Step | Hyp | Ref
| Expression |
1 | | simpll1 1214 |
. . . . . . 7
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) → 𝐷 ∈ (∞Met‘𝑋)) |
2 | | simprl 771 |
. . . . . . 7
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) → 𝑧 ∈ 𝐽) |
3 | | simprr 773 |
. . . . . . 7
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) → 𝐴 ∈ 𝑧) |
4 | | metdscn.j |
. . . . . . . 8
⊢ 𝐽 = (MetOpen‘𝐷) |
5 | 4 | mopni2 23418 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧) → ∃𝑟 ∈ ℝ+ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧) |
6 | 1, 2, 3, 5 | syl3anc 1373 |
. . . . . 6
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) → ∃𝑟 ∈ ℝ+ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧) |
7 | | simprr 773 |
. . . . . . . 8
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧) |
8 | 7 | ssrind 4165 |
. . . . . . 7
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → ((𝐴(ball‘𝐷)𝑟) ∩ 𝑆) ⊆ (𝑧 ∩ 𝑆)) |
9 | | rpgt0 12623 |
. . . . . . . . . 10
⊢ (𝑟 ∈ ℝ+
→ 0 < 𝑟) |
10 | | 0re 10860 |
. . . . . . . . . . 11
⊢ 0 ∈
ℝ |
11 | | rpre 12619 |
. . . . . . . . . . 11
⊢ (𝑟 ∈ ℝ+
→ 𝑟 ∈
ℝ) |
12 | | ltnle 10937 |
. . . . . . . . . . 11
⊢ ((0
∈ ℝ ∧ 𝑟
∈ ℝ) → (0 < 𝑟 ↔ ¬ 𝑟 ≤ 0)) |
13 | 10, 11, 12 | sylancr 590 |
. . . . . . . . . 10
⊢ (𝑟 ∈ ℝ+
→ (0 < 𝑟 ↔
¬ 𝑟 ≤
0)) |
14 | 9, 13 | mpbid 235 |
. . . . . . . . 9
⊢ (𝑟 ∈ ℝ+
→ ¬ 𝑟 ≤
0) |
15 | 14 | ad2antrl 728 |
. . . . . . . 8
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → ¬ 𝑟 ≤ 0) |
16 | | simpllr 776 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → (𝐹‘𝐴) = 0) |
17 | 16 | breq2d 5080 |
. . . . . . . . . . 11
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → (𝑟 ≤ (𝐹‘𝐴) ↔ 𝑟 ≤ 0)) |
18 | 1 | adantr 484 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → 𝐷 ∈ (∞Met‘𝑋)) |
19 | | simpl2 1194 |
. . . . . . . . . . . . 13
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) → 𝑆 ⊆ 𝑋) |
20 | 19 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → 𝑆 ⊆ 𝑋) |
21 | | simpl3 1195 |
. . . . . . . . . . . . 13
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) → 𝐴 ∈ 𝑋) |
22 | 21 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → 𝐴 ∈ 𝑋) |
23 | | rpxr 12620 |
. . . . . . . . . . . . 13
⊢ (𝑟 ∈ ℝ+
→ 𝑟 ∈
ℝ*) |
24 | 23 | ad2antrl 728 |
. . . . . . . . . . . 12
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → 𝑟 ∈ ℝ*) |
25 | | metdscn.f |
. . . . . . . . . . . . 13
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, <
)) |
26 | 25 | metdsge 23773 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝑟 ∈ ℝ*) → (𝑟 ≤ (𝐹‘𝐴) ↔ (𝑆 ∩ (𝐴(ball‘𝐷)𝑟)) = ∅)) |
27 | 18, 20, 22, 24, 26 | syl31anc 1375 |
. . . . . . . . . . 11
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → (𝑟 ≤ (𝐹‘𝐴) ↔ (𝑆 ∩ (𝐴(ball‘𝐷)𝑟)) = ∅)) |
28 | 17, 27 | bitr3d 284 |
. . . . . . . . . 10
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → (𝑟 ≤ 0 ↔ (𝑆 ∩ (𝐴(ball‘𝐷)𝑟)) = ∅)) |
29 | | incom 4130 |
. . . . . . . . . . 11
⊢ (𝑆 ∩ (𝐴(ball‘𝐷)𝑟)) = ((𝐴(ball‘𝐷)𝑟) ∩ 𝑆) |
30 | 29 | eqeq1i 2743 |
. . . . . . . . . 10
⊢ ((𝑆 ∩ (𝐴(ball‘𝐷)𝑟)) = ∅ ↔ ((𝐴(ball‘𝐷)𝑟) ∩ 𝑆) = ∅) |
31 | 28, 30 | bitrdi 290 |
. . . . . . . . 9
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → (𝑟 ≤ 0 ↔ ((𝐴(ball‘𝐷)𝑟) ∩ 𝑆) = ∅)) |
32 | 31 | necon3bbid 2979 |
. . . . . . . 8
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → (¬ 𝑟 ≤ 0 ↔ ((𝐴(ball‘𝐷)𝑟) ∩ 𝑆) ≠ ∅)) |
33 | 15, 32 | mpbid 235 |
. . . . . . 7
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → ((𝐴(ball‘𝐷)𝑟) ∩ 𝑆) ≠ ∅) |
