Step | Hyp | Ref
| Expression |
1 | | prdsxms.i |
. . . 4
⊢ (𝜑 → 𝐼 ∈ Fin) |
2 | | topnfn 16439 |
. . . . 5
⊢ TopOpen
Fn V |
3 | | prdsxms.r |
. . . . . . 7
⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp) |
4 | 3 | ffnd 6279 |
. . . . . 6
⊢ (𝜑 → 𝑅 Fn 𝐼) |
5 | | dffn2 6280 |
. . . . . 6
⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V) |
6 | 4, 5 | sylib 210 |
. . . . 5
⊢ (𝜑 → 𝑅:𝐼⟶V) |
7 | | fnfco 6306 |
. . . . 5
⊢ ((TopOpen
Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen
∘ 𝑅) Fn 𝐼) |
8 | 2, 6, 7 | sylancr 581 |
. . . 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 21744 |
. . . 4
⊢ ((𝐼 ∈ Fin ∧ (TopOpen
∘ 𝑅) Fn 𝐼) →
(∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶)) |
11 | 1, 8, 10 | syl2anc 579 |
. . 3
⊢ (𝜑 →
(∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶)) |
12 | | eldifsn 4536 |
. . . . . . . 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 22703 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
18 | | blrn 22584 |
. . . . . . . . . . 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 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝐷 ∈ (∞Met‘𝐵)) |
21 | | simprl 787 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝑝 ∈ 𝐵) |
22 | | simprr 789 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝑟 ∈
ℝ*) |
23 | | xbln0 22589 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟)) |
24 | 20, 21, 22, 23 | syl3anc 1494 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟)) |
25 | 1 | 3ad2ant1 1167 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝐼 ∈ Fin) |
26 | | mptexg 6740 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐼 ∈ Fin → (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∈ V) |
27 | 25, 26 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∈ V) |
28 | | ovex 6937 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) ∈ V |
29 | 28 | rgenw 3133 |
. . . . . . . . . . . . . . . . . 18
⊢
∀𝑛 ∈
𝐼 ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) ∈ V |
30 | | eqid 2825 |
. . . . . . . . . . . . . . . . . . 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)) |
31 | 30 | fnmpt 6253 |
. . . . . . . . . . . . . . . . . 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) |
32 | 29, 31 | 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) |
33 | 3 | 3ad2ant1 1167 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑅:𝐼⟶∞MetSp) |
34 | 33 | ffvelrnda 6608 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ ∞MetSp) |
35 | | prdsxms.v |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝑉 = (Base‘(𝑅‘𝑘)) |
36 | | prdsxms.e |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐸 = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉)) |
37 | 35, 36 | xmsxmet 22631 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅‘𝑘) ∈ ∞MetSp → 𝐸 ∈ (∞Met‘𝑉)) |
38 | 34, 37 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) |
39 | | eqid 2825 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) |
40 | | eqid 2825 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) |
41 | 14 | 3ad2ant1 1167 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑆 ∈ 𝑊) |
42 | 34 | ralrimiva 3175 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → ∀𝑘 ∈ 𝐼 (𝑅‘𝑘) ∈ ∞MetSp) |
43 | | simp2l 1260 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑝 ∈ 𝐵) |
44 | 33 | feqmptd 6496 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑅 = (𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) |
45 | 44 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) |
46 | 13, 45 | syl5eq 2873 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑌 = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) |
47 | 46 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
48 | 16, 47 | syl5eq 2873 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
49 | 43, 48 | eleqtrd 2908 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
50 | 39, 40, 41, 25, 42, 35, 49 | prdsbascl 16496 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → ∀𝑘 ∈ 𝐼 (𝑝‘𝑘) ∈ 𝑉) |
51 | 50 | r19.21bi 3141 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → (𝑝‘𝑘) ∈ 𝑉) |
52 | | simp2r 1261 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑟 ∈
ℝ*) |
53 | 52 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → 𝑟 ∈ ℝ*) |
