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| Description: Lemma for minveco 30904. 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 30834 | . . . 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 3966 | . . . . 5 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
| 6 | 4, 5 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | 
| 7 | 6 | simpld 494 | . . 3 ⊢ (𝜑 → 𝑊 ∈ (SubSp‘𝑈)) | 
| 8 | minveco.x | . . . 4 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 9 | minveco.y | . . . 4 ⊢ 𝑌 = (BaseSet‘𝑊) | |
| 10 | eqid 2736 | . . . 4 ⊢ (SubSp‘𝑈) = (SubSp‘𝑈) | |
| 11 | 8, 9, 10 | sspba 30747 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ (SubSp‘𝑈)) → 𝑌 ⊆ 𝑋) | 
| 12 | 3, 7, 11 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) | 
| 13 | minveco.d | . . . . . . . 8 ⊢ 𝐷 = (IndMet‘𝑈) | |
| 14 | 8, 13 | imsxmet 30712 | . . . . . . 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 24534 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus) | 
| 18 | 15, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Haus) | 
| 19 | lmfun 23390 | . . . . 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 30897 | . . . . 5 ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) | 
| 29 | eqid 2736 | . . . . . . 7 ⊢ (𝐽 ↾t 𝑌) = (𝐽 ↾t 𝑌) | |
| 30 | nnuz 12922 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
| 31 | 9 | fvexi 6919 | . . . . . . . 8 ⊢ 𝑌 ∈ V | 
| 32 | 31 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) | 
| 33 | 16 | mopntop 24451 | . . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) | 
| 34 | 15, 33 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) | 
| 35 | xmetres2 24372 | . . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌)) | |
| 36 | 15, 12, 35 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌)) | 
| 37 | eqid 2736 | . . . . . . . . . 10 ⊢ (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) | |
| 38 | 37 | mopntopon 24450 | . . . . . . . . 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 23306 | . . . . . . . 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 584 | . . . . . . 7 ⊢ (𝜑 → ((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ∈ 𝑌) | 
| 42 | 1zzd 12650 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ) | |
| 43 | 29, 30, 32, 34, 41, 42, 26 | lmss 23307 | . . . . . 6 ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹) ↔ 𝐹(⇝𝑡‘(𝐽 ↾t 𝑌))((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹))) | 
| 44 | eqid 2736 | . . . . . . . . . 10 ⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌)) | |
| 45 | 44, 16, 37 | metrest 24538 | . . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) | 
| 46 | 15, 12, 45 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) | 
| 47 | 46 | fveq2d 6909 | . . . . . . 7 ⊢ (𝜑 → (⇝𝑡‘(𝐽 ↾t 𝑌)) = (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) | 
| 48 | 47 | breqd 5153 | . . . . . 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 279 | . . . . 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 257 | . . . 4 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) | 
| 51 | funbrfv 6956 | . . . 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 2840 | . 2 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑌) | 
| 54 | 12, 53 | sseldd 3983 | 1 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ∩ cin 3949 ⊆ wss 3950 class class class wbr 5142 ↦ cmpt 5224 × cxp 5682 ran crn 5685 ↾ cres 5686 Fun wfun 6554 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 infcinf 9482 ℝcr 11155 1c1 11157 + caddc 11159 < clt 11296 ≤ cle 11297 / cdiv 11921 ℕcn 12267 2c2 12322 ↑cexp 14103 ↾t crest 17466 ∞Metcxmet 21350 MetOpencmopn 21355 Topctop 22900 TopOnctopon 22917 ⇝𝑡clm 23235 Hauscha 23317 NrmCVeccnv 30604 BaseSetcba 30606 −𝑣 cnsb 30609 normCVcnmcv 30610 IndMetcims 30611 SubSpcss 30741 CPreHilOLDccphlo 30832 CBanccbn 30882 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fi 9452 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ico 13394 df-icc 13395 df-fl 13833 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-rest 17468 df-topgen 17489 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-top 22901 df-topon 22918 df-bases 22954 df-ntr 23029 df-nei 23107 df-lm 23238 df-haus 23324 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-cfil 25290 df-cau 25291 df-cmet 25292 df-grpo 30513 df-gid 30514 df-ginv 30515 df-gdiv 30516 df-ablo 30565 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-0v 30618 df-vs 30619 df-nmcv 30620 df-ims 30621 df-ssp 30742 df-ph 30833 df-cbn 30883 | 
| This theorem is referenced by: minvecolem4 30900 | 
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