34 | | ssn0 4330 |
. . . . . . 7
⊢ ((((𝐴(ball‘𝐷)𝑟) ∩ 𝑆) ⊆ (𝑧 ∩ 𝑆) ∧ ((𝐴(ball‘𝐷)𝑟) ∩ 𝑆) ≠ ∅) → (𝑧 ∩ 𝑆) ≠ ∅) |
35 | 8, 33, 34 | syl2anc 587 |
. . . . . 6
⊢
(((((𝐷 ∈
(∞Met‘𝑋) ∧
𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) ∧ (𝑟 ∈ ℝ+ ∧ (𝐴(ball‘𝐷)𝑟) ⊆ 𝑧)) → (𝑧 ∩ 𝑆) ≠ ∅) |
36 | 6, 35 | rexlimddv 3218 |
. . . . 5
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ (𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧)) → (𝑧 ∩ 𝑆) ≠ ∅) |
37 | 36 | expr 460 |
. . . 4
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) ∧ 𝑧 ∈ 𝐽) → (𝐴 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)) |
38 | 37 | ralrimiva 3106 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) → ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅)) |
39 | 4 | mopntopon 23364 |
. . . . . . 7
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
40 | 39 | 3ad2ant1 1135 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
41 | 40 | adantr 484 |
. . . . 5
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) → 𝐽 ∈ (TopOn‘𝑋)) |
42 | | topontop 21837 |
. . . . 5
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
43 | 41, 42 | syl 17 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) → 𝐽 ∈ Top) |
44 | | toponuni 21838 |
. . . . . 6
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
45 | 41, 44 | syl 17 |
. . . . 5
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) → 𝑋 = ∪ 𝐽) |
46 | 19, 45 | sseqtrd 3956 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) → 𝑆 ⊆ ∪ 𝐽) |
47 | 21, 45 | eleqtrd 2841 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) → 𝐴 ∈ ∪ 𝐽) |
48 | | eqid 2738 |
. . . . 5
⊢ ∪ 𝐽 =
∪ 𝐽 |
49 | 48 | elcls 21997 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽
∧ 𝐴 ∈ ∪ 𝐽)
→ (𝐴 ∈
((cls‘𝐽)‘𝑆) ↔ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅))) |
50 | 43, 46, 47, 49 | syl3anc 1373 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) → (𝐴 ∈ ((cls‘𝐽)‘𝑆) ↔ ∀𝑧 ∈ 𝐽 (𝐴 ∈ 𝑧 → (𝑧 ∩ 𝑆) ≠ ∅))) |
51 | 38, 50 | mpbird 260 |
. 2
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) = 0) → 𝐴 ∈ ((cls‘𝐽)‘𝑆)) |
52 | | incom 4130 |
. . . . . . 7
⊢ ((𝐴(ball‘𝐷)(𝐹‘𝐴)) ∩ 𝑆) = (𝑆 ∩ (𝐴(ball‘𝐷)(𝐹‘𝐴))) |
53 | 25 | metdsf 23772 |
. . . . . . . . . . . 12
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) |
54 | 53 | ffvelrnda 6923 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ (0[,]+∞)) |
55 | 54 | 3impa 1112 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ (0[,]+∞)) |
56 | | eliccxr 13048 |
. . . . . . . . . 10
⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) → (𝐹‘𝐴) ∈
ℝ*) |
57 | 55, 56 | syl 17 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈
ℝ*) |
58 | 57 | xrleidd 12767 |
. . . . . . . 8
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ≤ (𝐹‘𝐴)) |
59 | 25 | metdsge 23773 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ (𝐹‘𝐴) ∈ ℝ*) → ((𝐹‘𝐴) ≤ (𝐹‘𝐴) ↔ (𝑆 ∩ (𝐴(ball‘𝐷)(𝐹‘𝐴))) = ∅)) |
60 | 57, 59 | mpdan 687 |
. . . . . . . 8
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴) ≤ (𝐹‘𝐴) ↔ (𝑆 ∩ (𝐴(ball‘𝐷)(𝐹‘𝐴))) = ∅)) |
61 | 58, 60 | mpbid 235 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝑆 ∩ (𝐴(ball‘𝐷)(𝐹‘𝐴))) = ∅) |
62 | 52, 61 | syl5eq 2791 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐴(ball‘𝐷)(𝐹‘𝐴)) ∩ 𝑆) = ∅) |
63 | 62 | adantr 484 |
. . . . 5
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((𝐴(ball‘𝐷)(𝐹‘𝐴)) ∩ 𝑆) = ∅) |