54 | | eqid 2825 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(MetOpen‘𝐸) =
(MetOpen‘𝐸) |
55 | 54 | blopn 22675 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑝‘𝑘) ∈ 𝑉 ∧ 𝑟 ∈ ℝ*) → ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸)) |
56 | 38, 51, 53, 55 | syl3anc 1494 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸)) |
57 | | 2fveq3 6438 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝑘 → (dist‘(𝑅‘𝑛)) = (dist‘(𝑅‘𝑘))) |
58 | | 2fveq3 6438 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑛 = 𝑘 → (Base‘(𝑅‘𝑛)) = (Base‘(𝑅‘𝑘))) |
59 | 58, 35 | syl6eqr 2879 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = 𝑘 → (Base‘(𝑅‘𝑛)) = 𝑉) |
60 | 59 | sqxpeqd 5374 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝑘 → ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛))) = (𝑉 × 𝑉)) |
61 | 57, 60 | reseq12d 5630 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑘 → ((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))) = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉))) |
62 | 61, 36 | syl6eqr 2879 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑘 → ((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))) = 𝐸) |
63 | 62 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑘 → (ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛))))) = (ball‘𝐸)) |
64 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑘 → (𝑝‘𝑛) = (𝑝‘𝑘)) |
65 | | eqidd 2826 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 = 𝑘 → 𝑟 = 𝑟) |
66 | 63, 64, 65 | oveq123d 6926 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑘 → ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) = ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
67 | | ovex 6937 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ V |
68 | 66, 30, 67 | fvmpt 6529 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ 𝐼 → ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) = ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
69 | 68 | adantl 475 |
. . . . . . . . . . . . . . . . . . 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)) |
70 | | fvco3 6522 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑅:𝐼⟶∞MetSp ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘))) |
71 | | prdsxms.k |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐾 = (TopOpen‘(𝑅‘𝑘)) |
72 | 70, 71 | syl6eqr 2879 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅:𝐼⟶∞MetSp ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾) |
73 | 33, 72 | sylan 575 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾) |
74 | 71, 35, 36 | xmstopn 22626 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑅‘𝑘) ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐸)) |
75 | 34, 74 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → 𝐾 = (MetOpen‘𝐸)) |
76 | 73, 75 | eqtrd 2861 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (MetOpen‘𝐸)) |
77 | 56, 69, 76 | 3eltr4d 2921 |
. . . . . . . . . . . . . . . . . 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)) |
78 | 77 | ralrimiva 3175 |
. . . . . . . . . . . . . . . . 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)) |
79 | 33 | feqmptd 6496 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑅 = (𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))) |
80 | 79 | oveq2d 6921 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) |
81 | 13, 80 | syl5eq 2873 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑌 = (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) |
82 | 81 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (dist‘𝑌) = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))) |
83 | 15, 82 | syl5eq 2873 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝐷 = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))) |
84 | 83 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (ball‘𝐷) =
(ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))) |
85 | 84 | oveqd 6922 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑝(ball‘𝐷)𝑟) = (𝑝(ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))𝑟)) |
86 | | fveq2 6433 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑛 = 𝑘 → (𝑅‘𝑛) = (𝑅‘𝑘)) |
