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Mirrors > Home > MPE Home > Th. List > minvecolem4b | Structured version Visualization version GIF version |
Description: Lemma for minveco 28965. 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 28895 | . . . 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 3882 | . . . . 5 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
6 | 4, 5 | sylib 221 | . . . 4 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
7 | 6 | simpld 498 | . . 3 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
8 | minveco.x | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
9 | minveco.y | . . . 4 ⊢ 𝑌 = (BaseSet‘𝑊) | |
10 | eqid 2737 | . . . 4 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
11 | 8, 9, 10 | sspba 28808 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋) |
12 | 3, 7, 11 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
13 | minveco.d | . . . . . . . 8 ⊢ 𝐷 = (IndMet‘𝑈) | |
14 | 8, 13 | imsxmet 28773 | . . . . . . 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 23418 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus) |
18 | 15, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Haus) |
19 | lmfun 22278 | . . . . 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 28958 | . . . . 5 ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) |
29 | eqid 2737 | . . . . . . 7 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌) | |
30 | nnuz 12477 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
31 | 9 | fvexi 6731 | . . . . . . . 8 ⊢ 𝑌 ∈ V |
32 | 31 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) |
33 | 16 | mopntop 23338 | . . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
34 | 15, 33 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) |
35 | xmetres2 23259 | . . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌)) | |
36 | 15, 12, 35 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌)) |
37 | eqid 2737 | . . . . . . . . . 10 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) | |
38 | 37 | mopntopon 23337 | . . . . . . . . 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 22194 | . . . . . . . 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 587 | . . . . . . 7 ⊢ (𝜑 → ((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ∈ 𝑌) |
42 | 1zzd 12208 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ) | |
43 | 29, 30, 32, 34, 41, 42, 26 | lmss 22195 | . . . . . 6 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ↔ 𝐹(⇝𝑡‘(𝐽 ↾t 𝑌))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))) |
44 | eqid 2737 | . . . . . . . . . 10 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌)) | |
45 | 44, 16, 37 | metrest 23422 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) |
46 | 15, 12, 45 | syl2anc 587 | . . . . . . . 8 ⊢ (𝜑 → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) |
47 | 46 | fveq2d 6721 | . . . . . . 7 ⊢ (𝜑 → (⇝𝑡‘(𝐽 ↾t 𝑌)) = (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
48 | 47 | breqd 5064 | . . . . . 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 282 | . . . . 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 260 | . . . 4 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) |
51 | funbrfv 6763 | . . . 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 2838 | . 2 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑌) |
54 | 12, 53 | sseldd 3902 | 1 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 Vcvv 3408 ∩ cin 3865 ⊆ wss 3866 class class class wbr 5053 ↦ cmpt 5135 × cxp 5549 ran crn 5552 ↾ cres 5553 Fun wfun 6374 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 infcinf 9057 ℝcr 10728 1c1 10730 + caddc 10732 < clt 10867 ≤ cle 10868 / cdiv 11489 ℕcn 11830 2c2 11885 ↑cexp 13635 ↾t crest 16925 ∞Metcxmet 20348 MetOpencmopn 20353 Topctop 21790 TopOnctopon 21807 ⇝𝑡clm 22123 Hauscha 22205 NrmCVeccnv 28665 BaseSetcba 28667 −𝑣 cnsb 28670 normCVcnmcv 28671 IndMetcims 28672 SubSpcss 28802 CPreHilOLDccphlo 28893 CBanccbn 28943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-addf 10808 ax-mulf 10809 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fi 9027 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-n0 12091 df-z 12177 df-uz 12439 df-q 12545 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-ico 12941 df-icc 12942 df-fl 13367 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-rest 16927 df-topgen 16948 df-psmet 20355 df-xmet 20356 df-met 20357 df-bl 20358 df-mopn 20359 df-fbas 20360 df-fg 20361 df-top 21791 df-topon 21808 df-bases 21843 df-ntr 21917 df-nei 21995 df-lm 22126 df-haus 22212 df-fil 22743 df-fm 22835 df-flim 22836 df-flf 22837 df-cfil 24152 df-cau 24153 df-cmet 24154 df-grpo 28574 df-gid 28575 df-ginv 28576 df-gdiv 28577 df-ablo 28626 df-vc 28640 df-nv 28673 df-va 28676 df-ba 28677 df-sm 28678 df-0v 28679 df-vs 28680 df-nmcv 28681 df-ims 28682 df-ssp 28803 df-ph 28894 df-cbn 28944 |
This theorem is referenced by: minvecolem4 28961 |
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