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Mirrors > Home > MPE Home > Th. List > minvecolem4a | Structured version Visualization version GIF version |
Description: Lemma for minveco 28257. 𝐹 is convergent in the subspace topology on 𝑌. (Contributed by Mario Carneiro, 7-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
minveco.x | ⊢ 𝑋 = (BaseSet‘𝑈) |
minveco.m | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
minveco.n | ⊢ 𝑁 = (normCV‘𝑈) |
minveco.y | ⊢ 𝑌 = (BaseSet‘𝑊) |
minveco.u | ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) |
minveco.w | ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) |
minveco.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
minveco.d | ⊢ 𝐷 = (IndMet‘𝑈) |
minveco.j | ⊢ 𝐽 = (MetOpen‘𝐷) |
minveco.r | ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) |
minveco.s | ⊢ 𝑆 = inf(𝑅, ℝ, < ) |
minveco.f | ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) |
minveco.1 | ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) |
Ref | Expression |
---|---|
minvecolem4a | ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minveco.u | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) | |
2 | phnv 28186 | . . . . . 6 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ NrmCVec) |
4 | minveco.w | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) | |
5 | elin 3992 | . . . . . . 7 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
6 | 4, 5 | sylib 210 | . . . . . 6 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
7 | 6 | simpld 489 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
8 | minveco.y | . . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊) | |
9 | minveco.d | . . . . . 6 ⊢ 𝐷 = (IndMet‘𝑈) | |
10 | eqid 2797 | . . . . . 6 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊) | |
11 | eqid 2797 | . . . . . 6 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
12 | 8, 9, 10, 11 | sspims 28111 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
13 | 3, 7, 12 | syl2anc 580 | . . . 4 ⊢ (𝜑 → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
14 | 6 | simprd 490 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ CBan) |
15 | eqid 2797 | . . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
16 | 15, 10 | cbncms 28238 | . . . . 5 ⊢ (𝑊 ∈ CBan → (IndMet‘𝑊) ∈ (CMet‘(BaseSet‘𝑊))) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (IndMet‘𝑊) ∈ (CMet‘(BaseSet‘𝑊))) |
18 | 13, 17 | eqeltrrd 2877 | . . 3 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘(BaseSet‘𝑊))) |
19 | minveco.x | . . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈) | |
20 | minveco.m | . . . . 5 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
21 | minveco.n | . . . . 5 ⊢ 𝑁 = (normCV‘𝑈) | |
22 | minveco.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
23 | minveco.j | . . . . 5 ⊢ 𝐽 = (MetOpen‘𝐷) | |
24 | minveco.r | . . . . 5 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) | |
25 | minveco.s | . . . . 5 ⊢ 𝑆 = inf(𝑅, ℝ, < ) | |
26 | minveco.f | . . . . 5 ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) | |
27 | minveco.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) | |
28 | 19, 20, 21, 8, 1, 4, 22, 9, 23, 24, 25, 26, 27 | minvecolem3 28249 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) |
29 | 19, 9 | imsmet 28063 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
30 | 1, 2, 29 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
31 | metxmet 22464 | . . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
32 | 30, 31 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
33 | causs 23421 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) | |
34 | 32, 26, 33 | syl2anc 580 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) |
35 | 28, 34 | mpbid 224 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) |
36 | eqid 2797 | . . . 4 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) | |
37 | 36 | cmetcau 23412 | . . 3 ⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘(BaseSet‘𝑊)) ∧ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
38 | 18, 35, 37 | syl2anc 580 | . 2 ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
39 | xmetres 22494 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌))) | |
40 | 36 | methaus 22650 | . . . 4 ⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌)) → (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) ∈ Haus) |
41 | 32, 39, 40 | 3syl 18 | . . 3 ⊢ (𝜑 → (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) ∈ Haus) |
42 | lmfun 21511 | . . 3 ⊢ ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) ∈ Haus → Fun (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) | |
43 | funfvbrb 6554 | . . 3 ⊢ (Fun (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) ↔ 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))) | |
44 | 41, 42, 43 | 3syl 18 | . 2 ⊢ (𝜑 → (𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) ↔ 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))) |
45 | 38, 44 | mpbid 224 | 1 ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∩ cin 3766 class class class wbr 4841 ↦ cmpt 4920 × cxp 5308 dom cdm 5310 ran crn 5311 ↾ cres 5312 Fun wfun 6093 ⟶wf 6095 ‘cfv 6099 (class class class)co 6876 infcinf 8587 ℝcr 10221 1c1 10223 + caddc 10225 < clt 10361 ≤ cle 10362 / cdiv 10974 ℕcn 11310 2c2 11364 ↑cexp 13110 ∞Metcxmet 20050 Metcmet 20051 MetOpencmopn 20055 ⇝𝑡clm 21356 Hauscha 21438 Cauccau 23376 CMetccmet 23377 NrmCVeccnv 27956 BaseSetcba 27958 −𝑣 cnsb 27961 normCVcnmcv 27962 IndMetcims 27963 SubSpcss 28093 CPreHilOLDccphlo 28184 CBanccbn 28235 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-pre-sup 10300 ax-addf 10301 ax-mulf 10302 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-om 7298 df-1st 7399 df-2nd 7400 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-er 7980 df-map 8095 df-pm 8096 df-en 8194 df-dom 8195 df-sdom 8196 df-sup 8588 df-inf 8589 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-n0 11577 df-z 11663 df-uz 11927 df-q 12030 df-rp 12071 df-xneg 12189 df-xadd 12190 df-xmul 12191 df-ico 12426 df-icc 12427 df-fl 12844 df-seq 13052 df-exp 13111 df-cj 14177 df-re 14178 df-im 14179 df-sqrt 14313 df-abs 14314 df-rest 16395 df-topgen 16416 df-psmet 20057 df-xmet 20058 df-met 20059 df-bl 20060 df-mopn 20061 df-fbas 20062 df-fg 20063 df-top 21024 df-topon 21041 df-bases 21076 df-ntr 21150 df-nei 21228 df-lm 21359 df-haus 21445 df-fil 21975 df-fm 22067 df-flim 22068 df-flf 22069 df-cfil 23378 df-cau 23379 df-cmet 23380 df-grpo 27865 df-gid 27866 df-ginv 27867 df-gdiv 27868 df-ablo 27917 df-vc 27931 df-nv 27964 df-va 27967 df-ba 27968 df-sm 27969 df-0v 27970 df-vs 27971 df-nmcv 27972 df-ims 27973 df-ssp 28094 df-ph 28185 df-cbn 28236 |
This theorem is referenced by: minvecolem4b 28251 minvecolem4 28253 |
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