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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 2737 | . 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 31246 | . . 3 ⊢ 𝐻 ∈ Sℋ | 
| 6 | 2, 3, 5 | hhssmet 31295 | . 2 ⊢ 𝐷 ∈ (Met‘𝐻) | 
| 7 | simpl 482 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘𝐷)) | |
| 8 | 2, 3, 5 | hhssims2 31294 | . . . . . . . . . . 11 ⊢ 𝐷 = ((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)) | 
| 9 | 8 | fveq2i 6909 | . . . . . . . . . 10 ⊢ (Cau‘𝐷) = (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))) | 
| 10 | 7, 9 | eleqtrdi 2851 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻)))) | 
| 11 | eqid 2737 | . . . . . . . . . . 11 ⊢ (normℎ ∘ −ℎ ) = (normℎ ∘ −ℎ ) | |
| 12 | 11 | hilxmet 31214 | . . . . . . . . . 10 ⊢ (normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) | 
| 13 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓:ℕ⟶𝐻) | |
| 14 | causs 25332 | . . . . . . . . . 10 ⊢ (((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) ∧ 𝑓:ℕ⟶𝐻) → (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ↔ 𝑓 ∈ (Cau‘((normℎ ∘ −ℎ ) ↾ (𝐻 × 𝐻))))) | |
| 15 | 12, 13, 14 | sylancr 587 | . . . . . . . . 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 31250 | . . . . . . . . . 10 ⊢ 𝐻 ⊆ ℋ | 
| 18 | fss 6752 | . . . . . . . . . 10 ⊢ ((𝑓:ℕ⟶𝐻 ∧ 𝐻 ⊆ ℋ) → 𝑓:ℕ⟶ ℋ) | |
| 19 | 13, 17, 18 | sylancl 586 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓:ℕ⟶ ℋ) | 
| 20 | ax-hilex 31018 | . . . . . . . . . 10 ⊢ ℋ ∈ V | |
| 21 | nnex 12272 | . . . . . . . . . 10 ⊢ ℕ ∈ V | |
| 22 | 20, 21 | elmap 8911 | . . . . . . . . 9 ⊢ (𝑓 ∈ ( ℋ ↑m ℕ) ↔ 𝑓:ℕ⟶ ℋ) | 
| 23 | 19, 22 | sylibr 234 | . . . . . . . 8 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ ( ℋ ↑m ℕ)) | 
| 24 | eqid 2737 | . . . . . . . . . 10 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
| 25 | 24, 11 | hhims 31191 | . . . . . . . . . 10 ⊢ (normℎ ∘ −ℎ ) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | 
| 26 | 24, 25 | hhcau 31217 | . . . . . . . . 9 ⊢ Cauchy = ((Cau‘(normℎ ∘ −ℎ )) ∩ ( ℋ ↑m ℕ)) | 
| 27 | 26 | elin2 4203 | . . . . . . . 8 ⊢ (𝑓 ∈ Cauchy ↔ (𝑓 ∈ (Cau‘(normℎ ∘ −ℎ )) ∧ 𝑓 ∈ ( ℋ ↑m ℕ))) | 
| 28 | 16, 23, 27 | sylanbrc 583 | . . . . . . 7 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ∈ Cauchy) | 
| 29 | ax-hcompl 31221 | . . . . . . 7 ⊢ (𝑓 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥) | |
| 30 | vex 3484 | . . . . . . . . 9 ⊢ 𝑓 ∈ V | |
| 31 | vex 3484 | . . . . . . . . 9 ⊢ 𝑥 ∈ V | |
| 32 | 30, 31 | breldm 5919 | . . . . . . . 8 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom ⇝𝑣 ) | 
| 33 | 32 | rexlimivw 3151 | . . . . . . 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 31256 | . . . . . . 7 ⊢ ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ | |
| 36 | ffun 6739 | . . . . . . 7 ⊢ ( ⇝𝑣 :dom ⇝𝑣 ⟶ ℋ → Fun ⇝𝑣 ) | |
| 37 | funfvbrb 7071 | . . . . . . 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 218 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓)) | 
| 40 | eqid 2737 | . . . . . . . 8 ⊢ (MetOpen‘(normℎ ∘ −ℎ )) = (MetOpen‘(normℎ ∘ −ℎ )) | |
| 41 | 24, 25, 40 | hhlm 31218 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) | 
| 42 | resss 6019 | . . . . . . 7 ⊢ ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) | |
