![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > minvecolem4b | Structured version Visualization version GIF version |
Description: Lemma for minveco 30124. The convergent point of the cauchy sequence 𝐹 is a member of the base space. (Contributed by Mario Carneiro, 16-Jun-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 |
---|---|
minvecolem4b | ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | minveco.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ CPreHilOLD) | |
2 | phnv 30054 | . . . 4 ⊢ (𝑈 ∈ CPreHilOLD → 𝑈 ∈ NrmCVec) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → 𝑈 ∈ NrmCVec) |
4 | minveco.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ ((SubSp‘𝑈) ∩ CBan)) | |
5 | elin 3963 | . . . . 5 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
6 | 4, 5 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
7 | 6 | simpld 495 | . . 3 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
8 | minveco.x | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
9 | minveco.y | . . . 4 ⊢ 𝑌 = (BaseSet‘𝑊) | |
10 | eqid 2732 | . . . 4 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
11 | 8, 9, 10 | sspba 29967 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋) |
12 | 3, 7, 11 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
13 | minveco.d | . . . . . . . 8 ⊢ 𝐷 = (IndMet‘𝑈) | |
14 | 8, 13 | imsxmet 29932 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (∞Met‘𝑋)) |
15 | 3, 14 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
16 | minveco.j | . . . . . . 7 ⊢ 𝐽 = (MetOpen‘𝐷) | |
17 | 16 | methaus 24020 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus) |
18 | 15, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Haus) |
19 | lmfun 22876 | . . . . 5 ⊢ (𝐽 ∈ Haus → Fun (⇝𝑡‘𝐽)) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (𝜑 → Fun (⇝𝑡‘𝐽)) |
21 | minveco.m | . . . . . 6 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
22 | minveco.n | . . . . . 6 ⊢ 𝑁 = (normCV‘𝑈) | |
23 | minveco.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
24 | minveco.r | . . . . . 6 ⊢ 𝑅 = ran (𝑦 ∈ 𝑌 ↦ (𝑁‘(𝐴𝑀𝑦))) | |
25 | minveco.s | . . . . . 6 ⊢ 𝑆 = inf(𝑅, ℝ, < ) | |
26 | minveco.f | . . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶𝑌) | |
27 | minveco.1 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐴𝐷(𝐹‘𝑛))↑2) ≤ ((𝑆↑2) + (1 / 𝑛))) | |
28 | 8, 21, 22, 9, 1, 4, 23, 13, 16, 24, 25, 26, 27 | minvecolem4a 30117 | . . . . 5 ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) |
29 | eqid 2732 | . . . . . . 7 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌) | |
30 | nnuz 12861 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
31 | 9 | fvexi 6902 | . . . . . . . 8 ⊢ 𝑌 ∈ V |
32 | 31 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) |
33 | 16 | mopntop 23937 | . . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
34 | 15, 33 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) |
35 | xmetres2 23858 | . . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌)) | |
36 | 15, 12, 35 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌)) |
37 | eqid 2732 | . . . . . . . . . 10 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) | |
38 | 37 | mopntopon 23936 | . . . . . . . . 9 ⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌) → (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) ∈ (TopOn‘𝑌)) |
39 | 36, 38 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) ∈ (TopOn‘𝑌)) |
40 | lmcl 22792 | . . . . . . . 8 ⊢ (((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) ∈ (TopOn‘𝑌) ∧ 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) → ((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ∈ 𝑌) | |
41 | 39, 28, 40 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → ((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ∈ 𝑌) |
42 | 1zzd 12589 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ) | |
43 | 29, 30, 32, 34, 41, 42, 26 | lmss 22793 | . . . . . 6 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ↔ 𝐹(⇝𝑡‘(𝐽 ↾t 𝑌))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))) |
44 | eqid 2732 | . . . . . . . . . 10 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌)) | |
45 | 44, 16, 37 | metrest 24024 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) |
46 | 15, 12, 45 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) |
47 | 46 | fveq2d 6892 | . . . . . . 7 ⊢ (𝜑 → (⇝𝑡‘(𝐽 ↾t 𝑌)) = (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
48 | 47 | breqd 5158 | . . . . . 6 ⊢ (𝜑 → (𝐹(⇝𝑡‘(𝐽 ↾t 𝑌))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ↔ 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))) |
49 | 43, 48 | bitrd 278 | . . . . 5 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ↔ 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))) |
50 | 28, 49 | mpbird 256 | . . . 4 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) |
51 | funbrfv 6939 | . . . 4 ⊢ (Fun (⇝𝑡‘𝐽) → (𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) → ((⇝𝑡‘𝐽)‘𝐹) = ((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))) | |
52 | 20, 50, 51 | sylc 65 | . . 3 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) = ((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) |
53 | 52, 41 | eqeltrd 2833 | . 2 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑌) |
54 | 12, 53 | sseldd 3982 | 1 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∩ cin 3946 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ran crn 5676 ↾ cres 5677 Fun wfun 6534 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 infcinf 9432 ℝcr 11105 1c1 11107 + caddc 11109 < clt 11244 ≤ cle 11245 / cdiv 11867 ℕcn 12208 2c2 12263 ↑cexp 14023 ↾t crest 17362 ∞Metcxmet 20921 MetOpencmopn 20926 Topctop 22386 TopOnctopon 22403 ⇝𝑡clm 22721 Hauscha 22803 NrmCVeccnv 29824 BaseSetcba 29826 −𝑣 cnsb 29829 normCVcnmcv 29830 IndMetcims 29831 SubSpcss 29961 CPreHilOLDccphlo 30052 CBanccbn 30102 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 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-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ico 13326 df-icc 13327 df-fl 13753 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-rest 17364 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-top 22387 df-topon 22404 df-bases 22440 df-ntr 22515 df-nei 22593 df-lm 22724 df-haus 22810 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-cfil 24763 df-cau 24764 df-cmet 24765 df-grpo 29733 df-gid 29734 df-ginv 29735 df-gdiv 29736 df-ablo 29785 df-vc 29799 df-nv 29832 df-va 29835 df-ba 29836 df-sm 29837 df-0v 29838 df-vs 29839 df-nmcv 29840 df-ims 29841 df-ssp 29962 df-ph 30053 df-cbn 30103 |
This theorem is referenced by: minvecolem4 30120 |
Copyright terms: Public domain | W3C validator |