| Step |
Hyp |
Ref |
Expression |
| 1 |
|
iscmet3.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
| 2 |
|
iscmet3.2 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
| 3 |
|
iscmet3.3 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
| 4 |
|
iscmet3.4 |
⊢ ( 𝜑 → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
| 5 |
|
iscmet3.6 |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ 𝑋 ) |
| 6 |
|
iscmet3.9 |
⊢ ( 𝜑 → ∀ 𝑘 ∈ ℤ ∀ 𝑢 ∈ ( 𝑆 ‘ 𝑘 ) ∀ 𝑣 ∈ ( 𝑆 ‘ 𝑘 ) ( 𝑢 𝐷 𝑣 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) |
| 7 |
|
iscmet3.10 |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝑍 ∀ 𝑛 ∈ ( 𝑀 ... 𝑘 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑆 ‘ 𝑛 ) ) |
| 8 |
|
iscmet3.7 |
⊢ ( 𝜑 → 𝐺 ∈ ( Fil ‘ 𝑋 ) ) |
| 9 |
|
iscmet3.8 |
⊢ ( 𝜑 → 𝑆 : ℤ ⟶ 𝐺 ) |
| 10 |
|
iscmet3.5 |
⊢ ( 𝜑 → 𝐹 ∈ dom ( ⇝𝑡 ‘ 𝐽 ) ) |
| 11 |
|
eldmg |
⊢ ( 𝐹 ∈ dom ( ⇝𝑡 ‘ 𝐽 ) → ( 𝐹 ∈ dom ( ⇝𝑡 ‘ 𝐽 ) ↔ ∃ 𝑥 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ) |
| 12 |
11
|
ibi |
⊢ ( 𝐹 ∈ dom ( ⇝𝑡 ‘ 𝐽 ) → ∃ 𝑥 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) |
| 13 |
10 12
|
syl |
⊢ ( 𝜑 → ∃ 𝑥 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) |
| 14 |
|
metxmet |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 15 |
4 14
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 16 |
2
|
mopntopon |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 17 |
15 16
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 18 |
|
lmcl |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) → 𝑥 ∈ 𝑋 ) |
| 19 |
17 18
|
sylan |
⊢ ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) → 𝑥 ∈ 𝑋 ) |
| 20 |
15
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 21 |
2
|
mopni2 |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐽 ∧ 𝑥 ∈ 𝑦 ) → ∃ 𝑟 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) |
| 22 |
21
|
3expia |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐽 ) → ( 𝑥 ∈ 𝑦 → ∃ 𝑟 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) |
| 23 |
20 22
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑦 ∈ 𝐽 ) → ( 𝑥 ∈ 𝑦 → ∃ 𝑟 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) |
| 24 |
8
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑦 ∈ 𝐽 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → 𝐺 ∈ ( Fil ‘ 𝑋 ) ) |
| 25 |
3
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → 𝑀 ∈ ℤ ) |
| 26 |
|
rphalfcl |
⊢ ( 𝑟 ∈ ℝ+ → ( 𝑟 / 2 ) ∈ ℝ+ ) |
| 27 |
26
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑟 / 2 ) ∈ ℝ+ ) |
| 28 |
1
|
iscmet3lem3 |
⊢ ( ( 𝑀 ∈ ℤ ∧ ( 𝑟 / 2 ) ∈ ℝ+ ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ) |
| 29 |
25 27 28
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ) |
| 30 |
20
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 31 |
19
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → 𝑥 ∈ 𝑋 ) |
| 32 |
|
blcntr |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ ( 𝑟 / 2 ) ∈ ℝ+ ) → 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 33 |
30 31 27 32
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → 𝑥 ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 34 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) |
| 35 |
27
|
rpxrd |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑟 / 2 ) ∈ ℝ* ) |
| 36 |
2
|
blopn |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ ( 𝑟 / 2 ) ∈ ℝ* ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∈ 𝐽 ) |
| 37 |
30 31 35 36
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ∈ 𝐽 ) |
| 38 |
1 33 25 34 37
|
lmcvg |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 39 |
