Metamath Proof Explorer


Theorem hhsscms

Description: The induced metric of a closed subspace is complete. (Contributed by NM, 10-Apr-2008) (Revised by Mario Carneiro, 14-May-2014) (New usage is discouraged.)

Ref Expression
Hypotheses hhssims2.1 𝑊 = ⟨ ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ , ( norm𝐻 ) ⟩
hhssims2.3 𝐷 = ( IndMet ‘ 𝑊 )
hhsscms.3 𝐻C
Assertion hhsscms 𝐷 ∈ ( CMet ‘ 𝐻 )

Proof

Step Hyp Ref Expression
1 hhssims2.1 𝑊 = ⟨ ⟨ ( + ↾ ( 𝐻 × 𝐻 ) ) , ( · ↾ ( ℂ × 𝐻 ) ) ⟩ , ( norm𝐻 ) ⟩
2 hhssims2.3 𝐷 = ( IndMet ‘ 𝑊 )
3 hhsscms.3 𝐻C
4 eqid ( MetOpen ‘ 𝐷 ) = ( MetOpen ‘ 𝐷 )
5 3 chshii 𝐻S
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 = ( ( norm ∘ − ) ↾ ( 𝐻 × 𝐻 ) )
9 8 fveq2i ( Cau ‘ 𝐷 ) = ( Cau ‘ ( ( norm ∘ − ) ↾ ( 𝐻 × 𝐻 ) ) )
10 7 9 eleqtrdi ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ∈ ( Cau ‘ ( ( norm ∘ − ) ↾ ( 𝐻 × 𝐻 ) ) ) )
11 eqid ( norm ∘ − ) = ( norm ∘ − )
12 11 hilxmet ( norm ∘ − ) ∈ ( ∞Met ‘ ℋ )
13 simpr ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 : ℕ ⟶ 𝐻 )
14 causs ( ( ( norm ∘ − ) ∈ ( ∞Met ‘ ℋ ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → ( 𝑓 ∈ ( Cau ‘ ( norm ∘ − ) ) ↔ 𝑓 ∈ ( Cau ‘ ( ( norm ∘ − ) ↾ ( 𝐻 × 𝐻 ) ) ) ) )
15 12 13 14 sylancr ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → ( 𝑓 ∈ ( Cau ‘ ( norm ∘ − ) ) ↔ 𝑓 ∈ ( Cau ‘ ( ( norm ∘ − ) ↾ ( 𝐻 × 𝐻 ) ) ) ) )
16 10 15 mpbird ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ∈ ( Cau ‘ ( norm ∘ − ) ) )
17 3 chssii 𝐻 ⊆ ℋ
18 fss ( ( 𝑓 : ℕ ⟶ 𝐻𝐻 ⊆ 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 ⟨ ⟨ + , · ⟩ , norm ⟩ = ⟨ ⟨ + , · ⟩ , norm
25 24 11 hhims ( norm ∘ − ) = ( IndMet ‘ ⟨ ⟨ + , · ⟩ , norm ⟩ )
26 24 25 hhcau Cauchy = ( ( Cau ‘ ( norm ∘ − ) ) ∩ ( ℋ ↑m ℕ ) )
27 26 elin2 ( 𝑓 ∈ Cauchy ↔ ( 𝑓 ∈ ( Cau ‘ ( norm ∘ − ) ) ∧ 𝑓 ∈ ( ℋ ↑m ℕ ) ) )
28 16 23 27 sylanbrc ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ∈ Cauchy )
29 ax-hcompl ( 𝑓 ∈ Cauchy → ∃ 𝑥 ∈ ℋ 𝑓𝑣 𝑥 )
30 vex 𝑓 ∈ V
31 vex 𝑥 ∈ V
32 30 31 breldm ( 𝑓𝑣 𝑥𝑓 ∈ dom ⇝𝑣 )
33 32 rexlimivw ( ∃ 𝑥 ∈ ℋ 𝑓𝑣 𝑥𝑓 ∈ 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 ⇝𝑣𝑓𝑣 ( ⇝𝑣𝑓 ) ) )
38 35 36 37 mp2b ( 𝑓 ∈ dom ⇝𝑣𝑓𝑣 ( ⇝𝑣𝑓 ) )
39 34 38 sylib ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓𝑣 ( ⇝𝑣𝑓 ) )
40 eqid ( MetOpen ‘ ( norm ∘ − ) ) = ( MetOpen ‘ ( norm ∘ − ) )
41 24 25 40 hhlm 𝑣 = ( ( ⇝𝑡 ‘ ( MetOpen ‘ ( norm ∘ − ) ) ) ↾ ( ℋ ↑m ℕ ) )
42 resss ( ( ⇝𝑡 ‘ ( MetOpen ‘ ( norm ∘ − ) ) ) ↾ ( ℋ ↑m ℕ ) ) ⊆ ( ⇝𝑡 ‘ ( MetOpen ‘ ( norm ∘ − ) ) )
43 41 42 eqsstri 𝑣 ⊆ ( ⇝𝑡 ‘ ( MetOpen ‘ ( norm ∘ − ) ) )
44 43 ssbri ( 𝑓𝑣 ( ⇝𝑣𝑓 ) → 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ ( norm ∘ − ) ) ) ( ⇝𝑣𝑓 ) )
45 39 44 syl ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ ( norm ∘ − ) ) ) ( ⇝𝑣𝑓 ) )
46 8 40 4 metrest ( ( ( norm ∘ − ) ∈ ( ∞Met ‘ ℋ ) ∧ 𝐻 ⊆ ℋ ) → ( ( MetOpen ‘ ( norm ∘ − ) ) ↾t 𝐻 ) = ( MetOpen ‘ 𝐷 ) )
47 12 17 46 mp2an ( ( MetOpen ‘ ( norm ∘ − ) ) ↾t 𝐻 ) = ( MetOpen ‘ 𝐷 )
48 47 eqcomi ( MetOpen ‘ 𝐷 ) = ( ( MetOpen ‘ ( norm ∘ − ) ) ↾t 𝐻 )
49 nnuz ℕ = ( ℤ ‘ 1 )
50 3 a1i ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝐻C )
51 40 mopntop ( ( norm ∘ − ) ∈ ( ∞Met ‘ ℋ ) → ( MetOpen ‘ ( norm ∘ − ) ) ∈ Top )
52 12 51 mp1i ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → ( MetOpen ‘ ( norm ∘ − ) ) ∈ Top )
53 fvex ( ⇝𝑣𝑓 ) ∈ V
54 53 chlimi ( ( 𝐻C𝑓 : ℕ ⟶ 𝐻𝑓𝑣 ( ⇝𝑣𝑓 ) ) → ( ⇝𝑣𝑓 ) ∈ 𝐻 )
55 50 13 39 54 syl3anc ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → ( ⇝𝑣𝑓 ) ∈ 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 ‘ ( norm ∘ − ) ) ) ( ⇝𝑣𝑓 ) ↔ 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ 𝐷 ) ) ( ⇝𝑣𝑓 ) ) )
58 45 57 mpbid ( ( 𝑓 ∈ ( Cau ‘ 𝐷 ) ∧ 𝑓 : ℕ ⟶ 𝐻 ) → 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ 𝐷 ) ) ( ⇝𝑣𝑓 ) )
59 30 53 breldm ( 𝑓 ( ⇝𝑡 ‘ ( MetOpen ‘ 𝐷 ) ) ( ⇝𝑣𝑓 ) → 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 )