| Step |
Hyp |
Ref |
Expression |
| 1 |
|
methaus.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
| 2 |
1
|
mopntop |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ Top ) |
| 3 |
2
|
adantr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → 𝐽 ∈ Top ) |
| 4 |
|
simpll |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 5 |
|
simplr1 |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴 ) ) → 𝐴 ⊆ 𝑋 ) |
| 6 |
|
simprr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴 ) ) → 𝑦 ∈ 𝐴 ) |
| 7 |
5 6
|
sseldd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴 ) ) → 𝑦 ∈ 𝑋 ) |
| 8 |
|
simprl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴 ) ) → 𝑥 ∈ ℕ ) |
| 9 |
8
|
nnrpd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴 ) ) → 𝑥 ∈ ℝ+ ) |
| 10 |
9
|
rpreccld |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴 ) ) → ( 1 / 𝑥 ) ∈ ℝ+ ) |
| 11 |
10
|
rpxrd |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴 ) ) → ( 1 / 𝑥 ) ∈ ℝ* ) |
| 12 |
1
|
blopn |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑦 ∈ 𝑋 ∧ ( 1 / 𝑥 ) ∈ ℝ* ) → ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ∈ 𝐽 ) |
| 13 |
4 7 11 12
|
syl3anc |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ∈ 𝐽 ) |
| 14 |
13
|
ralrimivva |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ∈ 𝐽 ) |
| 15 |
|
eqid |
⊢ ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) = ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) |
| 16 |
15
|
fmpo |
⊢ ( ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ∈ 𝐽 ↔ ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) : ( ℕ × 𝐴 ) ⟶ 𝐽 ) |
| 17 |
14 16
|
sylib |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) : ( ℕ × 𝐴 ) ⟶ 𝐽 ) |
| 18 |
17
|
frnd |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ⊆ 𝐽 ) |
| 19 |
|
simpll |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 20 |
|
simprl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) → 𝑢 ∈ 𝐽 ) |
| 21 |
|
simprr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) → 𝑧 ∈ 𝑢 ) |
| 22 |
1
|
mopni2 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) → ∃ 𝑟 ∈ ℝ+ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) |
| 23 |
19 20 21 22
|
syl3anc |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) → ∃ 𝑟 ∈ ℝ+ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) |
| 24 |
|
simprl |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ) → 𝑟 ∈ ℝ+ ) |
| 25 |
24
|
rphalfcld |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ) → ( 𝑟 / 2 ) ∈ ℝ+ ) |
| 26 |
|
elrp |
⊢ ( ( 𝑟 / 2 ) ∈ ℝ+ ↔ ( ( 𝑟 / 2 ) ∈ ℝ ∧ 0 < ( 𝑟 / 2 ) ) ) |
| 27 |
|
nnrecl |
⊢ ( ( ( 𝑟 / 2 ) ∈ ℝ ∧ 0 < ( 𝑟 / 2 ) ) → ∃ 𝑛 ∈ ℕ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) |
| 28 |
26 27
|
sylbi |
⊢ ( ( 𝑟 / 2 ) ∈ ℝ+ → ∃ 𝑛 ∈ ℕ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) |
| 29 |
25 28
|
syl |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ) → ∃ 𝑛 ∈ ℕ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) |
| 30 |
3
|
ad2antrr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝐽 ∈ Top ) |
| 31 |
|
simpr1 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → 𝐴 ⊆ 𝑋 ) |
| 32 |
31
|
ad2antrr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝐴 ⊆ 𝑋 ) |
| 33 |
1
|
mopnuni |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
| 34 |
33
|
ad3antrrr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝑋 = ∪ 𝐽 ) |
| 35 |
32 34
|
sseqtrd |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝐴 ⊆ ∪ 𝐽 ) |
| 36 |
|
simplrr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝑧 ∈ 𝑢 ) |
| 37 |
|
simplrl |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝑢 ∈ 𝐽 ) |
| 38 |
|
elunii |
⊢ ( ( 𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝐽 ) → 𝑧 ∈ ∪ 𝐽 ) |
| 39 |
36 37 38
|
syl2anc |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝑧 ∈ ∪ 𝐽 ) |
| 40 |
39 34
|
eleqtrrd |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝑧 ∈ 𝑋 ) |
| 41 |
|
simpr3 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) |
| 42 |
41
|
ad2antrr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) |
| 43 |
40 42
|
eleqtrrd |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝑧 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) |
| 44 |
19
|
adantr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 45 |
|
simprrl |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝑛 ∈ ℕ ) |
| 46 |
45
|
nnrpd |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝑛 ∈ ℝ+ ) |