1
|
rexanuz2 |
⊢ ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ↔ ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 40 |
1
|
r19.2uz |
⊢ ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) → ∃ 𝑘 ∈ 𝑍 ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 41 |
8
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑘 ∈ 𝑍 ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) ) → 𝐺 ∈ ( Fil ‘ 𝑋 ) ) |
| 42 |
9
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑘 ∈ 𝑍 ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) ) → 𝑆 : ℤ ⟶ 𝐺 ) |
| 43 |
|
eluzelz |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑘 ∈ ℤ ) |
| 44 |
43 1
|
eleq2s |
⊢ ( 𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ ) |
| 45 |
44
|
ad2antrl |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑘 ∈ 𝑍 ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) ) → 𝑘 ∈ ℤ ) |
| 46 |
|
ffvelrn |
⊢ ( ( 𝑆 : ℤ ⟶ 𝐺 ∧ 𝑘 ∈ ℤ ) → ( 𝑆 ‘ 𝑘 ) ∈ 𝐺 ) |
| 47 |
42 45 46
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑘 ∈ 𝑍 ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) ) → ( 𝑆 ‘ 𝑘 ) ∈ 𝐺 ) |
| 48 |
|
rpxr |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ* ) |
| 49 |
48
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → 𝑟 ∈ ℝ* ) |
| 50 |
|
blssm |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝑥 ∈ 𝑋 ∧ 𝑟 ∈ ℝ* ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑋 ) |
| 51 |
30 31 49 50
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑋 ) |
| 52 |
51
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑘 ∈ 𝑍 ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑋 ) |
| 53 |
44
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → 𝑘 ∈ ℤ ) |
| 54 |
|
1rp |
⊢ 1 ∈ ℝ+ |
| 55 |
|
rphalfcl |
⊢ ( 1 ∈ ℝ+ → ( 1 / 2 ) ∈ ℝ+ ) |
| 56 |
54 55
|
ax-mp |
⊢ ( 1 / 2 ) ∈ ℝ+ |
| 57 |
|
rpexpcl |
⊢ ( ( ( 1 / 2 ) ∈ ℝ+ ∧ 𝑘 ∈ ℤ ) → ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ+ ) |
| 58 |
56 57
|
mpan |
⊢ ( 𝑘 ∈ ℤ → ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ+ ) |
| 59 |
53 58
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ+ ) |
| 60 |
59
|
rpred |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ ) |
| 61 |
27
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝑟 / 2 ) ∈ ℝ+ ) |
| 62 |
61
|
rpred |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝑟 / 2 ) ∈ ℝ ) |
| 63 |
|
ltle |
⊢ ( ( ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ ∧ ( 𝑟 / 2 ) ∈ ℝ ) → ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) → ( ( 1 / 2 ) ↑ 𝑘 ) ≤ ( 𝑟 / 2 ) ) ) |
| 64 |
60 62 63
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) → ( ( 1 / 2 ) ↑ 𝑘 ) ≤ ( 𝑟 / 2 ) ) ) |
| 65 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝑆 ‘ 𝑛 ) = ( 𝑆 ‘ 𝑘 ) ) |
| 66 |
65
|
eleq2d |
⊢ ( 𝑛 = 𝑘 → ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑆 ‘ 𝑛 ) ↔ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑆 ‘ 𝑘 ) ) ) |
| 67 |
7
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ∀ 𝑛 ∈ ( 𝑀 ... 𝑘 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑆 ‘ 𝑛 ) ) |
| 68 |
|
eluzfz2 |
⊢ ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑘 ∈ ( 𝑀 ... 𝑘 ) ) |
| 69 |
68 1
|
eleq2s |
⊢ ( 𝑘 ∈ 𝑍 → 𝑘 ∈ ( 𝑀 ... 𝑘 ) ) |
| 70 |
69
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝑘 ∈ ( 𝑀 ... 𝑘 ) ) |
| 71 |
66 67 70
|
rspcdva |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑆 ‘ 𝑘 ) ) |
| 72 |
71
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑆 ‘ 𝑘 ) ) |
| 73 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) |
