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Mirrors > Home > MPE Home > Th. List > minvecolem4a | Structured version Visualization version GIF version |
Description: Lemma for minveco 28588. 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 28518 | . . . . . 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 4166 | . . . . . . 7 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
6 | 4, 5 | sylib 219 | . . . . . 6 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
7 | 6 | simpld 495 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
8 | minveco.y | . . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊) | |
9 | minveco.d | . . . . . 6 ⊢ 𝐷 = (IndMet‘𝑈) | |
10 | eqid 2818 | . . . . . 6 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊) | |
11 | eqid 2818 | . . . . . 6 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
12 | 8, 9, 10, 11 | sspims 28443 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
13 | 3, 7, 12 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
14 | 6 | simprd 496 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ CBan) |
15 | eqid 2818 | . . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
16 | 15, 10 | cbncms 28569 | . . . . 5 ⊢ (𝑊 ∈ CBan → (IndMet‘𝑊) ∈ (CMet‘(BaseSet‘𝑊))) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (IndMet‘𝑊) ∈ (CMet‘(BaseSet‘𝑊))) |
18 | 13, 17 | eqeltrrd 2911 | . . 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 28580 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) |
29 | 19, 9 | imsmet 28395 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
30 | 1, 2, 29 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
31 | metxmet 22871 | . . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
32 | 30, 31 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
33 | causs 23828 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) | |
34 | 32, 26, 33 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) |
35 | 28, 34 | mpbid 233 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) |
36 | eqid 2818 | . . . 4 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) | |
37 | 36 | cmetcau 23819 | . . 3 ⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘(BaseSet‘𝑊)) ∧ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
38 | 18, 35, 37 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
39 | xmetres 22901 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌))) | |
40 | 36 | methaus 23057 | . . . 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 21917 | . . 3 ⊢ ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) ∈ Haus → Fun (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) | |
43 | funfvbrb 6813 | . . 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 233 | 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 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 class class class wbr 5057 ↦ cmpt 5137 × cxp 5546 dom cdm 5548 ran crn 5549 ↾ cres 5550 Fun wfun 6342 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 infcinf 8893 ℝcr 10524 1c1 10526 + caddc 10528 < clt 10663 ≤ cle 10664 / cdiv 11285 ℕcn 11626 2c2 11680 ↑cexp 13417 ∞Metcxmet 20458 Metcmet 20459 MetOpencmopn 20463 ⇝𝑡clm 21762 Hauscha 21844 Cauccau 23783 CMetccmet 23784 NrmCVeccnv 28288 BaseSetcba 28290 −𝑣 cnsb 28293 normCVcnmcv 28294 IndMetcims 28295 SubSpcss 28425 CPreHilOLDccphlo 28516 CBanccbn 28566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-icc 12733 df-fl 13150 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-rest 16684 df-topgen 16705 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-top 21430 df-topon 21447 df-bases 21482 df-ntr 21556 df-nei 21634 df-lm 21765 df-haus 21851 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-cfil 23785 df-cau 23786 df-cmet 23787 df-grpo 28197 df-gid 28198 df-ginv 28199 df-gdiv 28200 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-vs 28303 df-nmcv 28304 df-ims 28305 df-ssp 28426 df-ph 28517 df-cbn 28567 |
This theorem is referenced by: minvecolem4b 28582 minvecolem4 28584 |
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