| 47 |
46
|
rpreccld |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ( 1 / 𝑛 ) ∈ ℝ+ ) |
| 48 |
47
|
rpxrd |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ( 1 / 𝑛 ) ∈ ℝ* ) |
| 49 |
1
|
blopn |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ 𝑋 ∧ ( 1 / 𝑛 ) ∈ ℝ* ) → ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∈ 𝐽 ) |
| 50 |
44 40 48 49
|
syl3anc |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∈ 𝐽 ) |
| 51 |
|
blcntr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑧 ∈ 𝑋 ∧ ( 1 / 𝑛 ) ∈ ℝ+ ) → 𝑧 ∈ ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) |
| 52 |
44 40 47 51
|
syl3anc |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝑧 ∈ ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) |
| 53 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
| 54 |
53
|
clsndisj |
⊢ ( ( ( 𝐽 ∈ Top ∧ 𝐴 ⊆ ∪ 𝐽 ∧ 𝑧 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) ) ∧ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∈ 𝐽 ∧ 𝑧 ∈ ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ) → ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ≠ ∅ ) |
| 55 |
30 35 43 50 52 54
|
syl32anc |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ≠ ∅ ) |
| 56 |
|
n0 |
⊢ ( ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ≠ ∅ ↔ ∃ 𝑡 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) |
| 57 |
55 56
|
sylib |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ∃ 𝑡 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) |
| 58 |
45
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝑛 ∈ ℕ ) |
| 59 |
|
simpr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) |
| 60 |
59
|
elin2d |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝑡 ∈ 𝐴 ) |
| 61 |
|
eqidd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) |
| 62 |
|
oveq2 |
⊢ ( 𝑥 = 𝑛 → ( 1 / 𝑥 ) = ( 1 / 𝑛 ) ) |
| 63 |
62
|
oveq2d |
⊢ ( 𝑥 = 𝑛 → ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) |
| 64 |
63
|
eqeq2d |
⊢ ( 𝑥 = 𝑛 → ( ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ↔ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ) |
| 65 |
|
oveq1 |
⊢ ( 𝑦 = 𝑡 → ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) |
| 66 |
65
|
eqeq2d |
⊢ ( 𝑦 = 𝑡 → ( ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ↔ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ) |
| 67 |
64 66
|
rspc2ev |
⊢ ( ( 𝑛 ∈ ℕ ∧ 𝑡 ∈ 𝐴 ∧ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) → ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ 𝐴 ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) |
| 68 |
58 60 61 67
|
syl3anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ 𝐴 ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) |
| 69 |
|
ovex |
⊢ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∈ V |
| 70 |
|
eqeq1 |
⊢ ( 𝑧 = ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) → ( 𝑧 = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ↔ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) |
| 71 |
70
|
2rexbidv |
⊢ ( 𝑧 = ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) → ( ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ 𝐴 𝑧 = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ↔ ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ 𝐴 ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) |
| 72 |
15
|
rnmpo |
⊢ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) = { 𝑧 ∣ ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ 𝐴 𝑧 = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) } |
| 73 |
69 71 72
|
elab2 |
⊢ ( ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ↔ ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ 𝐴 ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) = ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) |
| 74 |
68 73
|
sylibr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) |
| 75 |
59
|
elin1d |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝑡 ∈ ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) |
| 76 |
44
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 77 |
48
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 1 / 𝑛 ) ∈ ℝ* ) |
| 78 |
40
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝑧 ∈ 𝑋 ) |
| 79 |
32
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝐴 ⊆ 𝑋 ) |
| 80 |
79 60
|
sseldd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝑡 ∈ 𝑋 ) |
| 81 |
|