87 | 86 | cbvmptv 4973 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)) = (𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)) |
88 | 87 | oveq2i 6916 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) |
89 | | eqid 2825 |
. . . . . . . . . . . . . . . . . . 19
⊢
(Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) |
90 | | eqid 2825 |
. . . . . . . . . . . . . . . . . . 19
⊢
(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) |
91 | 81 | fveq2d 6437 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))) |
92 | 16, 91 | syl5eq 2873 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))) |
93 | 43, 92 | eleqtrd 2908 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))) |
94 | | simp3 1172 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → 0 < 𝑟) |
95 | 88, 89, 35, 36, 90, 41, 25, 34, 38, 93, 52, 94 | prdsbl 22666 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑝(ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
96 | 85, 95 | eqtrd 2861 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 <
𝑟) → (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
97 | | fneq1 6212 |
. . . . . . . . . . . . . . . . . . . . 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)) |
98 | | fveq1 6432 |
. . . . . . . . . . . . . . . . . . . . . . 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)) |
99 | 98 | eleq1d 2891 |
. . . . . . . . . . . . . . . . . . . . . 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))) |
100 | 99 | ralbidv 3195 |
. . . . . . . . . . . . . . . . . . . . 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))) |
101 | 97, 100 | anbi12d 624 |
. . . . . . . . . . . . . . . . . . . 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)))) |
102 | 98, 68 | sylan9eq 2881 |
. . . . . . . . . . . . . . . . . . . . . 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)) |
103 | 102 | ixpeq2dva 8190 |
. . . . . . . . . . . . . . . . . . . . 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)) |
104 | 103 | eqeq2d 2835 |
. . . . . . . . . . . . . . . . . . . 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))) |
105 | 101, 104 | anbi12d 624 |
. . . . . . . . . . . . . . . . . . 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)))) |
106 | 105 | spcegv 3511 |
. . . . . . . . . . . . . . . . . 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)))) |
107 | 106 | 3impib 1148 |
. . . . . . . . . . . . . . . . 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))) |
108 | 27, 32, 78, 96, 107 | syl121anc 1498 |
. . . . . . . . . . . . . . . 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))) |
109 | 108 | 3expia 1154 |
. . . . . . . . . . . . . . 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)))) |
110 | 24, 109 | sylbid 232 |
. . . . . . . . . . . . . 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)))) |
111 | 110 | adantr 474 |
. . . . . . . . . . . . 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)))) |
112 | | simpr 479 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → 𝑥 = (𝑝(ball‘𝐷)𝑟)) |
113 | 112 | neeq1d 3058 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ ↔ (𝑝(ball‘𝐷)𝑟) ≠ ∅)) |
114 | | ral0 4298 |
. . . . . . . . . . . . . . . . . . 19
⊢
∀𝑘 ∈
∅ (𝑔‘𝑘) = ∪
((TopOpen ∘ 𝑅)‘𝑘) |
115 | | difeq2 3949 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 = 𝐼 → (𝐼 ∖ 𝑧) = (𝐼 ∖ 𝐼)) |
116 | | difid 4178 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝐼 ∖ 𝐼) = ∅ |
117 | 115, 116 | syl6eq 2877 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = 𝐼 → (𝐼 ∖ 𝑧) = ∅) |
118 | 117 | raleqdv 3356 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = 𝐼 → (∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ ∅ (𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘))) |
119 | 118 | rspcev 3526 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐼 ∈ Fin ∧ ∀𝑘 ∈ ∅ (𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
120 | 1, 114, 119 | sylancl 580 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
121 | 120 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) →
∃𝑧 ∈ Fin
∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
122 | 121 | biantrud 527 |
. . . . . . . . . . . . . . . 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 | | df-3an 1113 |