| 43 | 41, 42 | eqsstri 4030 | . . . . . 6 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) | 
| 44 | 43 | ssbri 5188 | . . . . 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 24537 | . . . . . . 7 ⊢ (((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) ∧ 𝐻 ⊆ ℋ) → ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) = (MetOpen‘𝐷)) | 
| 47 | 12, 17, 46 | mp2an 692 | . . . . . 6 ⊢ ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) = (MetOpen‘𝐷) | 
| 48 | 47 | eqcomi 2746 | . . . . 5 ⊢ (MetOpen‘𝐷) = ((MetOpen‘(normℎ ∘ −ℎ )) ↾t 𝐻) | 
| 49 | nnuz 12921 | . . . . 5 ⊢ ℕ = (ℤ≥‘1) | |
| 50 | 4 | a1i 11 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝐻 ∈ Cℋ ) | 
| 51 | 40 | mopntop 24450 | . . . . . 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 6919 | . . . . . . 7 ⊢ ( ⇝𝑣 ‘𝑓) ∈ V | |
| 54 | 53 | chlimi 31253 | . . . . . 6 ⊢ ((𝐻 ∈ Cℋ ∧ 𝑓:ℕ⟶𝐻 ∧ 𝑓 ⇝𝑣 ( ⇝𝑣 ‘𝑓)) → ( ⇝𝑣 ‘𝑓) ∈ 𝐻) | 
| 55 | 50, 13, 39, 54 | syl3anc 1373 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → ( ⇝𝑣 ‘𝑓) ∈ 𝐻) | 
| 56 | 1zzd 12648 | . . . . 5 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 1 ∈ ℤ) | |
| 57 | 48, 49, 50, 52, 55, 56, 13 | lmss 23306 | . . . 4 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → (𝑓(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))( ⇝𝑣 ‘𝑓) ↔ 𝑓(⇝𝑡‘(MetOpen‘𝐷))( ⇝𝑣 ‘𝑓))) | 
| 58 | 45, 57 | mpbid 232 | . . 3 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶𝐻) → 𝑓(⇝𝑡‘(MetOpen‘𝐷))( ⇝𝑣 ‘𝑓)) | 
| 59 | 30, 53 | breldm 5919 | . . 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 25346 | 1 ⊢ 𝐷 ∈ (CMet‘𝐻) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 〈cop 4632 class class class wbr 5143 × cxp 5683 dom cdm 5685 ↾ cres 5687 ∘ ccom 5689 Fun wfun 6555 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 ℂcc 11153 1c1 11156 ℕcn 12266 ↾t crest 17465 ∞Metcxmet 21349 MetOpencmopn 21354 Topctop 22899 ⇝𝑡clm 23234 Cauccau 25287 CMetccmet 25288 IndMetcims 30610 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 normℎcno 30942 −ℎ cmv 30944 Cauchyccauold 30945 ⇝𝑣 chli 30946 Cℋ cch 30948 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 ax-mulf 11235 ax-hilex 31018 ax-hfvadd 31019 ax-hvcom 31020 ax-hvass 31021 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvmulass 31026 ax-hvdistr1 31027 ax-hvdistr2 31028 ax-hvmul0 31029 ax-hfi 31098 ax-his1 31101 ax-his2 31102 ax-his3 31103 ax-his4 31104 ax-hcompl 31221 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ico 13393 df-icc 13394 df-fz 13548 df-fl 13832 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-rest 17467 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-top 22900 df-topon 22917 df-bases 22953 df-ntr 23028 df-nei 23106 df-lm 23237 df-haus 23323 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-cfil 25289 df-cau 25290 df-cmet 25291 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 df-ablo 30564 df-vc 30578 df-nv 30611 df-va 30614 df-ba 30615 df-sm 30616 df-0v 30617 df-vs 30618 df-nmcv 30619 df-ims 30620 df-ssp 30741 df-hnorm 30987 df-hba 30988 df-hvsub 30990 df-hlim 30991 df-hcau 30992 df-sh 31226 df-ch 31240 df-ch0 31272 | 
| This theorem is referenced by: hhssbnOLD 31298 | 
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