| 74 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → ∀ 𝑘 ∈ ℤ ∀ 𝑢 ∈ ( 𝑆 ‘ 𝑘 ) ∀ 𝑣 ∈ ( 𝑆 ‘ 𝑘 ) ( 𝑢 𝐷 𝑣 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) |
| 75 |
44
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → 𝑘 ∈ ℤ ) |
| 76 |
|
rsp |
⊢ ( ∀ 𝑘 ∈ ℤ ∀ 𝑢 ∈ ( 𝑆 ‘ 𝑘 ) ∀ 𝑣 ∈ ( 𝑆 ‘ 𝑘 ) ( 𝑢 𝐷 𝑣 ) < ( ( 1 / 2 ) ↑ 𝑘 ) → ( 𝑘 ∈ ℤ → ∀ 𝑢 ∈ ( 𝑆 ‘ 𝑘 ) ∀ 𝑣 ∈ ( 𝑆 ‘ 𝑘 ) ( 𝑢 𝐷 𝑣 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) ) |
| 77 |
74 75 76
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → ∀ 𝑢 ∈ ( 𝑆 ‘ 𝑘 ) ∀ 𝑣 ∈ ( 𝑆 ‘ 𝑘 ) ( 𝑢 𝐷 𝑣 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) |
| 78 |
|
oveq1 |
⊢ ( 𝑢 = ( 𝐹 ‘ 𝑘 ) → ( 𝑢 𝐷 𝑣 ) = ( ( 𝐹 ‘ 𝑘 ) 𝐷 𝑣 ) ) |
| 79 |
78
|
breq1d |
⊢ ( 𝑢 = ( 𝐹 ‘ 𝑘 ) → ( ( 𝑢 𝐷 𝑣 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ↔ ( ( 𝐹 ‘ 𝑘 ) 𝐷 𝑣 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) ) |
| 80 |
|
oveq2 |
⊢ ( 𝑣 = 𝑦 → ( ( 𝐹 ‘ 𝑘 ) 𝐷 𝑣 ) = ( ( 𝐹 ‘ 𝑘 ) 𝐷 𝑦 ) ) |
| 81 |
80
|
breq1d |
⊢ ( 𝑣 = 𝑦 → ( ( ( 𝐹 ‘ 𝑘 ) 𝐷 𝑣 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ↔ ( ( 𝐹 ‘ 𝑘 ) 𝐷 𝑦 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) ) |
| 82 |
79 81
|
rspc2va |
⊢ ( ( ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑆 ‘ 𝑘 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) ∧ ∀ 𝑢 ∈ ( 𝑆 ‘ 𝑘 ) ∀ 𝑣 ∈ ( 𝑆 ‘ 𝑘 ) ( 𝑢 𝐷 𝑣 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) → ( ( 𝐹 ‘ 𝑘 ) 𝐷 𝑦 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) |
| 83 |
72 73 77 82
|
syl21anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → ( ( 𝐹 ‘ 𝑘 ) 𝐷 𝑦 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) |
| 84 |
15
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 85 |
44 58
|
syl |
⊢ ( 𝑘 ∈ 𝑍 → ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ+ ) |
| 86 |
85
|
rpxrd |
⊢ ( 𝑘 ∈ 𝑍 → ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ* ) |
| 87 |
86
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ* ) |
| 88 |
5
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) |
| 89 |
88
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) |
| 90 |
8
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 𝐺 ∈ ( Fil ‘ 𝑋 ) ) |
| 91 |
9 44 46
|
syl2an |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝑆 ‘ 𝑘 ) ∈ 𝐺 ) |
| 92 |
|
filelss |
⊢ ( ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ ( 𝑆 ‘ 𝑘 ) ∈ 𝐺 ) → ( 𝑆 ‘ 𝑘 ) ⊆ 𝑋 ) |
| 93 |
90 91 92
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝑆 ‘ 𝑘 ) ⊆ 𝑋 ) |
| 94 |
93
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → 𝑦 ∈ 𝑋 ) |
| 95 |
|
elbl2 |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ* ) ∧ ( ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ) ) → ( 𝑦 ∈ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ↔ ( ( 𝐹 ‘ 𝑘 ) 𝐷 𝑦 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) ) |
| 96 |
84 87 89 94 95
|
syl22anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → ( 𝑦 ∈ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ↔ ( ( 𝐹 ‘ 𝑘 ) 𝐷 𝑦 ) < ( ( 1 / 2 ) ↑ 𝑘 ) ) ) |
| 97 |
83 96
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) ∧ 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) ) → 𝑦 ∈ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ) |
| 98 |
97
|
ex |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝑦 ∈ ( 𝑆 ‘ 𝑘 ) → 𝑦 ∈ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ) ) |
| 99 |
98
|
ssrdv |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝑆 ‘ 𝑘 ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ) |
| 100 |
99
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝑆 ‘ 𝑘 ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ) |
