Step |
Hyp |
Ref |
Expression |
1 |
|
hhssims2.1 |
⊢ 𝑊 = 〈 〈 ( +ℎ ↾ ( 𝐻 × 𝐻 ) ) , ( ·ℎ ↾ ( ℂ × 𝐻 ) ) 〉 , ( normℎ ↾ 𝐻 ) 〉 |
2 |
|
hhssims2.3 |
⊢ 𝐷 = ( IndMet ‘ 𝑊 ) |
3 |
|
hhsscms.3 |
⊢ 𝐻 ∈ Cℋ |
4 |
|
eqid |
⊢ ( MetOpen ‘ 𝐷 ) = ( MetOpen ‘ 𝐷 ) |
5 |
3
|
chshii |
⊢ 𝐻 ∈ Sℋ |
6 |
1 2 5
|
hhssmet |
⊢ 𝐷 ∈ ( Met ‘ 𝐻 ) |
7 |
|
simpl |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ∈ ( Cau ‘ 𝐷 ) ) |
8 |
1 2 5
|
hhssims2 |
⊢ 𝐷 = ( ( normℎ ∘ −ℎ ) ↾ ( 𝐻 × 𝐻 ) ) |
9 |
8
|
fveq2i |
⊢ ( Cau ‘ 𝐷 ) = ( Cau ‘ ( ( normℎ ∘ −ℎ ) ↾ ( 𝐻 × 𝐻 ) ) ) |
10 |
7 9
|
eleqtrdi |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ∈ ( Cau ‘ ( ( normℎ ∘ −ℎ ) ↾ ( 𝐻 × 𝐻 ) ) ) ) |
11 |
|
eqid |
⊢ ( normℎ ∘ −ℎ ) = ( normℎ ∘ −ℎ ) |
12 |
11
|
hilxmet |
⊢ ( normℎ ∘ −ℎ ) ∈ ( ∞Met ‘ ℋ ) |
13 |
|
simpr |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 : ℕ ⟶ 𝐻 ) |
14 |
|
causs |
⊢ ( ( ( normℎ ∘ −ℎ ) ∈ ( ∞Met ‘ ℋ ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → ( 𝑓 ∈ ( Cau ‘ ( normℎ ∘ −ℎ ) ) ↔ 𝑓 ∈ ( Cau ‘ ( ( normℎ ∘ −ℎ ) ↾ ( 𝐻 × 𝐻 ) ) ) ) ) |
15 |
12 13 14
|
sylancr |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → ( 𝑓 ∈ ( Cau ‘ ( normℎ ∘ −ℎ ) ) ↔ 𝑓 ∈ ( Cau ‘ ( ( normℎ ∘ −ℎ ) ↾ ( 𝐻 × 𝐻 ) ) ) ) ) |
16 |
10 15
|
mpbird |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ∈ ( Cau ‘ ( normℎ ∘ −ℎ ) ) ) |
17 |
3
|
chssii |
⊢ 𝐻 ⊆ ℋ |
18 |
|
fss |
⊢ ( ( 𝑓 : ℕ ⟶ 𝐻 ∧ 𝐻 ⊆ ℋ ) → 𝑓 : ℕ ⟶ ℋ ) |
19 |
13 17 18
|
sylancl |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 : ℕ ⟶ ℋ ) |
20 |
|
ax-hilex |
⊢ ℋ ∈ V |
21 |
|
nnex |
⊢ ℕ ∈ V |
22 |
20 21
|
elmap |
⊢ ( 𝑓 ∈ ( ℋ ↑m ℕ ) ↔ 𝑓 : ℕ ⟶ ℋ ) |
23 |
19 22
|
sylibr |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ∈ ( ℋ ↑m ℕ ) ) |
24 |
|
eqid |
⊢ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 = 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 |
25 |
24 11
|
hhims |
⊢ ( normℎ ∘ −ℎ ) = ( IndMet ‘ 〈 〈 +ℎ , ·ℎ 〉 , normℎ 〉 ) |
26 |
24 25
|
hhcau |
⊢ Cauchy = ( ( Cau ‘ ( normℎ ∘ −ℎ ) ) ∩ ( ℋ ↑m ℕ ) ) |
27 |
26
|
elin2 |
⊢ ( 𝑓 ∈ Cauchy ↔ ( 𝑓 ∈ ( Cau ‘ ( normℎ ∘ −ℎ ) ) ∧ 𝑓 ∈ ( ℋ ↑m ℕ ) ) ) |
28 |
16 23 27
|
sylanbrc |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ∈ Cauchy ) |
29 |
|
ax-hcompl |
⊢ ( 𝑓 ∈ Cauchy → ∃ 𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥 ) |
30 |
|
vex |
⊢ 𝑓 ∈ V |
31 |
|
vex |
⊢ 𝑥 ∈ V |
32 |
30 31
|
breldm |
⊢ ( 𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom ⇝𝑣 ) |
33 |
32
|
rexlimivw |
⊢ ( ∃ 𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom ⇝𝑣 ) |
34 |
28 29 33
|
3syl |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ∈ dom ⇝𝑣 ) |
35 |
|
hlimf |
⊢ ⇝𝑣 : dom ⇝𝑣 ⟶ ℋ |
36 |
|
ffun |
⊢ ( ⇝𝑣 : dom ⇝𝑣 ⟶ ℋ → Fun ⇝𝑣 ) |
37 |
|
funfvbrb |
