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Mirrors > Home > HSE Home > Th. List > hhcms | Structured version Visualization version GIF version |
Description: The Hilbert space induced metric determines a complete metric space. (Contributed by NM, 10-Apr-2008.) (Proof shortened by Mario Carneiro, 14-May-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hhcms.1 | ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 |
hhcms.2 | ⊢ 𝐷 = (IndMet‘𝑈) |
Ref | Expression |
---|---|
hhcms | ⊢ 𝐷 ∈ (CMet‘ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . 2 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷) | |
2 | hhcms.1 | . . 3 ⊢ 𝑈 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
3 | hhcms.2 | . . 3 ⊢ 𝐷 = (IndMet‘𝑈) | |
4 | 2, 3 | hhmet 30896 | . 2 ⊢ 𝐷 ∈ (Met‘ ℋ) |
5 | 2, 3 | hhcau 30920 | . . . . . 6 ⊢ Cauchy = ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) |
6 | 5 | eleq2i 2817 | . . . . 5 ⊢ (𝑓 ∈ Cauchy ↔ 𝑓 ∈ ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ))) |
7 | elin 3956 | . . . . . 6 ⊢ (𝑓 ∈ ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) ↔ (𝑓 ∈ (Cau‘𝐷) ∧ 𝑓 ∈ ( ℋ ↑m ℕ))) | |
8 | ax-hilex 30721 | . . . . . . . 8 ⊢ ℋ ∈ V | |
9 | nnex 12215 | . . . . . . . 8 ⊢ ℕ ∈ V | |
10 | 8, 9 | elmap 8861 | . . . . . . 7 ⊢ (𝑓 ∈ ( ℋ ↑m ℕ) ↔ 𝑓:ℕ⟶ ℋ) |
11 | 10 | anbi2i 622 | . . . . . 6 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓 ∈ ( ℋ ↑m ℕ)) ↔ (𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶ ℋ)) |
12 | 7, 11 | bitri 275 | . . . . 5 ⊢ (𝑓 ∈ ((Cau‘𝐷) ∩ ( ℋ ↑m ℕ)) ↔ (𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶ ℋ)) |
13 | 6, 12 | bitri 275 | . . . 4 ⊢ (𝑓 ∈ Cauchy ↔ (𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶ ℋ)) |
14 | ax-hcompl 30924 | . . . 4 ⊢ (𝑓 ∈ Cauchy → ∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥) | |
15 | 13, 14 | sylbir 234 | . . 3 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶ ℋ) → ∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥) |
16 | 2, 3, 1 | hhlm 30921 | . . . . . . 7 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘𝐷)) ↾ ( ℋ ↑m ℕ)) |
17 | 16 | breqi 5144 | . . . . . 6 ⊢ (𝑓 ⇝𝑣 𝑥 ↔ 𝑓((⇝𝑡‘(MetOpen‘𝐷)) ↾ ( ℋ ↑m ℕ))𝑥) |
18 | vex 3470 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
19 | 18 | brresi 5980 | . . . . . 6 ⊢ (𝑓((⇝𝑡‘(MetOpen‘𝐷)) ↾ ( ℋ ↑m ℕ))𝑥 ↔ (𝑓 ∈ ( ℋ ↑m ℕ) ∧ 𝑓(⇝𝑡‘(MetOpen‘𝐷))𝑥)) |
20 | 17, 19 | bitri 275 | . . . . 5 ⊢ (𝑓 ⇝𝑣 𝑥 ↔ (𝑓 ∈ ( ℋ ↑m ℕ) ∧ 𝑓(⇝𝑡‘(MetOpen‘𝐷))𝑥)) |
21 | vex 3470 | . . . . . 6 ⊢ 𝑓 ∈ V | |
22 | 21, 18 | breldm 5898 | . . . . 5 ⊢ (𝑓(⇝𝑡‘(MetOpen‘𝐷))𝑥 → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
23 | 20, 22 | simplbiim 504 | . . . 4 ⊢ (𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
24 | 23 | rexlimivw 3143 | . . 3 ⊢ (∃𝑥 ∈ ℋ 𝑓 ⇝𝑣 𝑥 → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
25 | 15, 24 | syl 17 | . 2 ⊢ ((𝑓 ∈ (Cau‘𝐷) ∧ 𝑓:ℕ⟶ ℋ) → 𝑓 ∈ dom (⇝𝑡‘(MetOpen‘𝐷))) |
26 | 1, 4, 25 | iscmet3i 25162 | 1 ⊢ 𝐷 ∈ (CMet‘ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 ∩ cin 3939 〈cop 4626 class class class wbr 5138 dom cdm 5666 ↾ cres 5668 ⟶wf 6529 ‘cfv 6533 (class class class)co 7401 ↑m cmap 8816 ℕcn 12209 MetOpencmopn 21218 ⇝𝑡clm 23052 Cauccau 25103 CMetccmet 25104 IndMetcims 30313 ℋchba 30641 +ℎ cva 30642 ·ℎ csm 30643 normℎcno 30645 Cauchyccauold 30648 ⇝𝑣 chli 30649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186 ax-hilex 30721 ax-hfvadd 30722 ax-hvcom 30723 ax-hvass 30724 ax-hv0cl 30725 ax-hvaddid 30726 ax-hfvmul 30727 ax-hvmulid 30728 ax-hvmulass 30729 ax-hvdistr1 30730 ax-hvdistr2 30731 ax-hvmul0 30732 ax-hfi 30801 ax-his1 30804 ax-his2 30805 ax-his3 30806 ax-his4 30807 ax-hcompl 30924 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-oadd 8465 df-omul 8466 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-acn 9933 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-n0 12470 df-z 12556 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ico 13327 df-fz 13482 df-fl 13754 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-rest 17367 df-topgen 17388 df-psmet 21220 df-xmet 21221 df-met 21222 df-bl 21223 df-mopn 21224 df-fbas 21225 df-fg 21226 df-top 22718 df-topon 22735 df-bases 22771 df-ntr 22846 df-nei 22924 df-lm 23055 df-fil 23672 df-fm 23764 df-flim 23765 df-flf 23766 df-cfil 25105 df-cau 25106 df-cmet 25107 df-grpo 30215 df-gid 30216 df-ginv 30217 df-gdiv 30218 df-ablo 30267 df-vc 30281 df-nv 30314 df-va 30317 df-ba 30318 df-sm 30319 df-0v 30320 df-vs 30321 df-nmcv 30322 df-ims 30323 df-hnorm 30690 df-hvsub 30693 df-hlim 30694 df-hcau 30695 |
This theorem is referenced by: hhhl 30926 hilcms 30927 |
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