64 | 40 | ad2antrr 726 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → 𝐽 ∈ (TopOn‘𝑋)) |
65 | 64, 42 | syl 17 |
. . . . . . . 8
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → 𝐽 ∈ Top) |
66 | | simpll2 1215 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → 𝑆 ⊆ 𝑋) |
67 | 64, 44 | syl 17 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → 𝑋 = ∪ 𝐽) |
68 | 66, 67 | sseqtrd 3956 |
. . . . . . . 8
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → 𝑆 ⊆ ∪ 𝐽) |
69 | | simplr 769 |
. . . . . . . 8
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → 𝐴 ∈ ((cls‘𝐽)‘𝑆)) |
70 | | simpll1 1214 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → 𝐷 ∈ (∞Met‘𝑋)) |
71 | | simpll3 1216 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → 𝐴 ∈ 𝑋) |
72 | 57 | ad2antrr 726 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → (𝐹‘𝐴) ∈
ℝ*) |
73 | 4 | blopn 23425 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ (𝐹‘𝐴) ∈ ℝ*) → (𝐴(ball‘𝐷)(𝐹‘𝐴)) ∈ 𝐽) |
74 | 70, 71, 72, 73 | syl3anc 1373 |
. . . . . . . 8
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → (𝐴(ball‘𝐷)(𝐹‘𝐴)) ∈ 𝐽) |
75 | | simpr 488 |
. . . . . . . . 9
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → 0 < (𝐹‘𝐴)) |
76 | | xblcntr 23336 |
. . . . . . . . 9
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ ((𝐹‘𝐴) ∈ ℝ* ∧ 0 <
(𝐹‘𝐴))) → 𝐴 ∈ (𝐴(ball‘𝐷)(𝐹‘𝐴))) |
77 | 70, 71, 72, 75, 76 | syl112anc 1376 |
. . . . . . . 8
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → 𝐴 ∈ (𝐴(ball‘𝐷)(𝐹‘𝐴))) |
78 | 48 | clsndisj 21999 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽
∧ 𝐴 ∈
((cls‘𝐽)‘𝑆)) ∧ ((𝐴(ball‘𝐷)(𝐹‘𝐴)) ∈ 𝐽 ∧ 𝐴 ∈ (𝐴(ball‘𝐷)(𝐹‘𝐴)))) → ((𝐴(ball‘𝐷)(𝐹‘𝐴)) ∩ 𝑆) ≠ ∅) |
79 | 65, 68, 69, 74, 77, 78 | syl32anc 1380 |
. . . . . . 7
⊢ ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 0 < (𝐹‘𝐴)) → ((𝐴(ball‘𝐷)(𝐹‘𝐴)) ∩ 𝑆) ≠ ∅) |
80 | 79 | ex 416 |
. . . . . 6
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (0 < (𝐹‘𝐴) → ((𝐴(ball‘𝐷)(𝐹‘𝐴)) ∩ 𝑆) ≠ ∅)) |
81 | 80 | necon2bd 2957 |
. . . . 5
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((𝐴(ball‘𝐷)(𝐹‘𝐴)) ∩ 𝑆) = ∅ → ¬ 0 < (𝐹‘𝐴))) |
82 | 63, 81 | mpd 15 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ¬ 0 < (𝐹‘𝐴)) |
83 | | elxrge0 13070 |
. . . . . . . . 9
⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) ↔ ((𝐹‘𝐴) ∈ ℝ* ∧ 0 ≤
(𝐹‘𝐴))) |
84 | 83 | simprbi 500 |
. . . . . . . 8
⊢ ((𝐹‘𝐴) ∈ (0[,]+∞) → 0 ≤ (𝐹‘𝐴)) |
85 | 55, 84 | syl 17 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝐹‘𝐴)) |
86 | | 0xr 10905 |
. . . . . . . 8
⊢ 0 ∈
ℝ* |
87 | | xrleloe 12759 |
. . . . . . . 8
⊢ ((0
∈ ℝ* ∧ (𝐹‘𝐴) ∈ ℝ*) → (0 ≤
(𝐹‘𝐴) ↔ (0 < (𝐹‘𝐴) ∨ 0 = (𝐹‘𝐴)))) |
88 | 86, 57, 87 | sylancr 590 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (0 ≤ (𝐹‘𝐴) ↔ (0 < (𝐹‘𝐴) ∨ 0 = (𝐹‘𝐴)))) |
89 | 85, 88 | mpbid 235 |
. . . . . 6
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → (0 < (𝐹‘𝐴) ∨ 0 = (𝐹‘𝐴))) |
90 | 89 | adantr 484 |
. . . . 5
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (0 < (𝐹‘𝐴) ∨ 0 = (𝐹‘𝐴))) |
91 | 90 | ord 864 |
. . . 4
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (¬ 0 < (𝐹‘𝐴) → 0 = (𝐹‘𝐴))) |
92 | 82, 91 | mpd 15 |
. . 3
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 0 = (𝐹‘𝐴)) |
93 | 92 | eqcomd 2744 |
. 2
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹‘𝐴) = 0) |
94 | 51, 93 | impbida 801 |
1
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝐴) = 0 ↔ 𝐴 ∈ ((cls‘𝐽)‘𝑆))) |