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Mirrors > Home > HSE Home > Th. List > hhsscms | Structured version Visualization version GIF version |
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 |
---|---|
hhssims2.1 | ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 |
hhssims2.3 | ⊢ 𝐷 = (IndMet‘𝑊) |
hhsscms.3 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
hhsscms | ⊢ 𝐷 ∈ (CMet‘𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2731 | . 2 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
2 | hhssims2.1 | . . 3 ⊢ 𝑊 = 〈〈( +ℎ ↾ (𝐻 × 𝐻)), ( ·ℎ ↾ (ℂ × 𝐻))〉, (normℎ ↾ 𝐻)〉 | |
3 | hhssims2.3 | . . 3 ⊢ 𝐷 = (IndMet‘𝑊) | |
4 | hhsscms.3 | . . . 4 ⊢ 𝐻 ∈ Cℋ | |
5 | 4 | chshii 30913 | . . 3 ⊢ 𝐻 ∈ Sℋ |
6 | 2, 3, 5 | hhssmet 30962 | . 2 ⊢ 𝐷 ∈ (Met‘𝐻) |
7 | simpl 482 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘𝐷)) | |
8 | 2, 3, 5 | hhssims2 30961 | . . . . . . . . . . 11 ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) |
9 | 8 | fveq2i 6894 | . . . . . . . . . 10 ⊢ (Cau‘𝐷) = (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))) |
10 | 7, 9 | eleqtrdi 2842 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)))) |
11 | eqid 2731 | . . . . . . . . . . 11 ⊢ (normℎ ∘ −ℎ ) = (normℎ ∘ −ℎ ) | |
12 | 11 | hilxmet 30881 | . . . . . . . . . 10 ⊢ (normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) |
13 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓:ℕ⟶𝐻) | |
14 | causs 25146 | . . . . . . . . . 10 ⊢ (((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) ∧ 𝑓:ℕ⟶𝐻) → (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ↔ 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))))) | |
15 | 12, 13, 14 | sylancr 586 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ↔ 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))))) |
16 | 10, 15 | mpbird 257 | . . . . . . . 8 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘(normℎ ∘ −ℎ ))) |
17 | 4 | chssii 30917 | . . . . . . . . . 10 ⊢ 𝐻 ⊆ ℋ |
18 | fss 6734 | . . . . . . . . . 10 ⊢ ((𝑓:ℕ⟶𝐻 ∧ 𝐻 ⊆ ℋ) → 𝑓:ℕ⟶ ℋ) | |
19 | 13, 17, 18 | sylancl 585 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓:ℕ⟶ ℋ) |
20 | ax-hilex 30685 | . . . . . . . . . 10 ⊢ ℋ ∈ V | |
21 | nnex 12225 | . . . . . . . . . 10 ⊢ ℕ ∈ V | |
22 | 20, 21 | elmap 8871 | . . . . . . . . 9 ⊢ (𝑓 ∈ ( ℋ ↑m ℕ) ↔ 𝑓:ℕ⟶ ℋ) |
23 | 19, 22 | sylibr 233 | . . . . . . . 8 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ ( ℋ ↑m ℕ)) |
24 | eqid 2731 | . . . . . . . . . 10 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
25 | 24, 11 | hhims 30858 | . . . . . . . . . 10 ⊢ (normℎ ∘ −ℎ ) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
26 | 24, 25 | hhcau 30884 | . . . . . . . . 9 ⊢ Cauchy = ((Cau‘(normℎ ∘ −ℎ )) ∩ ( ℋ ↑m ℕ)) |
27 | 26 | elin2 4197 | . . . . . . . 8 ⊢ (𝑓 ∈ Cauchy ↔ (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ∧ 𝑓 ∈ ( ℋ ↑m ℕ))) |
28 | 16, 23, 27 | sylanbrc 582 | . . . . . . 7 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ Cauchy) |
29 | ax-hcompl 30888 | . . . . . . 7 ⊢ (𝑓 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥) | |
30 | vex 3477 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
31 | vex 3477 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
32 | 30, 31 | breldm 5908 | . . . . . . . 8 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom ⇝𝑣 ) |
33 | 32 | rexlimivw 3150 | . . . . . . 7 ⊢ (∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom ⇝𝑣 ) |
34 | 28, 29, 33 | 3syl 18 | . . . . . 6 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ dom ⇝𝑣 ) |
35 | hlimf 30923 | . . . . . . 7 ⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ | |
36 | ffun 6720 | . . . . . . 7 ⊢ ( ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ → Fun ⇝𝑣 ) | |
37 | funfvbrb 7052 | . . . . . . 7 ⊢ (Fun ⇝𝑣 → (𝑓 ∈ dom ⇝𝑣 ↔ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓))) | |
38 | 35, 36, 37 | mp2b 10 | . . . . . 6 ⊢ (𝑓 ∈ dom ⇝𝑣 ↔ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓)) |
39 | 34, 38 | sylib 217 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓)) |
40 | eqid 2731 | . . . . . . . 8 ⊢ (MetOpen‘(normℎ ∘ −ℎ )) = (MetOpen‘(normℎ ∘ −ℎ )) | |
41 | 24, 25, 40 | hhlm 30885 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) |
42 | resss 6006 | . . . . . . 7 ⊢ ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) | |