| 101 |
30
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |
| 102 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → 𝐹 : 𝑍 ⟶ 𝑋 ) |
| 103 |
102
|
ffvelrnda |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) |
| 104 |
59
|
rpxrd |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ* ) |
| 105 |
35
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝑟 / 2 ) ∈ ℝ* ) |
| 106 |
|
ssbl |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ* ∧ ( 𝑟 / 2 ) ∈ ℝ* ) ∧ ( ( 1 / 2 ) ↑ 𝑘 ) ≤ ( 𝑟 / 2 ) ) → ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 107 |
106
|
3expia |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) ∈ ℝ* ∧ ( 𝑟 / 2 ) ∈ ℝ* ) ) → ( ( ( 1 / 2 ) ↑ 𝑘 ) ≤ ( 𝑟 / 2 ) → ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 108 |
101 103 104 105 107
|
syl22anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( ( 1 / 2 ) ↑ 𝑘 ) ≤ ( 𝑟 / 2 ) → ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 109 |
|
sstr |
⊢ ( ( ( 𝑆 ‘ 𝑘 ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ∧ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( ( 1 / 2 ) ↑ 𝑘 ) ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) → ( 𝑆 ‘ 𝑘 ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 110 |
100 108 109
|
syl6an |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( ( 1 / 2 ) ↑ 𝑘 ) ≤ ( 𝑟 / 2 ) → ( 𝑆 ‘ 𝑘 ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 111 |
64 110
|
syld |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) → ( 𝑆 ‘ 𝑘 ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 112 |
111
|
adantrd |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) → ( 𝑆 ‘ 𝑘 ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 113 |
112
|
impr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑘 ∈ 𝑍 ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) ) → ( 𝑆 ‘ 𝑘 ) ⊆ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) |
| 114 |
31
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → 𝑥 ∈ 𝑋 ) |
| 115 |
|
blcom |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝑟 / 2 ) ∈ ℝ* ) ∧ ( 𝑥 ∈ 𝑋 ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ↔ 𝑥 ∈ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 116 |
101 105 114 103 115
|
syl22anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ↔ 𝑥 ∈ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) |
| 117 |
|
rpre |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ ) |
| 118 |
117
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → 𝑟 ∈ ℝ ) |
| 119 |
|
blhalf |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) ∧ ( 𝑟 ∈ ℝ ∧ 𝑥 ∈ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) → ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) |
| 120 |
119
|
expr |
⊢ ( ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ 𝑋 ) ∧ 𝑟 ∈ ℝ ) → ( 𝑥 ∈ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) → ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) |
| 121 |
101 103 118 120
|
syl21anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝑥 ∈ ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) → ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) |
| 122 |
116 121
|
sylbid |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) → ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) |
| 123 |
122
|
adantld |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ 𝑘 ∈ 𝑍 ) → ( ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) |
| 124 |
123
|
impr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑘 ∈ 𝑍 ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) ) → ( ( 𝐹 ‘ 𝑘 ) ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) |
