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Mirrors > Home > MPE Home > Th. List > minvecolem4a | Structured version Visualization version GIF version |
Description: Lemma for minveco 28663. 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 28593 | . . . . . 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 4171 | . . . . . . 7 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
6 | 4, 5 | sylib 220 | . . . . . 6 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
7 | 6 | simpld 497 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
8 | minveco.y | . . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊) | |
9 | minveco.d | . . . . . 6 ⊢ 𝐷 = (IndMet‘𝑈) | |
10 | eqid 2823 | . . . . . 6 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊) | |
11 | eqid 2823 | . . . . . 6 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
12 | 8, 9, 10, 11 | sspims 28518 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
13 | 3, 7, 12 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
14 | 6 | simprd 498 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ CBan) |
15 | eqid 2823 | . . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
16 | 15, 10 | cbncms 28644 | . . . . 5 ⊢ (𝑊 ∈ CBan → (IndMet‘𝑊) ∈ (CMet‘(BaseSet‘𝑊))) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (IndMet‘𝑊) ∈ (CMet‘(BaseSet‘𝑊))) |
18 | 13, 17 | eqeltrrd 2916 | . . 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 28655 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) |
29 | 19, 9 | imsmet 28470 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
30 | 1, 2, 29 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
31 | metxmet 22946 | . . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
32 | 30, 31 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
33 | causs 23903 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) | |
34 | 32, 26, 33 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) |
35 | 28, 34 | mpbid 234 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) |
36 | eqid 2823 | . . . 4 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) | |
37 | 36 | cmetcau 23894 | . . 3 ⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘(BaseSet‘𝑊)) ∧ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
38 | 18, 35, 37 | syl2anc 586 | . 2 ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
39 | xmetres 22976 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌))) | |
40 | 36 | methaus 23132 | . . . 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 21991 | . . 3 ⊢ ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) ∈ Haus → Fun (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) | |
43 | funfvbrb 6823 | . . 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 234 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 dom cdm 5557 ran crn 5558 ↾ cres 5559 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 infcinf 8907 ℝcr 10538 1c1 10540 + caddc 10542 < clt 10677 ≤ cle 10678 / cdiv 11299 ℕcn 11640 2c2 11695 ↑cexp 13432 ∞Metcxmet 20532 Metcmet 20533 MetOpencmopn 20537 ⇝𝑡clm 21836 Hauscha 21918 Cauccau 23858 CMetccmet 23859 NrmCVeccnv 28363 BaseSetcba 28365 −𝑣 cnsb 28368 normCVcnmcv 28369 IndMetcims 28370 SubSpcss 28500 CPreHilOLDccphlo 28591 CBanccbn 28641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ico 12747 df-icc 12748 df-fl 13165 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-rest 16698 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-top 21504 df-topon 21521 df-bases 21556 df-ntr 21630 df-nei 21708 df-lm 21839 df-haus 21925 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-cfil 23860 df-cau 23861 df-cmet 23862 df-grpo 28272 df-gid 28273 df-ginv 28274 df-gdiv 28275 df-ablo 28324 df-vc 28338 df-nv 28371 df-va 28374 df-ba 28375 df-sm 28376 df-0v 28377 df-vs 28378 df-nmcv 28379 df-ims 28380 df-ssp 28501 df-ph 28592 df-cbn 28642 |
This theorem is referenced by: minvecolem4b 28657 minvecolem4 28659 |
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