blcom |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 1 / 𝑛 ) ∈ ℝ* ) ∧ ( 𝑧 ∈ 𝑋 ∧ 𝑡 ∈ 𝑋 ) ) → ( 𝑡 ∈ ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ↔ 𝑧 ∈ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ) |
| 82 |
76 77 78 80 81
|
syl22anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 𝑡 ∈ ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ↔ 𝑧 ∈ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ) |
| 83 |
75 82
|
mpbid |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝑧 ∈ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) |
| 84 |
|
simprll |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → 𝑟 ∈ ℝ+ ) |
| 85 |
84
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝑟 ∈ ℝ+ ) |
| 86 |
85
|
rphalfcld |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 𝑟 / 2 ) ∈ ℝ+ ) |
| 87 |
86
|
rpxrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 𝑟 / 2 ) ∈ ℝ* ) |
| 88 |
|
simprrr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) |
| 89 |
84
|
rphalfcld |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ( 𝑟 / 2 ) ∈ ℝ+ ) |
| 90 |
|
rpre |
⊢ ( ( 1 / 𝑛 ) ∈ ℝ+ → ( 1 / 𝑛 ) ∈ ℝ ) |
| 91 |
|
rpre |
⊢ ( ( 𝑟 / 2 ) ∈ ℝ+ → ( 𝑟 / 2 ) ∈ ℝ ) |
| 92 |
|
ltle |
⊢ ( ( ( 1 / 𝑛 ) ∈ ℝ ∧ ( 𝑟 / 2 ) ∈ ℝ ) → ( ( 1 / 𝑛 ) < ( 𝑟 / 2 ) → ( 1 / 𝑛 ) ≤ ( 𝑟 / 2 ) ) ) |
| 93 |
90 91 92
|
syl2an |
⊢ ( ( ( 1 / 𝑛 ) ∈ ℝ+ ∧ ( 𝑟 / 2 ) ∈ ℝ+ ) → ( ( 1 / 𝑛 ) < ( 𝑟 / 2 ) → ( 1 / 𝑛 ) ≤ ( 𝑟 / 2 ) ) ) |
| 94 |
47 89 93
|
syl2anc |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ( ( 1 / 𝑛 ) < ( 𝑟 / 2 ) → ( 1 / 𝑛 ) ≤ ( 𝑟 / 2 ) ) ) |
| 95 |
88 94
|
mpd |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ( 1 / 𝑛 ) ≤ ( 𝑟 / 2 ) ) |
| 96 |
95
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 1 / 𝑛 ) ≤ ( 𝑟 / 2 ) ) |
| 97 |
|
ssbl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑡 ∈ 𝑋 ) ∧ ( ( 1 / 𝑛 ) ∈ ℝ* ∧ ( 𝑟 / 2 ) ∈ ℝ* ) ∧ ( 1 / 𝑛 ) ≤ ( 𝑟 / 2 ) ) → ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ⊆ ( 𝑡 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 98 |
76 80 77 87 96 97
|
syl221anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ⊆ ( 𝑡 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 99 |
85
|
rpred |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝑟 ∈ ℝ ) |
| 100 |
98 83
|
sseldd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → 𝑧 ∈ ( 𝑡 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 101 |
|
blhalf |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑡 ∈ 𝑋 ) ∧ ( 𝑟 ∈ ℝ ∧ 𝑧 ∈ ( 𝑡 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) → ( 𝑡 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ) |
| 102 |
76 80 99 100 101
|
syl22anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 𝑡 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ) |
| 103 |
|
simprlr |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) |
| 104 |
103
|
adantr |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) |
| 105 |
102 104
|
sstrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 𝑡 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ 𝑢 ) |
| 106 |
98 105
|
sstrd |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ⊆ 𝑢 ) |
| 107 |
|
eleq2 |
⊢ ( 𝑤 = ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) → ( 𝑧 ∈ 𝑤 ↔ 𝑧 ∈ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ) ) |
| 108 |
|
sseq1 |
⊢ ( 𝑤 = ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) → ( 𝑤 ⊆ 𝑢 ↔ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ⊆ 𝑢 ) ) |
| 109 |
107 108
|
anbi12d |
⊢ ( 𝑤 = ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) → ( ( 𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢 ) ↔ ( 𝑧 ∈ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∧ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ⊆ 𝑢 ) ) ) |
| 110 |
109
|
rspcev |
⊢ ( ( ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ∧ ( 𝑧 ∈ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∧ ( 𝑡 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ⊆ 𝑢 ) ) → ∃ 𝑤 ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ( 𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢 ) ) |
| 111 |
74 83 106 110
|
syl12anc |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) ∧ 𝑡 ∈ ( ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / 𝑛 ) ) ∩ 𝐴 ) ) → ∃ 𝑤 ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ( 𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢 ) ) |
| 112 |
57 111
|
exlimddv |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) ) → ∃ 𝑤 ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ( 𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢 ) ) |
| 113 |
112
|
anassrs |
⊢ ( ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ) ∧ ( 𝑛 ∈ ℕ ∧ ( 1 / 𝑛 ) < ( 𝑟 / 2 ) ) ) → ∃ 𝑤 ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ( 𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢 ) ) |
| 114 |
29 113
|
rexlimddv |
⊢ ( ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑧 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑢 ) ) → ∃ 𝑤 ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ( 𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢 ) ) |
| 115 |
23 114
|
rexlimddv |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) ∧ ( 𝑢 ∈ 𝐽 ∧ 𝑧 ∈ 𝑢 ) ) → ∃ 𝑤 ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ( 𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢 ) ) |
| 116 |
115
|
ralrimivva |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ∀ 𝑢 ∈ 𝐽 ∀ 𝑧 ∈ 𝑢 ∃ 𝑤 ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ( 𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢 ) ) |
| 117 |
|
basgen2 |
⊢ ( ( 𝐽 ∈ Top ∧ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ⊆ 𝐽 ∧ ∀ 𝑢 ∈ 𝐽 ∀ 𝑧 ∈ 𝑢 ∃ 𝑤 ∈ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ( 𝑧 ∈ 𝑤 ∧ 𝑤 ⊆ 𝑢 ) ) → ( topGen ‘ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) = 𝐽 ) |
| 118 |
3 18 116 117
|
syl3anc |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ( topGen ‘ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) = 𝐽 ) |
| 119 |
118 3
|
eqeltrd |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ( topGen ‘ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) ∈ Top ) |
| 120 |
|
tgclb |
⊢ ( ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ∈ TopBases ↔ ( topGen ‘ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) ∈ Top ) |
| 121 |
119 120
|
sylibr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ∈ TopBases ) |
| 122 |
|
omelon |
⊢ ω ∈ On |
| 123 |
|
simpr2 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → 𝐴 ≼ ω ) |
| 124 |
|
nnex |
⊢ ℕ ∈ V |
| 125 |
124
|
xpdom2 |
⊢ ( 𝐴 ≼ ω → ( ℕ × 𝐴 ) ≼ ( ℕ × ω ) ) |
| 126 |
123 125
|
syl |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ( ℕ × 𝐴 ) ≼ ( ℕ × ω ) ) |
| 127 |
|
nnenom |
⊢ ℕ ≈ ω |
| 128 |
|
omex |
⊢ ω ∈ V |
| 129 |
128
|
enref |
⊢ ω ≈ ω |
| 130 |
|
xpen |
⊢ ( ( ℕ ≈ ω ∧ ω ≈ ω ) → ( ℕ × ω ) ≈ ( ω × ω ) ) |
| 131 |
127 129 130
|
mp2an |
⊢ ( ℕ × ω ) ≈ ( ω × ω ) |
| 132 |
|
xpomen |
⊢ ( ω × ω ) ≈ ω |
| 133 |
131 132
|
entri |
⊢ ( ℕ × ω ) ≈ ω |
| 134 |
|
domentr |
⊢ ( ( ( ℕ × 𝐴 ) ≼ ( ℕ × ω ) ∧ ( ℕ × ω ) ≈ ω ) → ( ℕ × 𝐴 ) ≼ ω ) |
| 135 |
126 133 134
|
sylancl |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ( ℕ × 𝐴 ) ≼ ω ) |
| 136 |
|
ondomen |
⊢ ( ( ω ∈ On ∧ ( ℕ × 𝐴 ) ≼ ω ) → ( ℕ × 𝐴 ) ∈ dom card ) |
| 137 |
122 135 136
|
sylancr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ( ℕ × 𝐴 ) ∈ dom card ) |
| 138 |
17
|
ffnd |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) Fn ( ℕ × 𝐴 ) ) |
| 139 |
|
dffn4 |
⊢ ( ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) Fn ( ℕ × 𝐴 ) ↔ ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) : ( ℕ × 𝐴 ) –onto→ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) |
| 140 |
138 139
|
sylib |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) : ( ℕ × 𝐴 ) –onto→ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) |
| 141 |
|
fodomnum |
⊢ ( ( ℕ × 𝐴 ) ∈ dom card → ( ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) : ( ℕ × 𝐴 ) –onto→ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) → ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ≼ ( ℕ × 𝐴 ) ) ) |
| 142 |
137 140 141
|
sylc |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ≼ ( ℕ × 𝐴 ) ) |
| 143 |
|
domtr |
⊢ ( ( ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ≼ ( ℕ × 𝐴 ) ∧ ( ℕ × 𝐴 ) ≼ ω ) → ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ≼ ω ) |
| 144 |
142 135 143
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ≼ ω ) |
| 145 |
|
2ndci |
⊢ ( ( ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ∈ TopBases ∧ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ≼ ω ) → ( topGen ‘ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) ∈ 2ndω ) |
| 146 |
121 144 145
|
syl2anc |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → ( topGen ‘ ran ( 𝑥 ∈ ℕ , 𝑦 ∈ 𝐴 ↦ ( 𝑦 ( ball ‘ 𝐷 ) ( 1 / 𝑥 ) ) ) ) ∈ 2ndω ) |
| 147 |
118 146
|
eqeltrrd |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐴 ⊆ 𝑋 ∧ 𝐴 ≼ ω ∧ ( ( cls ‘ 𝐽 ) ‘ 𝐴 ) = 𝑋 ) ) → 𝐽 ∈ 2ndω ) |