| 125 |
113 124
|
sstrd |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑘 ∈ 𝑍 ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) ) → ( 𝑆 ‘ 𝑘 ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) |
| 126 |
|
filss |
⊢ ( ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ ( ( 𝑆 ‘ 𝑘 ) ∈ 𝐺 ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑋 ∧ ( 𝑆 ‘ 𝑘 ) ⊆ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐺 ) |
| 127 |
41 47 52 125 126
|
syl13anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) ∧ ( 𝑘 ∈ 𝑍 ∧ ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐺 ) |
| 128 |
127
|
rexlimdvaa |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → ( ∃ 𝑘 ∈ 𝑍 ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐺 ) ) |
| 129 |
40 128
|
syl5 |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐺 ) ) |
| 130 |
39 129
|
syl5bir |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → ( ( ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( ( 1 / 2 ) ↑ 𝑘 ) < ( 𝑟 / 2 ) ∧ ∃ 𝑗 ∈ 𝑍 ∀ 𝑘 ∈ ( ℤ≥ ‘ 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑥 ( ball ‘ 𝐷 ) ( 𝑟 / 2 ) ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐺 ) ) |
| 131 |
29 38 130
|
mp2and |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑟 ∈ ℝ+ ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐺 ) |
| 132 |
131
|
ad2ant2r |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑦 ∈ 𝐽 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐺 ) |
| 133 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
| 134 |
|
toponss |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑦 ∈ 𝐽 ) → 𝑦 ⊆ 𝑋 ) |
| 135 |
133 134
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑦 ∈ 𝐽 ) → 𝑦 ⊆ 𝑋 ) |
| 136 |
135
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑦 ∈ 𝐽 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → 𝑦 ⊆ 𝑋 ) |
| 137 |
|
simprr |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑦 ∈ 𝐽 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) |
| 138 |
|
filss |
⊢ ( ( 𝐺 ∈ ( Fil ‘ 𝑋 ) ∧ ( ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ∈ 𝐺 ∧ 𝑦 ⊆ 𝑋 ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → 𝑦 ∈ 𝐺 ) |
| 139 |
24 132 136 137 138
|
syl13anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑦 ∈ 𝐽 ) ∧ ( 𝑟 ∈ ℝ+ ∧ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 ) ) → 𝑦 ∈ 𝐺 ) |
| 140 |
139
|
rexlimdvaa |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑦 ∈ 𝐽 ) → ( ∃ 𝑟 ∈ ℝ+ ( 𝑥 ( ball ‘ 𝐷 ) 𝑟 ) ⊆ 𝑦 → 𝑦 ∈ 𝐺 ) ) |
| 141 |
23 140
|
syld |
⊢ ( ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) ∧ 𝑦 ∈ 𝐽 ) → ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐺 ) ) |
| 142 |
141
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) → ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐺 ) ) |
| 143 |
|
flimopn |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐺 ∈ ( Fil ‘ 𝑋 ) ) → ( 𝑥 ∈ ( 𝐽 fLim 𝐺 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐺 ) ) ) ) |
| 144 |
17 8 143
|
syl2anc |
⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐽 fLim 𝐺 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐺 ) ) ) ) |
| 145 |
144
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) → ( 𝑥 ∈ ( 𝐽 fLim 𝐺 ) ↔ ( 𝑥 ∈ 𝑋 ∧ ∀ 𝑦 ∈ 𝐽 ( 𝑥 ∈ 𝑦 → 𝑦 ∈ 𝐺 ) ) ) ) |
| 146 |
19 142 145
|
mpbir2and |
⊢ ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) → 𝑥 ∈ ( 𝐽 fLim 𝐺 ) ) |
| 147 |
146
|
ne0d |
⊢ ( ( 𝜑 ∧ 𝐹 ( ⇝𝑡 ‘ 𝐽 ) 𝑥 ) → ( 𝐽 fLim 𝐺 ) ≠ ∅ ) |
| 148 |
13 147
|
exlimddv |
⊢ ( 𝜑 → ( 𝐽 fLim 𝐺 ) ≠ ∅ ) |