43 | 41, 42 | eqsstri 4016 | . . . . . 6 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) |
44 | 43 | ssbri 5193 | . . . . 5 ⊢ (𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓) → 𝑓(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))( ⇝𝑣 ‘𝑓)) |
45 | 39, 44 | syl 17 | . . . 4 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))( ⇝𝑣 ‘𝑓)) |
46 | 8, 40, 1 | metrest 24353 | . . . . . . 7 ⊢ (((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) ∧ 𝐻 ⊆ ℋ) → ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) = (MetOpen‘𝐷)) |
47 | 12, 17, 46 | mp2an 689 | . . . . . 6 ⊢ ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) = (MetOpen‘𝐷) |
48 | 47 | eqcomi 2740 | . . . . 5 ⊢ (MetOpen‘𝐷) = ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) |
49 | nnuz 12872 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
50 | 4 | a1i 11 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝐻 ∈ Cℋ ) |
51 | 40 | mopntop 24266 | . . . . . 6 ⊢ ((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) → (MetOpen‘(normℎ ∘ −ℎ )) ∈ Top) |
52 | 12, 51 | mp1i 13 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → (MetOpen‘(normℎ ∘ −ℎ )) ∈ Top) |
53 | fvex 6904 | . . . . . . 7 ⊢ ( ⇝𝑣 ‘𝑓) ∈ V | |
54 | 53 | chlimi 30920 | . . . . . 6 ⊢ ((𝐻 ∈ Cℋ ∧ 𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓)) → ( ⇝𝑣 ‘𝑓) ∈ 𝐻) |
55 | 50, 13, 39, 54 | syl3anc 1370 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → ( ⇝𝑣 ‘𝑓) ∈ 𝐻) |
56 | 1zzd 12600 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 1 ∈ ℤ) | |
57 | 48, 49, 50, 52, 55, 56, 13 | lmss 23122 | . . . 4 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → (𝑓(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))( ⇝𝑣 ‘𝑓) ↔ 𝑓(⇝𝑡‘(MetOpen‘𝐷))( ⇝𝑣 ‘𝑓))) |
58 | 45, 57 | mpbid 231 | . . 3 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓(⇝𝑡‘(MetOpen‘𝐷))( ⇝𝑣 ‘𝑓)) |
59 | 30, 53 | breldm 5908 | . . 3 ⊢ (𝑓(⇝𝑡‘(MetOpen‘𝐷))( ⇝𝑣 ‘𝑓) → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
60 | 58, 59 | syl 17 | . 2 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
61 | 1, 6, 60 | iscmet3i 25160 | 1 ⊢ 𝐷 ∈ (CMet‘𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∃wrex 3069 ⊆ wss 3948 〈cop 4634 class class class wbr 5148 × cxp 5674 dom cdm 5676 ↾ cres 5678 ∘ ccom 5680 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ↑m cmap 8826 ℂcc 11114 1c1 11117 ℕcn 12219 ↾t crest 17373 ∞Metcxmet 21218 MetOpencmopn 21223 Topctop 22715 ⇝𝑡clm 23050 Cauccau 25101 CMetccmet 25102 IndMetcims 30277 ℋchba 30605 +ℎ cva 30606 ·ℎ csm 30607 normℎcno 30609 −ℎ cmv 30611 Cauchyccauold 30612 ⇝𝑣 chli 30613 Cℋ cch 30615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9642 ax-cc 10436 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 ax-pre-sup 11194 ax-addf 11195 ax-mulf 11196 ax-hilex 30685 ax-hfvadd 30686 ax-hvcom 30687 ax-hvass 30688 ax-hv0cl 30689 ax-hvaddid 30690 ax-hfvmul 30691 ax-hvmulid 30692 ax-hvmulass 30693 ax-hvdistr1 30694 ax-hvdistr2 30695 ax-hvmul0 30696 ax-hfi 30765 ax-his1 30768 ax-his2 30769 ax-his3 30770 ax-his4 30771 ax-hcompl 30888 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-omul 8477 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fi 9412 df-sup 9443 df-inf 9444 df-oi 9511 df-card 9940 df-acn 9943 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-n0 12480 df-z 12566 df-uz 12830 df-q 12940 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-ico 13337 df-icc 13338 df-fz 13492 df-fl 13764 df-seq 13974 df-exp 14035 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-clim 15439 df-rlim 15440 df-rest 17375 df-topgen 17396 df-psmet 21225 df-xmet 21226 df-met 21227 df-bl 21228 df-mopn 21229 df-fbas 21230 df-fg 21231 df-top 22716 df-topon 22733 df-bases 22769 df-ntr 22844 df-nei 22922 df-lm 23053 df-haus 23139 df-fil 23670 df-fm 23762 df-flim 23763 df-flf 23764 df-cfil 25103 df-cau 25104 df-cmet 25105 df-grpo 30179 df-gid 30180 df-ginv 30181 df-gdiv 30182 df-ablo 30231 df-vc 30245 df-nv 30278 df-va 30281 df-ba 30282 df-sm 30283 df-0v 30284 df-vs 30285 df-nmcv 30286 df-ims 30287 df-ssp 30408 df-hnorm 30654 df-hba 30655 df-hvsub 30657 df-hlim 30658 df-hcau 30659 df-sh 30893 df-ch 30907 df-ch0 30939 |
This theorem is referenced by: hhssbnOLD 30965 |
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