. . . . . . . . . . . . . . . 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))) |
124 | 122, 123 | syl6rbbr 282 |
. . . . . . . . . . . . . . 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)))) |
125 | | eqeq1 2829 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘))) |
126 | 124, 125 | bi2anan9 629 |
. . . . . . . . . . . . . 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)))) |
127 | 126 | exbidv 2020 |
. . . . . . . . . . . . 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)))) |
128 | 111, 113,
127 | 3imtr4d 286 |
. . . . . . . . . . . 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)))) |
129 | 128 | ex 403 |
. . . . . . . . . . 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))))) |
130 | 129 | rexlimdvva 3248 |
. . . . . . . . . 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))))) |
131 | 19, 130 | sylbid 232 |
. . . . . . . . 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))))) |
132 | 131 | impd 400 |
. . . . . . . 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)))) |
133 | 12, 132 | syl5bi 234 |
. . . . . . 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)))) |
134 | 133 | alrimiv 2026 |
. . . . . 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)))) |
135 | | ssab 3897 |
. . . . . 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)))) |
136 | 134, 135 | sylibr 226 |
. . . . 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))}) |
137 | 136, 9 | syl6sseqr 3877 |
. . . 4
⊢ (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆
𝐶) |
138 | | ssv 3850 |
. . . . . . . . . 10
⊢
∞MetSp ⊆ V |
139 | | fnssres 6237 |
. . . . . . . . . 10
⊢ ((TopOpen
Fn V ∧ ∞MetSp ⊆ V) → (TopOpen ↾ ∞MetSp) Fn
∞MetSp) |
140 | 2, 138, 139 | mp2an 683 |
. . . . . . . . 9
⊢ (TopOpen
↾ ∞MetSp) Fn ∞MetSp |
141 | | fvres 6452 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∞MetSp →
((TopOpen ↾ ∞MetSp)‘𝑥) = (TopOpen‘𝑥)) |
142 | | xmstps 22628 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ∞MetSp →
𝑥 ∈
TopSp) |
143 | | eqid 2825 |
. . . . . . . . . . . . 13
⊢
(TopOpen‘𝑥) =
(TopOpen‘𝑥) |
144 | 143 | tpstop 21112 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ TopSp →
(TopOpen‘𝑥) ∈
Top) |
145 | 142, 144 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ∞MetSp →
(TopOpen‘𝑥) ∈
Top) |
146 | 141, 145 | eqeltrd 2906 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ∞MetSp →
((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top) |
147 | 146 | rgen 3131 |
. . . . . . . . 9
⊢
∀𝑥 ∈
∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top |
148 | | ffnfv 6637 |
. . . . . . . . 9
⊢ ((TopOpen
↾ ∞MetSp):∞MetSp⟶Top ↔ ((TopOpen ↾
∞MetSp) Fn ∞MetSp ∧ ∀𝑥 ∈ ∞MetSp ((TopOpen ↾
∞MetSp)‘𝑥)
∈ Top)) |
149 | 140, 147,
148 | mpbir2an 702 |
. . . . . . . 8
⊢ (TopOpen
↾ ∞MetSp):∞MetSp⟶Top |
150 | | fco2 6296 |
. . . . . . . 8
⊢
(((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen
∘ 𝑅):𝐼⟶Top) |
151 | 149, 3, 150 | sylancr 581 |
. . . . . . 7
⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top) |
152 | | eqid 2825 |
. . . . . . . 8
⊢ X𝑛 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) = X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) |
153 | 9, 152 | ptbasfi 21755 |
. . . . . . 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))))) |
154 | 1, 151, 153 | syl2anc 579 |
. . . . . 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))))) |
155 | | eqid 2825 |
. . . . . . . . 9
⊢
(MetOpen‘𝐷) =
(MetOpen‘𝐷) |
156 | 155 | mopntop 22615 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Top) |
157 | 17, 156 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (MetOpen‘𝐷) ∈ Top) |
158 | 13, 16, 14, 1, 4 | prdsbas2 16482 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 = X𝑘 ∈ 𝐼 (Base‘(𝑅‘𝑘))) |
159 | 3, 72 | sylan 575 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾) |
160 | 3 | ffvelrnda 6608 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ ∞MetSp) |
161 | | xmstps 22628 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑅‘𝑘) ∈ ∞MetSp → (𝑅‘𝑘) ∈ TopSp) |
162 | 160, 161 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ TopSp) |
163 | 35, 71 | istps 21109 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑅‘𝑘) ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑉)) |
