Step | Hyp | Ref
| Expression |
1 | | eqid 2610 |
. . . 4
⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) |
2 | | eqid 2610 |
. . . 4
⊢
(Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) |
3 | | prdsbnd.v |
. . . 4
⊢ 𝑉 = (Base‘(𝑅‘𝑥)) |
4 | | prdsbnd.e |
. . . 4
⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉)) |
5 | | eqid 2610 |
. . . 4
⊢
(dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) |
6 | | prdsbnd.s |
. . . 4
⊢ (𝜑 → 𝑆 ∈ 𝑊) |
7 | | prdsbnd.i |
. . . 4
⊢ (𝜑 → 𝐼 ∈ Fin) |
8 | | fvex 6113 |
. . . . 5
⊢ (𝑅‘𝑥) ∈ V |
9 | 8 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V) |
10 | | prdstotbnd.m |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (TotBnd‘𝑉)) |
11 | | totbndmet 32741 |
. . . . 5
⊢ (𝐸 ∈ (TotBnd‘𝑉) → 𝐸 ∈ (Met‘𝑉)) |
12 | 10, 11 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉)) |
13 | 1, 2, 3, 4, 5, 6, 7, 9, 12 | prdsmet 21985 |
. . 3
⊢ (𝜑 → (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) ∈ (Met‘(Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))) |
14 | | prdsbnd.d |
. . . 4
⊢ 𝐷 = (dist‘𝑌) |
15 | | prdsbnd.y |
. . . . . 6
⊢ 𝑌 = (𝑆Xs𝑅) |
16 | | prdsbnd.r |
. . . . . . . 8
⊢ (𝜑 → 𝑅 Fn 𝐼) |
17 | | dffn5 6151 |
. . . . . . . 8
⊢ (𝑅 Fn 𝐼 ↔ 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) |
18 | 16, 17 | sylib 207 |
. . . . . . 7
⊢ (𝜑 → 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) |
19 | 18 | oveq2d 6565 |
. . . . . 6
⊢ (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) |
20 | 15, 19 | syl5eq 2656 |
. . . . 5
⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) |
21 | 20 | fveq2d 6107 |
. . . 4
⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))) |
22 | 14, 21 | syl5eq 2656 |
. . 3
⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))) |
23 | | prdsbnd.b |
. . . . 5
⊢ 𝐵 = (Base‘𝑌) |
24 | 20 | fveq2d 6107 |
. . . . 5
⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))) |
25 | 23, 24 | syl5eq 2656 |
. . . 4
⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))) |
26 | 25 | fveq2d 6107 |
. . 3
⊢ (𝜑 → (Met‘𝐵) =
(Met‘(Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))) |
27 | 13, 22, 26 | 3eltr4d 2703 |
. 2
⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵)) |
28 | 7 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 𝐼 ∈ Fin) |
29 | | istotbnd3 32740 |
. . . . . . . . . . 11
⊢ (𝐸 ∈ (TotBnd‘𝑉) ↔ (𝐸 ∈ (Met‘𝑉) ∧ ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)) |
30 | 29 | simprbi 479 |
. . . . . . . . . 10
⊢ (𝐸 ∈ (TotBnd‘𝑉) → ∀𝑟 ∈ ℝ+
∃𝑤 ∈ (𝒫
𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉) |
31 | 10, 30 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉) |
32 | 31 | r19.21bi 2916 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑟 ∈ ℝ+) →
∃𝑤 ∈ (𝒫
𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉) |
33 | | df-rex 2902 |
. . . . . . . . 9
⊢
(∃𝑤 ∈
(𝒫 𝑉 ∩
Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 ↔ ∃𝑤(𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)) |
34 | | rexv 3193 |
. . . . . . . . 9
⊢
(∃𝑤 ∈ V
(𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉) ↔ ∃𝑤(𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)) |
35 | 33, 34 | bitr4i 266 |
. . . . . . . 8
⊢
(∃𝑤 ∈
(𝒫 𝑉 ∩
Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 ↔ ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)) |
36 | 32, 35 | sylib 207 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑟 ∈ ℝ+) →
∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)) |
37 | 36 | an32s 842 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑥 ∈ 𝐼) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)) |
38 | 37 | ralrimiva 2949 |
. . . . 5
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) →
∀𝑥 ∈ 𝐼 ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)) |
39 | | eleq1 2676 |
. . . . . . 7
⊢ (𝑤 = (𝑓‘𝑥) → (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ↔ (𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin))) |
40 | | iuneq1 4470 |
. . . . . . . 8
⊢ (𝑤 = (𝑓‘𝑥) → ∪
𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟)) |
41 | 40 | eqeq1d 2612 |
. . . . . . 7
⊢ (𝑤 = (𝑓‘𝑥) → (∪
𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 ↔ ∪
𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉)) |
42 | 39, 41 | anbi12d 743 |
. . . . . 6
⊢ (𝑤 = (𝑓‘𝑥) → ((𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉) ↔ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) |
43 | 42 | ac6sfi 8089 |
. . . . 5
⊢ ((𝐼 ∈ Fin ∧ ∀𝑥 ∈ 𝐼 ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)) → ∃𝑓(𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) |
44 | 28, 38, 43 | syl2anc 691 |
. . . 4
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) →
∃𝑓(𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) |
45 | | elfpw 8151 |
. . . . . . . . . . . 12
⊢ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((𝑓‘𝑥) ⊆ 𝑉 ∧ (𝑓‘𝑥) ∈ Fin)) |
46 | 45 | simplbi 475 |
. . . . . . . . . . 11
⊢ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓‘𝑥) ⊆ 𝑉) |
47 | 46 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓‘𝑥) ⊆ 𝑉) |
48 | 47 | ralimi 2936 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝑉) |
49 | 48 | ad2antll 761 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝑉) |
50 | | ss2ixp 7807 |
. . . . . . . 8
⊢
(∀𝑥 ∈
𝐼 (𝑓‘𝑥) ⊆ 𝑉 → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ X𝑥 ∈ 𝐼 𝑉) |
51 | 49, 50 | syl 17 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ X𝑥 ∈ 𝐼 𝑉) |
52 | | fnfi 8123 |
. . . . . . . . . . 11
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → 𝑅 ∈ Fin) |
53 | 16, 7, 52 | syl2anc 691 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑅 ∈ Fin) |
54 | | fndm 5904 |
. . . . . . . . . . 11
⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼) |
55 | 16, 54 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → dom 𝑅 = 𝐼) |
56 | 15, 6, 53, 23, 55 | prdsbas 15940 |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
57 | 3 | rgenw 2908 |
. . . . . . . . . 10
⊢
∀𝑥 ∈
𝐼 𝑉 = (Base‘(𝑅‘𝑥)) |
58 | | ixpeq2 7808 |
. . . . . . . . . 10
⊢
(∀𝑥 ∈
𝐼 𝑉 = (Base‘(𝑅‘𝑥)) → X𝑥 ∈ 𝐼 𝑉 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))) |
59 | 57, 58 | ax-mp 5 |
. . . . . . . . 9
⊢ X𝑥 ∈
𝐼 𝑉 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)) |
60 | 56, 59 | syl6eqr 2662 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝑉) |
61 | 60 | ad2antrr 758 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐵 = X𝑥 ∈ 𝐼 𝑉) |
62 | 51, 61 | sseqtr4d 3605 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝐵) |
63 | 28 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐼 ∈ Fin) |
64 | 45 | simprbi 479 |
. . . . . . . . . 10
⊢ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓‘𝑥) ∈ Fin) |
65 | 64 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓‘𝑥) ∈ Fin) |
66 | 65 | ralimi 2936 |
. . . . . . . 8
⊢
(∀𝑥 ∈
𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin) |
67 | 66 | ad2antll 761 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin) |
68 | | ixpfi 8146 |
. . . . . . 7
⊢ ((𝐼 ∈ Fin ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin) → X𝑥 ∈
𝐼 (𝑓‘𝑥) ∈ Fin) |
69 | 63, 67, 68 | syl2anc 691 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin) |
70 | | elfpw 8151 |
. . . . . 6
⊢ (X𝑥 ∈
𝐼 (𝑓‘𝑥) ∈ (𝒫 𝐵 ∩ Fin) ↔ (X𝑥 ∈
𝐼 (𝑓‘𝑥) ⊆ 𝐵 ∧ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin)) |
71 | 62, 69, 70 | sylanbrc 695 |
. . . . 5
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ (𝒫 𝐵 ∩ Fin)) |
72 | | metxmet 21949 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ (Met‘𝐵) → 𝐷 ∈ (∞Met‘𝐵)) |
73 | 27, 72 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
74 | | rpxr 11716 |
. . . . . . . . . 10
⊢ (𝑟 ∈ ℝ+
→ 𝑟 ∈
ℝ*) |
75 | | blssm 22033 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵) |
76 | 75 | 3expa 1257 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵) |
77 | 76 | an32s 842 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) ∧ 𝑦 ∈ 𝐵) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵) |
78 | 77 | ralrimiva 2949 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) →
∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵) |
79 | 73, 74, 78 | syl2an 493 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) →
∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵) |
80 | 79 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵) |
81 | | ssralv 3629 |
. . . . . . . 8
⊢ (X𝑥 ∈
𝐼 (𝑓‘𝑥) ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵 → ∀𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)) |
82 | 62, 80, 81 | sylc 63 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵) |
83 | | iunss 4497 |
. . . . . . 7
⊢ (∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵 ↔ ∀𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵) |
84 | 82, 83 | sylibr 223 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵) |
85 | 63 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ Fin) |
86 | 61 | eleq2d 2673 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔 ∈ 𝐵 ↔ 𝑔 ∈ X𝑥 ∈ 𝐼 𝑉)) |
87 | | vex 3176 |
. . . . . . . . . . . . . . . 16
⊢ 𝑔 ∈ V |
88 | 87 | elixp 7801 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 ∈ X𝑥 ∈
𝐼 𝑉 ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)) |
89 | 88 | simprbi 479 |
. . . . . . . . . . . . . 14
⊢ (𝑔 ∈ X𝑥 ∈
𝐼 𝑉 → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉) |
90 | | df-rex 2902 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∃𝑧 ∈
(𝑓‘𝑥)(𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧(𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))) |
91 | | eliun 4460 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑔‘𝑥) ∈ ∪
𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ (𝑓‘𝑥)(𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) |
92 | | rexv 3193 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(∃𝑧 ∈ V
(𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ∃𝑧(𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))) |
93 | 90, 91, 92 | 3bitr4i 291 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑔‘𝑥) ∈ ∪
𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))) |
94 | | eleq2 2677 |
. . . . . . . . . . . . . . . . . . 19
⊢ (∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔‘𝑥) ∈ ∪
𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) ↔ (𝑔‘𝑥) ∈ 𝑉)) |
95 | 93, 94 | syl5bbr 273 |
. . . . . . . . . . . . . . . . . 18
⊢ (∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → (∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ (𝑔‘𝑥) ∈ 𝑉)) |
96 | 95 | biimprd 237 |
. . . . . . . . . . . . . . . . 17
⊢ (∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔‘𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))) |
97 | 96 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ((𝑔‘𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))) |
98 | 97 | ral2imi 2931 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑥 ∈
𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉 → ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))) |
99 | 98 | ad2antll 761 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉 → ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))) |
100 | 89, 99 | syl5 33 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔 ∈ X𝑥 ∈ 𝐼 𝑉 → ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))) |
101 | 86, 100 | sylbid 229 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔 ∈ 𝐵 → ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))) |
102 | 101 | imp 444 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))) |
103 | | eleq1 2676 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (𝑦‘𝑥) → (𝑧 ∈ (𝑓‘𝑥) ↔ (𝑦‘𝑥) ∈ (𝑓‘𝑥))) |
104 | | oveq1 6556 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = (𝑦‘𝑥) → (𝑧(ball‘𝐸)𝑟) = ((𝑦‘𝑥)(ball‘𝐸)𝑟)) |
105 | 104 | eleq2d 2673 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (𝑦‘𝑥) → ((𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) |
106 | 103, 105 | anbi12d 743 |
. . . . . . . . . . . 12
⊢ (𝑧 = (𝑦‘𝑥) → ((𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) |
107 | 106 | ac6sfi 8089 |
. . . . . . . . . . 11
⊢ ((𝐼 ∈ Fin ∧ ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))) → ∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) |
108 | 85, 102, 107 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → ∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) |
