| Step |
Hyp |
Ref |
Expression |
| 1 |
|
metdscn.f |
⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ inf ( ran ( 𝑦 ∈ 𝑆 ↦ ( 𝑥 𝐷 𝑦 ) ) , ℝ* , < ) ) |
| 2 |
|
metdscn.j |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
| 3 |
|
simpll1 |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 4 |
|
simprl |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) → 𝑧 ∈ 𝐽 ) |
| 5 |
|
simprr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) → 𝐴 ∈ 𝑧 ) |
| 6 |
2
|
mopni2 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) → ∃ 𝑟 ∈ ℝ+ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) |
| 7 |
3 4 5 6
|
syl3anc |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) → ∃ 𝑟 ∈ ℝ+ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) |
| 8 |
|
simprr |
⊢ ( ( ( ( ( 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 ) |
| 9 |
8
|
ssrind |
⊢ ( ( ( ( ( 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 ) ) |
| 10 |
|
rpgt0 |
⊢ ( 𝑟 ∈ ℝ+ → 0 < 𝑟 ) |
| 11 |
|
0re |
⊢ 0 ∈ ℝ |
| 12 |
|
rpre |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ ) |
| 13 |
|
ltnle |
⊢ ( ( 0 ∈ ℝ ∧ 𝑟 ∈ ℝ ) → ( 0 < 𝑟 ↔ ¬ 𝑟 ≤ 0 ) ) |
| 14 |
11 12 13
|
sylancr |
⊢ ( 𝑟 ∈ ℝ+ → ( 0 < 𝑟 ↔ ¬ 𝑟 ≤ 0 ) ) |
| 15 |
10 14
|
mpbid |
⊢ ( 𝑟 ∈ ℝ+ → ¬ 𝑟 ≤ 0 ) |
| 16 |
15
|
ad2antrl |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) → ¬ 𝑟 ≤ 0 ) |
| 17 |
|
simpllr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) → ( 𝐹 ‘ 𝐴 ) = 0 ) |
| 18 |
17
|
breq2d |
⊢ ( ( ( ( ( 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 ) ) |
| 19 |
3
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 20 |
|
simpl2 |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) → 𝑆 ⊆ 𝑋 ) |
| 21 |
20
|
ad2antrr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) → 𝑆 ⊆ 𝑋 ) |
| 22 |
|
simpl3 |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) → 𝐴 ∈ 𝑋 ) |
| 23 |
22
|
ad2antrr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) → 𝐴 ∈ 𝑋 ) |
| 24 |
|
rpxr |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ* ) |
| 25 |
24
|
ad2antrl |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) → 𝑟 ∈ ℝ* ) |
| 26 |
1
|
metdsge |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝑟 ∈ ℝ* ) → ( 𝑟 ≤ ( 𝐹 ‘ 𝐴 ) ↔ ( 𝑆 ∩ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ) = ∅ ) ) |
| 27 |
19 21 23 25 26
|
syl31anc |
⊢ ( ( ( ( ( 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 |
18 27
|
bitr3d |
⊢ ( ( ( ( ( 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 |
⊢ ( 𝑆 ∩ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ) = ( ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ∩ 𝑆 ) |
| 30 |
29
|
eqeq1i |
⊢ ( ( 𝑆 ∩ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ) = ∅ ↔ ( ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ∩ 𝑆 ) = ∅ ) |
| 31 |
28 30
|
bitrdi |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( ( 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 |
16 32
|
mpbid |
⊢ ( ( ( ( ( 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 |
⊢ ( ( ( ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ∩ 𝑆 ) ⊆ ( 𝑧 ∩ 𝑆 ) ∧ ( ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ∩ 𝑆 ) ≠ ∅ ) → ( 𝑧 ∩ 𝑆 ) ≠ ∅ ) |
| 35 |
9 33 34
|
syl2anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝐴 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑧 ) ) → ( 𝑧 ∩ 𝑆 ) ≠ ∅ ) |
| 36 |
7 35
|
rexlimddv |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ ( 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑧 ) ) → ( 𝑧 ∩ 𝑆 ) ≠ ∅ ) |
| 37 |
36
|
expr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) ∧ 𝑧 ∈ 𝐽 ) → ( 𝐴 ∈ 𝑧 → ( 𝑧 ∩ 𝑆 ) ≠ ∅ ) ) |
| 38 |
37
|
ralrimiva |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) → ∀ 𝑧 ∈ 𝐽 ( 𝐴 ∈ 𝑧 → ( 𝑧 ∩ 𝑆 ) ≠ ∅ ) ) |
| 39 |
2
|
mopntopon |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 40 |
39
|
3ad2ant1 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 41 |
40
|
adantr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 42 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
| 43 |
41 42
|
syl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) → 𝐽 ∈ Top ) |
| 44 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
| 45 |
41 44
|
syl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) → 𝑋 = ∪ 𝐽 ) |
| 46 |
20 45
|
sseqtrd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) → 𝑆 ⊆ ∪ 𝐽 ) |
| 47 |
22 45
|
eleqtrd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) → 𝐴 ∈ ∪ 𝐽 ) |
| 48 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
| 49 |
48
|
elcls |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ 𝐴 ∈ ∪ 𝐽 ) → ( 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↔ ∀ 𝑧 ∈ 𝐽 ( 𝐴 ∈ 𝑧 → ( 𝑧 ∩ 𝑆 ) ≠ ∅ ) ) ) |
| 50 |
43 46 47 49
|
syl3anc |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) → ( 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ↔ ∀ 𝑧 ∈ 𝐽 ( 𝐴 ∈ 𝑧 → ( 𝑧 ∩ 𝑆 ) ≠ ∅ ) ) ) |
| 51 |
38 50
|