164 | 162, 163 | sylib 210 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐾 ∈ (TopOn‘𝑉)) |
165 | 159, 164 | eqeltrd 2906 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉)) |
166 | | toponuni 21089 |
. . . . . . . . . . . . . . 15
⊢
(((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉) → 𝑉 = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
167 | 165, 166 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑉 = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
168 | 35, 167 | syl5eqr 2875 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (Base‘(𝑅‘𝑘)) = ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
169 | 168 | ixpeq2dva 8190 |
. . . . . . . . . . . 12
⊢ (𝜑 → X𝑘 ∈
𝐼 (Base‘(𝑅‘𝑘)) = X𝑘 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
170 | 158, 169 | eqtrd 2861 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 = X𝑘 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑘)) |
171 | | fveq2 6433 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑛 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛)) |
172 | 171 | unieqd 4668 |
. . . . . . . . . . . 12
⊢ (𝑘 = 𝑛 → ∪
((TopOpen ∘ 𝑅)‘𝑘) = ∪ ((TopOpen
∘ 𝑅)‘𝑛)) |
173 | 172 | cbvixpv 8193 |
. . . . . . . . . . 11
⊢ X𝑘 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑘) = X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) |
174 | 170, 173 | syl6eq 2877 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 = X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛)) |
175 | 155 | mopntopon 22614 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ (TopOn‘𝐵)) |
176 | 17, 175 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → (MetOpen‘𝐷) ∈ (TopOn‘𝐵)) |
177 | | toponmax 21101 |
. . . . . . . . . . 11
⊢
((MetOpen‘𝐷)
∈ (TopOn‘𝐵)
→ 𝐵 ∈
(MetOpen‘𝐷)) |
178 | 176, 177 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ (MetOpen‘𝐷)) |
179 | 174, 178 | eqeltrrd 2907 |
. . . . . . . . 9
⊢ (𝜑 → X𝑛 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ∈ (MetOpen‘𝐷)) |
180 | 179 | snssd 4558 |
. . . . . . . 8
⊢ (𝜑 → {X𝑛 ∈
𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ⊆ (MetOpen‘𝐷)) |
181 | 174 | mpteq1d 4961 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘))) |
182 | 181 | ad2antrr 717 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘))) |
183 | 182 | cnveqd 5530 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘))) |
184 | 183 | imaeq1d 5706 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) |
185 | | fveq1 6432 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 = 𝑝 → (𝑤‘𝑘) = (𝑝‘𝑘)) |
186 | 185 | eleq1d 2891 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑤 = 𝑝 → ((𝑤‘𝑘) ∈ 𝑢 ↔ (𝑝‘𝑘) ∈ 𝑢)) |
187 | | eqid 2825 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) |
188 | 187 | mptpreima 5869 |
. . . . . . . . . . . . . . . . . . 19
⊢ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝐵 ∣ (𝑤‘𝑘) ∈ 𝑢} |
189 | 186, 188 | elrab2 3589 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢)) |
190 | 160, 37 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) |
191 | 190 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝐸 ∈ (∞Met‘𝑉)) |
192 | | simprl 787 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑢 ∈ 𝐾) |
193 | 160, 74 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐾 = (MetOpen‘𝐸)) |
194 | 193 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝐾 = (MetOpen‘𝐸)) |
195 | 192, 194 | eleqtrd 2908 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑢 ∈ (MetOpen‘𝐸)) |
196 | | simprrr 800 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → (𝑝‘𝑘) ∈ 𝑢) |
197 | 54 | mopni2 22668 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑢 ∈ (MetOpen‘𝐸) ∧ (𝑝‘𝑘) ∈ 𝑢) → ∃𝑟 ∈ ℝ+ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢) |
198 | 191, 195,
196, 197 | syl3anc 1494 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → ∃𝑟 ∈ ℝ+ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢) |
199 | 17 | ad3antrrr 721 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝐷 ∈ (∞Met‘𝐵)) |
200 | | simprrl 799 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑝 ∈ 𝐵) |
201 | 200 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ 𝐵) |
202 | | rpxr 12123 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑟 ∈ ℝ+
