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Mirrors > Home > MPE Home > Th. List > minvecolem4a | Structured version Visualization version GIF version |
Description: Lemma for minveco 30913. 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 30843 | . . . . . 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 3979 | . . . . . . 7 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
6 | 4, 5 | sylib 218 | . . . . . 6 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
7 | 6 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) |
8 | minveco.y | . . . . . 6 ⊢ 𝑌 = (BaseSet‘𝑊) | |
9 | minveco.d | . . . . . 6 ⊢ 𝐷 = (IndMet‘𝑈) | |
10 | eqid 2735 | . . . . . 6 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊) | |
11 | eqid 2735 | . . . . . 6 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
12 | 8, 9, 10, 11 | sspims 30768 | . . . . 5 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
13 | 3, 7, 12 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (IndMet‘𝑊) = (𝐷 ↾ (𝑌 × 𝑌))) |
14 | 6 | simprd 495 | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ CBan) |
15 | eqid 2735 | . . . . . 6 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
16 | 15, 10 | cbncms 30894 | . . . . 5 ⊢ (𝑊 ∈ CBan → (IndMet‘𝑊) ∈ (CMet‘(BaseSet‘𝑊))) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (IndMet‘𝑊) ∈ (CMet‘(BaseSet‘𝑊))) |
18 | 13, 17 | eqeltrrd 2840 | . . 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 30905 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) |
29 | 19, 9 | imsmet 30720 | . . . . . . 7 ⊢ (𝑈 ∈ NrmCVec → 𝐷 ∈ (Met‘𝑋)) |
30 | 1, 2, 29 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) |
31 | metxmet 24360 | . . . . . 6 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) | |
32 | 30, 31 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
33 | causs 25346 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) | |
34 | 32, 26, 33 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))))) |
35 | 28, 34 | mpbid 232 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) |
36 | eqid 2735 | . . . 4 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) | |
37 | 36 | cmetcau 25337 | . . 3 ⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘(BaseSet‘𝑊)) ∧ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
38 | 18, 35, 37 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
39 | xmetres 24390 | . . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋 ∩ 𝑌))) | |
40 | 36 | methaus 24549 | . . . 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 23405 | . . 3 ⊢ ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) ∈ Haus → Fun (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) | |
43 | funfvbrb 7071 | . . 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 232 | 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 class class class wbr 5148 ↦ cmpt 5231 × cxp 5687 dom cdm 5689 ran crn 5690 ↾ cres 5691 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 infcinf 9479 ℝcr 11152 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 / cdiv 11918 ℕcn 12264 2c2 12319 ↑cexp 14099 ∞Metcxmet 21367 Metcmet 21368 MetOpencmopn 21372 ⇝𝑡clm 23250 Hauscha 23332 Cauccau 25301 CMetccmet 25302 NrmCVeccnv 30613 BaseSetcba 30615 −𝑣 cnsb 30618 normCVcnmcv 30619 IndMetcims 30620 SubSpcss 30750 CPreHilOLDccphlo 30841 CBanccbn 30891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12612 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ico 13390 df-icc 13391 df-fl 13829 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-rest 17469 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-top 22916 df-topon 22933 df-bases 22969 df-ntr 23044 df-nei 23122 df-lm 23253 df-haus 23339 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-cfil 25303 df-cau 25304 df-cmet 25305 df-grpo 30522 df-gid 30523 df-ginv 30524 df-gdiv 30525 df-ablo 30574 df-vc 30588 df-nv 30621 df-va 30624 df-ba 30625 df-sm 30626 df-0v 30627 df-vs 30628 df-nmcv 30629 df-ims 30630 df-ssp 30751 df-ph 30842 df-cbn 30892 |
This theorem is referenced by: minvecolem4b 30907 minvecolem4 30909 |
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