109 | | ffn 5958 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦:𝐼⟶V → 𝑦 Fn 𝐼) |
110 | | simpl 472 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → (𝑦‘𝑥) ∈ (𝑓‘𝑥)) |
111 | 110 | ralimi 2936 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑥 ∈
𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (𝑓‘𝑥)) |
112 | 109, 111 | anim12i 588 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → (𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (𝑓‘𝑥))) |
113 | | vex 3176 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑦 ∈ V |
114 | 113 | elixp 7801 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ X𝑥 ∈
𝐼 (𝑓‘𝑥) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (𝑓‘𝑥))) |
115 | 112, 114 | sylibr 223 |
. . . . . . . . . . . . . . 15
⊢ ((𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → 𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥)) |
116 | 115 | adantl 481 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥)) |
117 | 86 | biimpa 500 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ X𝑥 ∈ 𝐼 𝑉) |
118 | | ixpfn 7800 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑔 ∈ X𝑥 ∈
𝐼 𝑉 → 𝑔 Fn 𝐼) |
119 | 117, 118 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → 𝑔 Fn 𝐼) |
120 | 119 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑔 Fn 𝐼) |
121 | | simpr 476 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) |
122 | 121 | ralimi 2936 |
. . . . . . . . . . . . . . . . 17
⊢
(∀𝑥 ∈
𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) |
123 | 122 | ad2antll 761 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) |
124 | 87 | elixp 7801 |
. . . . . . . . . . . . . . . 16
⊢ (𝑔 ∈ X𝑥 ∈
𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟) ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) |
125 | 120, 123,
124 | sylanbrc 695 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑔 ∈ X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟)) |
126 | | simp-4l 802 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝜑) |
127 | 51 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ X𝑥 ∈ 𝐼 𝑉) |
128 | 127, 116 | sseldd 3569 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑦 ∈ X𝑥 ∈ 𝐼 𝑉) |
129 | 126, 60 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝐵 = X𝑥 ∈ 𝐼 𝑉) |
130 | 128, 129 | eleqtrrd 2691 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑦 ∈ 𝐵) |
131 | | simp-4r 803 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑟 ∈ ℝ+) |
132 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑦 = 𝑥 → (𝑅‘𝑦) = (𝑅‘𝑥)) |
133 | 132 | cbvmptv 4678 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)) = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)) |
134 | 133 | oveq2i 6560 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) |
135 | 20, 134 | syl6eqr 2662 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) |
136 | 135 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))) |
137 | 14, 136 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))) |
138 | 137 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (ball‘𝐷) =
(ball‘(dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))) |
139 | 138 | oveqdr 6573 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘(dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))𝑟)) |
140 | | eqid 2610 |
. . . . . . . . . . . . . . . . . 18
⊢
(Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) |
141 | | eqid 2610 |
. . . . . . . . . . . . . . . . . 18
⊢
(dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) = (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) |
142 | 6 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑆 ∈ 𝑊) |
143 | 7 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝐼 ∈ Fin) |
144 | 8 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V) |
145 | | metxmet 21949 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉)) |
146 | 12, 145 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) |
147 | 146 | adantlr 747 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) |
148 | | simprl 790 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑦 ∈ 𝐵) |