⊢ ( Fun ⇝𝑣 → ( 𝑓 ∈ dom ⇝𝑣 ↔ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘ 𝑓 ) ) ) |
38 |
35 36 37
|
mp2b |
⊢ ( 𝑓 ∈ dom ⇝𝑣 ↔ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘ 𝑓 ) ) |
39 |
34 38
|
sylib |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ⇝𝑣 ( ⇝𝑣 ‘ 𝑓 ) ) |
40 |
|
eqid |
⊢ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) = ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) |
41 |
24 25 40
|
hhlm |
⊢ ⇝𝑣 = ( ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ↾ ( ℋ ↑m ℕ ) ) |
42 |
|
resss |
⊢ ( ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ↾ ( ℋ ↑m ℕ ) ) ⊆ ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) |
43 |
41 42
|
eqsstri |
⊢ ⇝𝑣 ⊆ ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) |
44 |
43
|
ssbri |
⊢ ( 𝑓 ⇝𝑣 ( ⇝𝑣 ‘ 𝑓 ) → 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ( ⇝𝑣 ‘ 𝑓 ) ) |
45 |
39 44
|
syl |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ( ⇝𝑣 ‘ 𝑓 ) ) |
46 |
8 40 4
|
metrest |
⊢ ( ( ( normℎ ∘ −ℎ ) ∈ ( ∞Met ‘ ℋ ) ∧ 𝐻 ⊆ ℋ ) → ( ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ↾t 𝐻 ) = ( MetOpen ‘ 𝐷 ) ) |
47 |
12 17 46
|
mp2an |
⊢ ( ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ↾t 𝐻 ) = ( MetOpen ‘ 𝐷 ) |
48 |
47
|
eqcomi |
⊢ ( MetOpen ‘ 𝐷 ) = ( ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ↾t 𝐻 ) |
49 |
|
nnuz |
⊢ ℕ = ( ℤ≥ ‘ 1 ) |
50 |
3
|
a1i |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝐻 ∈ Cℋ ) |
51 |
40
|
mopntop |
⊢ ( ( normℎ ∘ −ℎ ) ∈ ( ∞Met ‘ ℋ ) → ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ∈ Top ) |
52 |
12 51
|
mp1i |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ∈ Top ) |
53 |
|
fvex |
⊢ ( ⇝𝑣 ‘ 𝑓 ) ∈ V |
54 |
53
|
chlimi |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝑓 : ℕ ⟶ 𝐻 ∧ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘ 𝑓 ) ) → ( ⇝𝑣 ‘ 𝑓 ) ∈ 𝐻 ) |
55 |
50 13 39 54
|
syl3anc |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → ( ⇝𝑣 ‘ 𝑓 ) ∈ 𝐻 ) |
56 |
|
1zzd |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 1 ∈ ℤ ) |
57 |
48 49 50 52 55 56 13
|
lmss |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → ( 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ ( normℎ ∘ −ℎ ) ) ) ( ⇝𝑣 ‘ 𝑓 ) ↔ 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ 𝐷 ) ) ( ⇝𝑣 ‘ 𝑓 ) ) ) |
58 |
45 57
|
mpbid |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ 𝐷 ) ) ( ⇝𝑣 ‘ 𝑓 ) ) |
59 |
30 53
|
breldm |
⊢ ( 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ 𝐷 ) ) ( ⇝𝑣 ‘ 𝑓 ) → 𝑓 ∈ dom ( ⇝𝑡 ‘ ( MetOpen ‘ 𝐷 ) ) ) |
60 |
58 59
|
syl |
⊢ ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ∈ dom ( ⇝𝑡 ‘ ( MetOpen ‘ 𝐷 ) ) ) |
61 |
4 6 60
|
iscmet3i |
⊢ 𝐷 ∈ ( CMet ‘ 𝐻 ) |