→ 𝑟 ∈
ℝ*) |
203 | 202 | ad2antrl 719 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ*) |
204 | 155 | blopn 22675 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷)) |
205 | 199, 201,
203, 204 | syl3anc 1494 |
. . . . . . . . . . . . . . . . . . . . 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)) |
206 | | simprl 787 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ+) |
207 | | blcntr 22588 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟)) |
208 | 199, 201,
206, 207 | syl3anc 1494 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟)) |
209 | | blssm 22593 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵) |
210 | 199, 201,
203, 209 | syl3anc 1494 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵) |
211 | | simplrr 796 |
. . . . . . . . . . . . . . . . . . . . . . . 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) |
212 | | simplll 791 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝜑) |
213 | | rpgt0 12126 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑟 ∈ ℝ+
→ 0 < 𝑟) |
214 | 213 | ad2antrl 719 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 0 < 𝑟) |
215 | 212, 201,
203, 214, 96 | syl121anc 1498 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 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)) |
216 | 215 | eleq2d 2892 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 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))) |
217 | 216 | biimpa 470 |
. . . . . . . . . . . . . . . . . . . . . . . . . 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)) |
218 | | vex 3417 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 𝑤 ∈ V |
219 | 218 | elixp 8182 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑤 ∈ X𝑘 ∈
𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟) ↔ (𝑤 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟))) |
220 | 219 | simprbi 492 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑤 ∈ X𝑘 ∈
𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟) → ∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟)) |
221 | 217, 220 | 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)) |
222 | | simp-4r 803 |
. . . . . . . . . . . . . . . . . . . . . . . . 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) |
223 | | rsp 3138 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(∀𝑘 ∈
𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟) → (𝑘 ∈ 𝐼 → (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟))) |
224 | 221, 222,
223 | 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)) |
225 | 211, 224 | sseldd 3828 |
. . . . . . . . . . . . . . . . . . . . . . 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) |
226 | 210, 225 | ssrabdv 3906 |
. . . . . . . . . . . . . . . . . . . . . 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}) |
227 | 226, 188 | syl6sseqr 3877 |
. . . . . . . . . . . . . . . . . . . . 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)) |
228 | | eleq2 2895 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑝 ∈ 𝑦 ↔ 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))) |
229 | | sseq1 3851 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑝(ball‘𝐷)𝑟) ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))) |
230 | 228, 229 | anbi12d 624 |
. . . . . . . . . . . . . . . . . . . . . 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)))) |
231 | 230 | rspcev 3526 |
. . . . . . . . . . . . . . . . . . . . 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))) |
232 | 205, 208,
227, 231 | syl12anc 870 |
. . . . . . . . . . . . . . . . . . . 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))) |
233 | 198, 232 | rexlimddv 3245 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))) |
234 | 233 | expr 450 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ((𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))) |
235 | 189, 234 | syl5bi 234 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))) |
236 | 235 | ralrimiv 3174 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ∀𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))) |
237 | 157 | ad2antrr 717 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (MetOpen‘𝐷) ∈ Top) |
