Step |
Hyp |
Ref |
Expression |
1 |
|
axhil.1 |
⊢ 𝑈 = 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 |
2 |
|
axhil.2 |
⊢ 𝑈 ∈ CHilOLD |
3 |
|
simpl |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ) |
4 |
|
eqid |
⊢ ( IndMet ‘ 𝑈 ) = ( IndMet ‘ 𝑈 ) |
5 |
|
eqid |
⊢ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) = ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) |
6 |
4 5
|
hlcompl |
⊢ ( ( 𝑈 ∈ CHilOLD ∧ 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ) → 𝐹 ∈ dom ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ) |
7 |
2 3 6
|
sylancr |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → 𝐹 ∈ dom ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ) |
8 |
|
eldm2g |
⊢ ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) → ( 𝐹 ∈ dom ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ↔ ∃ 𝑥 〈 𝐹 , 𝑥 〉 ∈ ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → ( 𝐹 ∈ dom ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ↔ ∃ 𝑥 〈 𝐹 , 𝑥 〉 ∈ ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ) ) |
10 |
7 9
|
mpbid |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → ∃ 𝑥 〈 𝐹 , 𝑥 〉 ∈ ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ) |
11 |
|
df-br |
⊢ ( 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 ↔ 〈 𝐹 , 𝑥 〉 ∈ ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ) |
12 |
2
|
hlnvi |
⊢ 𝑈 ∈ NrmCVec |
13 |
|
df-hba |
⊢ ℋ = ( BaseSet ‘ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 ) |
14 |
1
|
fveq2i |
⊢ ( BaseSet ‘ 𝑈 ) = ( BaseSet ‘ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 ) |
15 |
13 14
|
eqtr4i |
⊢ ℋ = ( BaseSet ‘ 𝑈 ) |
16 |
15 4
|
imsxmet |
⊢ ( 𝑈 ∈ NrmCVec → ( IndMet ‘ 𝑈 ) ∈ ( ∞Met ‘ ℋ ) ) |
17 |
5
|
mopntopon |
⊢ ( ( IndMet ‘ 𝑈 ) ∈ ( ∞Met ‘ ℋ ) → ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ∈ ( TopOn ‘ ℋ ) ) |
18 |
12 16 17
|
mp2b |
⊢ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ∈ ( TopOn ‘ ℋ ) |
19 |
|
lmcl |
⊢ ( ( ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ∈ ( TopOn ‘ ℋ ) ∧ 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 ) → 𝑥 ∈ ℋ ) |
20 |
18 19
|
mpan |
⊢ ( 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 → 𝑥 ∈ ℋ ) |
21 |
20
|
a1i |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → ( 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 → 𝑥 ∈ ℋ ) ) |
22 |
1 12 15 4 5
|
h2hlm |
⊢ ⇝𝑣 = ( ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ↾ ( ℋ ↑m ℕ ) ) |
23 |
22
|
breqi |
⊢ ( 𝐹 ⇝𝑣 𝑥 ↔ 𝐹 ( ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ↾ ( ℋ ↑m ℕ ) ) 𝑥 ) |
24 |
|
brres |
⊢ ( 𝑥 ∈ V → ( 𝐹 ( ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ↾ ( ℋ ↑m ℕ ) ) 𝑥 ↔ ( 𝐹 ∈ ( ℋ ↑m ℕ ) ∧ 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 ) ) ) |
25 |
24
|
elv |
⊢ ( 𝐹 ( ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) ↾ ( ℋ ↑m ℕ ) ) 𝑥 ↔ ( 𝐹 ∈ ( ℋ ↑m ℕ ) ∧ 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 ) ) |
26 |
23 25
|
bitri |
⊢ ( 𝐹 ⇝𝑣 𝑥 ↔ ( 𝐹 ∈ ( ℋ ↑m ℕ ) ∧ 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 ) ) |
27 |
26
|
baib |
⊢ ( 𝐹 ∈ ( ℋ ↑m ℕ ) → ( 𝐹 ⇝𝑣 𝑥 ↔ 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 ) ) |
28 |
27
|
adantl |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → ( 𝐹 ⇝𝑣 𝑥 ↔ 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 ) ) |
29 |
28
|
biimprd |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → ( 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 → 𝐹 ⇝𝑣 𝑥 ) ) |
30 |
21 29
|
jcad |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → ( 𝐹 ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) 𝑥 → ( 𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥 ) ) ) |
31 |
11 30
|
syl5bir |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → ( 〈 𝐹 , 𝑥 〉 ∈ ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) → ( 𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥 ) ) ) |
32 |
31
|
eximdv |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → ( ∃ 𝑥 〈 𝐹 , 𝑥 〉 ∈ ( ⇝𝑡 ‘ ( MetOpen ‘ ( IndMet ‘ 𝑈 ) ) ) → ∃ 𝑥 ( 𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥 ) ) ) |
33 |
10 32
|
mpd |
⊢ ( ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) → ∃ 𝑥 ( 𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥 ) ) |
34 |
|
elin |
⊢ ( 𝐹 ∈ ( ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∩ ( ℋ ↑m ℕ ) ) ↔ ( 𝐹 ∈ ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∧ 𝐹 ∈ ( ℋ ↑m ℕ ) ) ) |
35 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥 ↔ ∃ 𝑥 ( 𝑥 ∈ ℋ ∧ 𝐹 ⇝𝑣 𝑥 ) ) |
36 |
33 34 35
|
3imtr4i |
⊢ ( 𝐹 ∈ ( ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∩ ( ℋ ↑m ℕ ) ) → ∃ 𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥 ) |
37 |
1 12 15 4
|
h2hcau |
⊢ Cauchy = ( ( Cau ‘ ( IndMet ‘ 𝑈 ) ) ∩ ( ℋ ↑m ℕ ) ) |
38 |
36 37
|
eleq2s |
⊢ ( 𝐹 ∈ Cauchy → ∃ 𝑥 ∈ ℋ 𝐹 ⇝𝑣 𝑥 ) |