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Mirrors > Home > MPE Home > Th. List > minvecolem4b | Structured version Visualization version GIF version |
Description: Lemma for minveco 29826. 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 29756 | . . . 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 3926 | . . . . 5 ⊢ (𝑊 ∈ ((SubSp‘𝑈) ∩ CBan) ↔ (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) | |
6 | 4, 5 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝑊 ∈ (SubSp‘𝑈) ∧ 𝑊 ∈ CBan)) |
7 | 6 | simpld 495 | . . 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 29669 | . . 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 29634 | . . . . . . 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 23876 | . . . . . 6 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus) |
18 | 15, 17 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ Haus) |
19 | lmfun 22732 | . . . . 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 29819 | . . . . 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 12806 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
31 | 9 | fvexi 6856 | . . . . . . . 8 ⊢ 𝑌 ∈ V |
32 | 31 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ V) |
33 | 16 | mopntop 23793 | . . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
34 | 15, 33 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) |
35 | xmetres2 23714 | . . . . . . . . . 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 23792 | . . . . . . . . 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 22648 | . . . . . . . 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 12534 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℤ) | |
43 | 29, 30, 32, 34, 41, 42, 26 | lmss 22649 | . . . . . 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 23880 | . . . . . . . . 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 6846 | . . . . . . 7 ⊢ (𝜑 → (⇝𝑡‘(𝐽 ↾t 𝑌)) = (⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))) |
48 | 47 | breqd 5116 | . . . . . 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 278 | . . . . 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 256 | . . . 4 ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)((⇝𝑡‘(MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))))‘𝐹)) |
51 | funbrfv 6893 | . . . 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 2838 | . 2 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑌) |
54 | 12, 53 | sseldd 3945 | 1 ⊢ (𝜑 → ((⇝𝑡‘𝐽)‘𝐹) ∈ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ran crn 5634 ↾ cres 5635 Fun wfun 6490 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 infcinf 9377 ℝcr 11050 1c1 11052 + caddc 11054 < clt 11189 ≤ cle 11190 / cdiv 11812 ℕcn 12153 2c2 12208 ↑cexp 13967 ↾t crest 17302 ∞Metcxmet 20781 MetOpencmopn 20786 Topctop 22242 TopOnctopon 22259 ⇝𝑡clm 22577 Hauscha 22659 NrmCVeccnv 29526 BaseSetcba 29528 −𝑣 cnsb 29531 normCVcnmcv 29532 IndMetcims 29533 SubSpcss 29663 CPreHilOLDccphlo 29754 CBanccbn 29804 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fi 9347 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ico 13270 df-icc 13271 df-fl 13697 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-rest 17304 df-topgen 17325 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-top 22243 df-topon 22260 df-bases 22296 df-ntr 22371 df-nei 22449 df-lm 22580 df-haus 22666 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-cfil 24619 df-cau 24620 df-cmet 24621 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 df-ims 29543 df-ssp 29664 df-ph 29755 df-cbn 29805 |
This theorem is referenced by: minvecolem4 29822 |
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