}) ⊆ 𝐶 ∧
𝐶 ⊆
(topGen‘(ran (ball‘𝐷) ∖ {∅}))) →
(topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶)) |
271 | 136, 269,
270 | syl2anc 584 |
. . 3
⊢ (𝜑 → (topGen‘(ran
(ball‘𝐷) ∖
{∅})) = (topGen‘𝐶)) |
272 | 11, 271 | eqtr4d 2782 |
. 2
⊢ (𝜑 →
(∏t‘(TopOpen ∘ 𝑅)) = (topGen‘(ran (ball‘𝐷) ∖
{∅}))) |
273 | | prdsxms.j |
. . 3
⊢ 𝐽 = (TopOpen‘𝑌) |
274 | 13, 14, 1, 4, 273 | prdstopn 22788 |
. 2
⊢ (𝜑 → 𝐽 = (∏t‘(TopOpen
∘ 𝑅))) |
275 | 272, 274,
267 | 3eqtr4d 2789 |
1
⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷)) |