149 | 135 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))) |
150 | 23, 149 | syl5eq 2656 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))) |
151 | 150 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))) |
152 | 148, 151 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑦 ∈ (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))) |
153 | 74 | ad2antll 761 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈
ℝ*) |
154 | | rpgt0 11720 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑟 ∈ ℝ+
→ 0 < 𝑟) |
155 | 154 | ad2antll 761 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 0 <
𝑟) |
156 | 134, 140,
3, 4, 141, 142, 143, 144, 147, 152, 153, 155 | prdsbl 22106 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → (𝑦(ball‘(dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))𝑟) = X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟)) |
157 | 139, 156 | eqtrd 2644 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟)) |
158 | 126, 130,
131, 157 | syl12anc 1316 |
. . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → (𝑦(ball‘𝐷)𝑟) = X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟)) |
159 | 125, 158 | eleqtrrd 2691 |
. . . . . . . . . . . . . 14
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑔 ∈ (𝑦(ball‘𝐷)𝑟)) |
160 | 116, 159 | jca 553 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑟 ∈ ℝ+)
∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → (𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟))) |
161 | 160 | ex 449 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → ((𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → (𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))) |
162 | 161 | eximdv 1833 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → (∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → ∃𝑦(𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))) |
163 | | df-rex 2902 |
. . . . . . . . . . 11
⊢
(∃𝑦 ∈
X 𝑥
∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦(𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟))) |
164 | 162, 163 | syl6ibr 241 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → (∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → ∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟))) |
165 | 108, 164 | mpd 15 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → ∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟)) |
166 | 165 | ex 449 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔 ∈ 𝐵 → ∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟))) |
167 | | eliun 4460 |
. . . . . . . 8
⊢ (𝑔 ∈ ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟)) |
168 | 166, 167 | syl6ibr 241 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔 ∈ 𝐵 → 𝑔 ∈ ∪
𝑦 ∈ X 𝑥 ∈
𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟))) |
169 | 168 | ssrdv 3574 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐵 ⊆ ∪
𝑦 ∈ X 𝑥 ∈
𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟)) |
170 | 84, 169 | eqssd 3585 |
. . . . 5
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵) |
171 | | iuneq1 4470 |
. . . . . . 7
⊢ (𝑣 = X𝑥 ∈ 𝐼 (𝑓‘𝑥) → ∪
𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = ∪ 𝑦 ∈ X
𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟)) |
172 | 171 | eqeq1d 2612 |
. . . . . 6
⊢ (𝑣 = X𝑥 ∈ 𝐼 (𝑓‘𝑥) → (∪
𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵 ↔ ∪
𝑦 ∈ X 𝑥 ∈
𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵)) |
173 | 172 | rspcev 3282 |
. . . . 5
⊢ ((X𝑥 ∈
𝐼 (𝑓‘𝑥) ∈ (𝒫 𝐵 ∩ Fin) ∧ ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵) |
174 | 71, 170, 173 | syl2anc 691 |
. . . 4
⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵) |
175 | 44, 174 | exlimddv 1850 |
. . 3
⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) →
∃𝑣 ∈ (𝒫
𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵) |
176 | 175 | ralrimiva 2949 |
. 2
⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵) |
177 | | istotbnd3 32740 |
. 2
⊢ (𝐷 ∈ (TotBnd‘𝐵) ↔ (𝐷 ∈ (Met‘𝐵) ∧ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)) |
178 | 27, 176, 177 | sylanbrc 695 |
1
⊢ (𝜑 → 𝐷 ∈ (TotBnd‘𝐵)) |