![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > minvecolem4b | Structured version Visualization version GIF version |
Description: Lemma for minveco 28080. 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 28009 | . . . 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 3947 | . . . . 5 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
6 | 4, 5 | sylib 208 | . . . 4 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
7 | 6 | simpld 482 | . . 3 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
8 | minveco.x | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
9 | minveco.y | . . . 4 ⊢ 𝑌 = (BaseSet‘𝑊) | |
10 | eqid 2771 | . . . 4 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
11 | 8, 9, 10 | sspba 27922 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋) |
12 | 3, 7, 11 | syl2anc 573 | . 2 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
13 | minveco.d | . . . . . . . 8 ⊢ 𝐷 = (IndMet‘𝑈) | |
14 | 8, 13 | imsxmet 27887 | . . . . . . 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 22545 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus) |
18 | 15, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Haus) |
19 | lmfun 21406 | . . . . 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 28073 | . . . . 5 ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) |
29 | eqid 2771 | . . . . . . 7 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌) | |
30 | nnuz 11930 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
31 | 9 | fvexi 6345 | . . . . . . . 8 ⊢ 𝑌 ∈ V |
32 | 31 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) |
33 | 16 | mopntop 22465 | . . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
34 | 15, 33 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) |
35 | xmetres2 22386 | . . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌)) | |
36 | 15, 12, 35 | syl2anc 573 | . . . . . . . . 9 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌)) |
37 | eqid 2771 | . . . . . . . . . 10 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) | |
38 | 37 | mopntopon 22464 | . . . . . . . . 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 21322 | . . . . . . . 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 573 | . . . . . . 7 ⊢ (𝜑 → ((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ∈ 𝑌) |
42 | 1zzd 11615 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ) | |
43 | 29, 30, 32, 34, 41, 42, 26 | lmss 21323 | . . . . . 6 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ↔ 𝐹(⇝𝑡‘(𝐽 ↾t 𝑌))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))) |
44 | eqid 2771 | . . . . . . . . . 10 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌)) | |
45 | 44, 16, 37 | metrest 22549 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) |
46 | 15, 12, 45 | syl2anc 573 | . . . . . . . 8 ⊢ (𝜑 → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) |
47 | 46 | fveq2d 6337 | . . . . . . 7 ⊢ (𝜑 → (⇝𝑡‘(𝐽 ↾t 𝑌)) = (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
48 | 47 | breqd 4798 | . . . . . 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 268 | . . . . 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 247 | . . . 4 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) |
51 | funbrfv 6377 | . . . 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 2850 | . 2 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑌) |
54 | 12, 53 | sseldd 3753 | 1 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∩ cin 3722 ⊆ wss 3723 class class class wbr 4787 ↦ cmpt 4864 × cxp 5248 ran crn 5251 ↾ cres 5252 Fun wfun 6024 ⟶wf 6026 ‘cfv 6030 (class class class)co 6796 infcinf 8507 ℝcr 10141 1c1 10143 + caddc 10145 < clt 10280 ≤ cle 10281 / cdiv 10890 ℕcn 11226 2c2 11276 ↑cexp 13067 ↾t crest 16289 ∞Metcxmt 19946 MetOpencmopn 19951 Topctop 20918 TopOnctopon 20935 ⇝𝑡clm 21251 Hauscha 21333 NrmCVeccnv 27779 BaseSetcba 27781 −𝑣 cnsb 27784 normCVcnmcv 27785 IndMetcims 27786 SubSpcss 27916 CPreHilOLDccphlo 28007 CBanccbn 28058 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 ax-cnex 10198 ax-resscn 10199 ax-1cn 10200 ax-icn 10201 ax-addcl 10202 ax-addrcl 10203 ax-mulcl 10204 ax-mulrcl 10205 ax-mulcom 10206 ax-addass 10207 ax-mulass 10208 ax-distr 10209 ax-i2m1 10210 ax-1ne0 10211 ax-1rid 10212 ax-rnegex 10213 ax-rrecex 10214 ax-cnre 10215 ax-pre-lttri 10216 ax-pre-lttrn 10217 ax-pre-ltadd 10218 ax-pre-mulgt0 10219 ax-pre-sup 10220 ax-addf 10221 ax-mulf 10222 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-om 7217 df-1st 7319 df-2nd 7320 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-oadd 7721 df-er 7900 df-map 8015 df-pm 8016 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fi 8477 df-sup 8508 df-inf 8509 df-pnf 10282 df-mnf 10283 df-xr 10284 df-ltxr 10285 df-le 10286 df-sub 10474 df-neg 10475 df-div 10891 df-nn 11227 df-2 11285 df-3 11286 df-4 11287 df-n0 11500 df-z 11585 df-uz 11894 df-q 11997 df-rp 12036 df-xneg 12151 df-xadd 12152 df-xmul 12153 df-ico 12386 df-icc 12387 df-fl 12801 df-seq 13009 df-exp 13068 df-cj 14047 df-re 14048 df-im 14049 df-sqrt 14183 df-abs 14184 df-rest 16291 df-topgen 16312 df-psmet 19953 df-xmet 19954 df-met 19955 df-bl 19956 df-mopn 19957 df-fbas 19958 df-fg 19959 df-top 20919 df-topon 20936 df-bases 20971 df-ntr 21045 df-nei 21123 df-lm 21254 df-haus 21340 df-fil 21870 df-fm 21962 df-flim 21963 df-flf 21964 df-cfil 23272 df-cau 23273 df-cmet 23274 df-grpo 27687 df-gid 27688 df-ginv 27689 df-gdiv 27690 df-ablo 27739 df-vc 27754 df-nv 27787 df-va 27790 df-ba 27791 df-sm 27792 df-0v 27793 df-vs 27794 df-nmcv 27795 df-ims 27796 df-ssp 27917 df-ph 28008 df-cbn 28059 |
This theorem is referenced by: minvecolem4 28076 |
Copyright terms: Public domain | W3C validator |