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Mirrors > Home > MPE Home > Th. List > minvecolem4a | Structured version Visualization version GIF version |
Description: Lemma for minveco 28667. 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 28597 | . . . . . 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 3897 | . . . . . . 7 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
6 | 4, 5 | sylib 221 | . . . . . 6 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
7 | 6 | simpld 498 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
8 | minveco.y | . . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊) | |
9 | minveco.d | . . . . . 6 ⊢ 𝐷 = (IndMet‘𝑈) | |
10 | eqid 2798 | . . . . . 6 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊) | |
11 | eqid 2798 | . . . . . 6 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
12 | 8, 9, 10, 11 | sspims 28522 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
13 | 3, 7, 12 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
14 | 6 | simprd 499 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ CBan) |
15 | eqid 2798 | . . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
16 | 15, 10 | cbncms 28648 | . . . . 5 ⊢ (𝑊 ∈ CBan → (IndMet‘𝑊) ∈ (CMet‘(BaseSet‘𝑊))) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (IndMet‘𝑊) ∈ (CMet‘(BaseSet‘𝑊))) |
18 | 13, 17 | eqeltrrd 2891 | . . 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 28659 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) |
29 | 19, 9 | imsmet 28474 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
30 | 1, 2, 29 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
31 | metxmet 22941 | . . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
32 | 30, 31 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
33 | causs 23902 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) | |
34 | 32, 26, 33 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) |
35 | 28, 34 | mpbid 235 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) |
36 | eqid 2798 | . . . 4 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) | |
37 | 36 | cmetcau 23893 | . . 3 ⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘(BaseSet‘𝑊)) ∧ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
38 | 18, 35, 37 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
39 | xmetres 22971 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌))) | |
40 | 36 | methaus 23127 | . . . 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 21986 | . . 3 ⊢ ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) ∈ Haus → Fun (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) | |
43 | funfvbrb 6798 | . . 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 235 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 class class class wbr 5030 ↦ cmpt 5110 × cxp 5517 dom cdm 5519 ran crn 5520 ↾ cres 5521 Fun wfun 6318 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 infcinf 8889 ℝcr 10525 1c1 10527 + caddc 10529 < clt 10664 ≤ cle 10665 / cdiv 11286 ℕcn 11625 2c2 11680 ↑cexp 13425 ∞Metcxmet 20076 Metcmet 20077 MetOpencmopn 20081 ⇝𝑡clm 21831 Hauscha 21913 Cauccau 23857 CMetccmet 23858 NrmCVeccnv 28367 BaseSetcba 28369 −𝑣 cnsb 28372 normCVcnmcv 28373 IndMetcims 28374 SubSpcss 28504 CPreHilOLDccphlo 28595 CBanccbn 28645 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 13157 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-rest 16688 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-top 21499 df-topon 21516 df-bases 21551 df-ntr 21625 df-nei 21703 df-lm 21834 df-haus 21920 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-cfil 23859 df-cau 23860 df-cmet 23861 df-grpo 28276 df-gid 28277 df-ginv 28278 df-gdiv 28279 df-ablo 28328 df-vc 28342 df-nv 28375 df-va 28378 df-ba 28379 df-sm 28380 df-0v 28381 df-vs 28382 df-nmcv 28383 df-ims 28384 df-ssp 28505 df-ph 28596 df-cbn 28646 |
This theorem is referenced by: minvecolem4b 28661 minvecolem4 28663 |
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