238 | | eltop2 21150 |
. . . . . . . . . . . . . . . . 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)))) |
239 | 237, 238 | 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)))) |
240 | 236, 239 | mpbird 249 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
241 | 184, 240 | eqeltrrd 2907 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
242 | 241 | ralrimiva 3175 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑢 ∈ 𝐾 (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
243 | 159 | raleqdv 3356 |
. . . . . . . . . . . . 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))) |
244 | 242, 243 | mpbird 249 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
245 | 244 | ralrimiva 3175 |
. . . . . . . . . . 11
⊢ (𝜑 → ∀𝑘 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
246 | | fveq2 6433 |
. . . . . . . . . . . . 13
⊢ (𝑘 = 𝑚 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑚)) |
247 | | fveq2 6433 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑚 → (𝑤‘𝑘) = (𝑤‘𝑚)) |
248 | 247 | mpteq2dv 4968 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝑚 → (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚))) |
249 | 248 | cnveqd 5530 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑚 → ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚))) |
250 | 249 | imaeq1d 5706 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝑚 → (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)) |
251 | 250 | eleq1d 2891 |
. . . . . . . . . . . . 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))) |
252 | 246, 251 | raleqbidv 3364 |
. . . . . . . . . . . 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))) |
253 | 252 | cbvralv 3383 |
. . . . . . . . . . 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)) |
254 | 245, 253 | sylib 210 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑚 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷)) |
255 | | eqid 2825 |
. . . . . . . . . . 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)) |
256 | 255 | fmpt2x 7499 |
. . . . . . . . . 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)) |
257 | 254, 256 | sylib 210 |
. . . . . . . . 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)) |
258 | 257 | frnd 6285 |
. . . . . . . 8
⊢ (𝜑 → ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen
∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)) ⊆ (MetOpen‘𝐷)) |
259 | 180, 258 | unssd 4016 |
. . . . . . 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)) |
260 | | fiss 8599 |
. . . . . . 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))) |
261 | 157, 259,
260 | syl2anc 579 |
. . . . . 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))) |
262 | 154, 261 | eqsstrd 3864 |
. . . . 5
⊢ (𝜑 → 𝐶 ⊆ (fi‘(MetOpen‘𝐷))) |
263 | | fitop 21075 |
. . . . . . 7
⊢
((MetOpen‘𝐷)
∈ Top → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷)) |
264 | 157, 263 | syl 17 |
. . . . . 6
⊢ (𝜑 →
(fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷)) |
265 | 155 | mopnval 22613 |
. . . . . . . 8
⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) = (topGen‘ran
(ball‘𝐷))) |
266 | 17, 265 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (MetOpen‘𝐷) = (topGen‘ran
(ball‘𝐷))) |
267 | | tgdif0 21167 |
. . . . . . 7
⊢
(topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘ran
(ball‘𝐷)) |
268 | 266, 267 | syl6eqr 2879 |
. . . . . 6
⊢ (𝜑 → (MetOpen‘𝐷) = (topGen‘(ran
(ball‘𝐷) ∖
{∅}))) |
269 | 264, 268 | eqtrd 2861 |
. . . . 5
⊢ (𝜑 →
(fi‘(MetOpen‘𝐷)) = (topGen‘(ran (ball‘𝐷) ∖
{∅}))) |
270 | 262, 269 | sseqtrd 3866 |
. . . 4
⊢ (𝜑 → 𝐶 ⊆ (topGen‘(ran
(ball‘𝐷) ∖
{∅}))) |
271 | | 2basgen 21165 |
. . . 4
⊢ (((ran
(ball‘𝐷) ∖
{∅}) ⊆ 𝐶 ∧
𝐶 ⊆
(topGen‘(ran (ball‘𝐷) ∖ {∅}))) →
(topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶)) |
272 | 137, 270,
271 | syl2anc 579 |
. . 3
⊢ (𝜑 → (topGen‘(ran
(ball‘𝐷) ∖
{∅})) = (topGen‘𝐶)) |
273 | 11, 272 | eqtr4d 2864 |
. 2
⊢ (𝜑 →
(∏t‘(TopOpen ∘ 𝑅)) = (topGen‘(ran (ball‘𝐷) ∖
{∅}))) |
274 | | prdsxms.j |
. . 3
⊢ 𝐽 = (TopOpen‘𝑌) |
275 | 13, 14, 1, 4, 274 | prdstopn 21802 |
. 2
⊢ (𝜑 → 𝐽 = (∏t‘(TopOpen
∘ 𝑅))) |
276 | 273, 275,
268 | 3eqtr4d 2871 |
1
⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷)) |