mpbird |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) = 0 ) → 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
| 52 |
|
incom |
⊢ ( ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ∩ 𝑆 ) = ( 𝑆 ∩ ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ) |
| 53 |
1
|
metdsf |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → 𝐹 : 𝑋 ⟶ ( 0 [,] +∞ ) ) |
| 54 |
53
|
ffvelrnda |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 55 |
54
|
3impa |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ) |
| 56 |
|
eliccxr |
⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) → ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ) |
| 57 |
55 56
|
syl |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ) |
| 58 |
57
|
xrleidd |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐹 ‘ 𝐴 ) ≤ ( 𝐹 ‘ 𝐴 ) ) |
| 59 |
1
|
metdsge |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ) → ( ( 𝐹 ‘ 𝐴 ) ≤ ( 𝐹 ‘ 𝐴 ) ↔ ( 𝑆 ∩ ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ) = ∅ ) ) |
| 60 |
57 59
|
mpdan |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝐴 ) ≤ ( 𝐹 ‘ 𝐴 ) ↔ ( 𝑆 ∩ ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ) = ∅ ) ) |
| 61 |
58 60
|
mpbid |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 𝑆 ∩ ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ) = ∅ ) |
| 62 |
52 61
|
eqtrid |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ∩ 𝑆 ) = ∅ ) |
| 63 |
62
|
adantr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) → ( ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ∩ 𝑆 ) = ∅ ) |
| 64 |
40
|
ad2antrr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 65 |
64 42
|
syl |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → 𝐽 ∈ Top ) |
| 66 |
|
simpll2 |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → 𝑆 ⊆ 𝑋 ) |
| 67 |
64 44
|
syl |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → 𝑋 = ∪ 𝐽 ) |
| 68 |
66 67
|
sseqtrd |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → 𝑆 ⊆ ∪ 𝐽 ) |
| 69 |
|
simplr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) |
| 70 |
|
simpll1 |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 71 |
|
simpll3 |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → 𝐴 ∈ 𝑋 ) |
| 72 |
57
|
ad2antrr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ) |
| 73 |
2
|
blopn |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ) → ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ∈ 𝐽 ) |
| 74 |
70 71 72 73
|
syl3anc |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ∈ 𝐽 ) |
| 75 |
|
simpr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → 0 < ( 𝐹 ‘ 𝐴 ) ) |
| 76 |
|
xblcntr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐴 ∈ 𝑋 ∧ ( ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) ) → 𝐴 ∈ ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ) |
| 77 |
70 71 72 75 76
|
syl112anc |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → 𝐴 ∈ ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ) |
| 78 |
48
|
clsndisj |
⊢ ( ( ( 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 |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ∧ 0 < ( 𝐹 ‘ 𝐴 ) ) → ( ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ∩ 𝑆 ) ≠ ∅ ) |
| 80 |
79
|
ex |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) → ( 0 < ( 𝐹 ‘ 𝐴 ) → ( ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ∩ 𝑆 ) ≠ ∅ ) ) |
| 81 |
80
|
necon2bd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) → ( ( ( 𝐴 ( ball ‘ 𝐷 ) ( 𝐹 ‘ 𝐴 ) ) ∩ 𝑆 ) = ∅ → ¬ 0 < ( 𝐹 ‘ 𝐴 ) ) ) |
| 82 |
63 81
|
mpd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) → ¬ 0 < ( 𝐹 ‘ 𝐴 ) ) |
| 83 |
|
elxrge0 |
⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) ↔ ( ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ∧ 0 ≤ ( 𝐹 ‘ 𝐴 ) ) ) |
| 84 |
83
|
simprbi |
⊢ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 0 [,] +∞ ) → 0 ≤ ( 𝐹 ‘ 𝐴 ) ) |
| 85 |
55 84
|
syl |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → 0 ≤ ( 𝐹 ‘ 𝐴 ) ) |
| 86 |
|
0xr |
⊢ 0 ∈ ℝ* |
| 87 |
|
xrleloe |
⊢ ( ( 0 ∈ ℝ* ∧ ( 𝐹 ‘ 𝐴 ) ∈ ℝ* ) → ( 0 ≤ ( 𝐹 ‘ 𝐴 ) ↔ ( 0 < ( 𝐹 ‘ 𝐴 ) ∨ 0 = ( 𝐹 ‘ 𝐴 ) ) ) ) |
| 88 |
86 57 87
|
sylancr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 0 ≤ ( 𝐹 ‘ 𝐴 ) ↔ ( 0 < ( 𝐹 ‘ 𝐴 ) ∨ 0 = ( 𝐹 ‘ 𝐴 ) ) ) ) |
| 89 |
85 88
|
mpbid |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( 0 < ( 𝐹 ‘ 𝐴 ) ∨ 0 = ( 𝐹 ‘ 𝐴 ) ) ) |
| 90 |
89
|
adantr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) → ( 0 < ( 𝐹 ‘ 𝐴 ) ∨ 0 = ( 𝐹 ‘ 𝐴 ) ) ) |
| 91 |
90
|
ord |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) → ( ¬ 0 < ( 𝐹 ‘ 𝐴 ) → 0 = ( 𝐹 ‘ 𝐴 ) ) ) |
| 92 |
82 91
|
mpd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) → 0 = ( 𝐹 ‘ 𝐴 ) ) |
| 93 |
92
|
eqcomd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) ∧ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) → ( 𝐹 ‘ 𝐴 ) = 0 ) |
| 94 |
51 93
|
impbida |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐹 ‘ 𝐴 ) = 0 ↔ 𝐴 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑆 ) ) ) |